Open AccessArticle

A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

Zizhao Zhou

¹,

Ahmad Aziz Al Ahmadi

^2,*

Alina Alb Lupas

^3,* and

Khalil Hadi Hakami

⁴

Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK

Department of Electrical Engineering, College of Engineering, Taif University, Taif 21944, Saudi Arabia

Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania

⁴

Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia

Authors to whom correspondence should be addressed.

Axioms 2024, 13(10), 666; https://doi.org/10.3390/axioms13100666

Submission received: 18 August 2024 / Revised: 23 September 2024 / Accepted: 25 September 2024 / Published: 26 September 2024

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

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Abstract

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate.

Keywords:

coordinated fuzzy mappings over; ỽ-convexity over coordinates; two-dimensional Riemann–Liouville fractional integral operator; fractional inequalities

MSC:

26A33; 26A51; 26D10

1. Introduction

Focused on fractional-order integrals, derivatives and applications over real and complex domains, fractional calculus is a branch of mathematics. The use of classical analysis arithmetic is necessary to produce findings from fractional analysis that are more realistic. The use of fractional differential equations and integral equations allows for the solving of a wide range of mathematical models. Due to the fact that classical mathematical models are specific examples of fractional-order mathematical models, the findings drawn from fractional-order mathematical models are more comprehensive and correct. Fractional theory allows for the processing of any number of orders, real or integer, as opposed to just integer orders, making it a more suitable approach. In the modern day, fractional calculus techniques and tools have an impact on almost every nonlinear subject or field of research. Electrical engineering, control theory, mechanical engineering, viscoelasticity, rheology, optics and physics are only a few of the many domains of engineering that have numerous and fruitful applications; for further information, see References [1,2,3,4,5,6,7,8].

Rounding and measuring mistakes are reduced in mathematical computations using a mathematical approach known as interval analysis. The most influential study on interval analysis was written by the Japanese scientist Teruo Sunag, as referenced in [9]. In this article, the laws governing fundamental operations with intervals are not only thoroughly investigated but also mathematically analyzed. The range of rational functions can be calculated by using only the terminal points of rational functions over intervals. As multidimensional intervals, interval vectors are also discussed along with the appropriate operations. By enclosing the residual term and confining the value of a definite integral, interval arithmetic tools are used to calculate a pointwise enclosure for the solution of an initial value problem. A compact interval has developed over the past three decades into an autonomous object in numerical analysis for confirming or containing solutions to a variety of mathematical problems or demonstrating that such problems cannot be solved in a specific domain. See References [10,11] for additional information on some real-world uses of interval analysis in various linear and nonlinear disciplines. The H-H-inequality, as we are all aware, is the first geometric interpretation of a convex function and frequently employed in fields involving convex optimization. We also know that inequalities must be incorporated in order to confirm their existence, uniqueness and stability when evaluating the accuracies of various mathematical models based on actual events. In order to address uncertainty and assess the stability of differential models, writers have devised fractional forms of inequality since they frequently utilize mathematical models based on natural phenomena to portray results with greater accuracy. For more information, see [12,13] and the references therein.

But, both in practical and pure analysis, generalized convexity mapping can handle a wide range of issues. There are a number of well-known inequalities that have been created utilizing related classes of convexity, including the Simpson, Ostrowski, Opial, Hardy and well-known H-H models, that have been extended to interval-valued functions. Different concepts of convex classes and integral operators, including the traditional Riemann integral, Caputo Fabrizo, Riemann–Liouville and k-fractional operators, are used by different authors to construct these inequalities. Hadamard fractions and Riemann–Liouville fractions were used by Wang et al. [14] to study various identities for differentiable functions, and they also used these identities to show some inequalities based on s-, r-, m-, (s, m)- and other convex functions. Additionally, Iscan [15] used fractional operators to generate a number of new H-H forms using the preinvexity concept. For the product of two convex maps, Pachpette constructed the refined form of H-H-inequality using a fractional integral in [16]. Chan developed a number of H-H-inequalities for the products of convex functions based on Riemann–Liouville fractional integrals (see Reference [17]). For harmonically convex mappings in the context of fuzzy interval-valued setting, Khan et al. [18] proposed inequalities in an enlarged version using the Riemann–Liouville fractional integral operator. Shi et al.’s [19] work was then expanded to include coordinated convex interval-valued mappings (

Ι V M

s) by fractional integrals. They started by creating H-H-inequality results utilizing h-convex and harmonically h-convex functions. Dragomir created Riemann–Liouville fractional integrals with the concept of h-convexity in order to create H-H-inequalities, as seen in Ref. [20]. By means of Riemann–Liouville integral operators, Khan and his coworkers produced H-H-inequalities for left–right set-valued functions [21]. Inequalities of the H-H-Fejer type for harmonically convex functions were created by Kunt et al. [22]. H-H-inequalities with bounds were constructed by Moshin et al. [23] using q-calculus in a variety of ways. Refer to references [24,25,26,27,28,29] for some recent advancements regarding developed inequities. In 2014, Bhunia used a variety of interval measures to investigate the optimality of situations with multiple objectives. Using the radius and interval midpoints, he also created the idea of the center-radius order [30]. In a recent paper by the following authors, Bhunia’s idea was used to more precisely create several kinds of inequalities based on various integral operators [31,32,33,34]. The majority of these inequities, according to our evaluation of the literature, are brought about by partial-order or inclusion relations. The fundamental benefit of fuzzy-order relations for coordinated h-convex functions over the fuzzy codomain is that the inequality term may be predicted more precisely. This claim can be supported by useful illustrations of suggested theorems. It is crucial to comprehend the application of fuzzy total order relations to the analysis of various coordinated convex mapping classes. For more information related to coordinated interval calculus and coordinated fuzzy calculus, see [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58] and the references therein.

Motivated by the recent research, we introduce convexity for the fuzzy-number-valued mappings of two variables in order to create a new set of H-H-type inequalities for the

F N V M

s that are coordinated. The main advantage of these inequalities is their ability to be transformed into new fuzzy Riemann–Liouville fractional H-H inequalities for coordinated convex

F N V M

s. For more information related to classical H-H inequalities for coordinated convex interval-valued mappings (

Ι V M

s), see [59].

For our research, we consulted Refs. [60,61,62,63,64] for examples of high-caliber books and papers that serve as inspiration. We created a few different H-H-inequalities on fuzzy-number-valued mappings using newly established class-coordinated

ỽ

-convex mappings over fuzzy codomain and fractional integrals. For the purpose of illustrating their accuracy, we also made several samples. The essay concludes in the manner described below. Section 2 provides a succinct historical context. Section 3 talks about the main results, and in Section 4, a succinct conclusion is provided.

2. Preliminaries

We will go over the fundamental phrases and ideas that aid in understanding the ideas behind our recent discoveries in this section. In upcoming definitions, remarks, examples and findings, we will use the notions

R_{I}

R_{I}^{+}

F_{0}

and

F_{0}^{+}

and

R_{I}

for a set of fuzzy numbers, set of positive fuzzy numbers, set of intervals and collection of positive intervals, respectively.

Definition 1

([38,39]). Given

\tilde{Ỿ} \in F_{0}

, the level sets or cut sets are provided by

{[\tilde{Ỿ}]}^{γ} = \{x \in R | \tilde{Ỿ} (x) > γ\},

(1)

\forall

γ \in [0, 1]

and by

{[\tilde{Ỿ}]}^{0} = c l (\{x \in R | \tilde{Ỿ} (x) > 0\}) .

(2)

These sets are known as

γ

-level sets or

γ

-cut sets of

\tilde{Ỿ}

Proposition 1

([62]). Let

\tilde{Ỿ}, \tilde{₼} \in F_{0}

. Then, relation

“ \leq_{F} ”

is provided on

F_{0}

\tilde{Ỿ} \leq_{F} \tilde{₼}

when and only when

{{[\tilde{Ỿ}]}^{γ} \leq}_{I} {[\tilde{₼}]}^{γ}

, for every

γ \in [0, 1],

which are left- and right-order relations.

Proposition 2

([61]). Let

\tilde{Ỿ}, \tilde{₼} \in F_{0}

. Then, relation

“ \supseteq_{F} ”

is provided on

F_{0}

\tilde{Ỿ} \supseteq_{F} \tilde{₼}

when and only when

{[\tilde{Ỿ}]}^{γ} \supseteq_{I} {[\tilde{₼}]}^{γ}

, for every

γ \in [0, 1],

which is the

U D -

order relation on

F_{0}

Recall the approaching notions, which are offered in the literature. If

\tilde{Ỿ}, \tilde{₼} \in F_{0}

and

t \in R

, then, for every

γ \in [0, 1],

the arithmetic operations addition “

\oplus ”

, multiplication “

\otimes ”

and scaler multiplication “

⊙ ”

are established by

{[\tilde{Ỿ} \oplus \tilde{₼}]}^{γ} = {[\tilde{Ỿ}]}^{γ} + {[\tilde{₼}]}^{γ},

(3)

{[\tilde{Ỿ} \otimes \tilde{₼}]}^{γ} = {[\tilde{Ỿ}]}^{γ} \times {[\tilde{₼}]}^{γ},

(4)

{[t ⊙ \tilde{Ỿ}]}^{γ} = t {[\tilde{Ỿ}]}^{γ},

(5)

For these results, Equations (4)–(6) have immediate implications.

Lemma 1

([40]). Let

\tilde{G} : [p, q] \subset R \to F_{0}

be an

F N V M

, and its

Ι V M

s are classified according to their

γ

-levels

G_{γ} : [p, q] \subset R \to R_{I}

, which are provided by

G_{γ} (x) = [G_{*} (x, γ), G^{*} (x, γ)]

\forall

x \in [p, q]

and

\forall

γ \in (0, 1] .

Then,

\tilde{G}

F A

-integrable over

[p, q]

if and only if

G_{*} (x, γ)

and

G^{*} (x, γ)

are both

A

-integrable over

[p, q]

. Moreover, if

\tilde{G}

F A

-integrable over

[p, q],

then

{[(F A) \int_{p}^{q} \tilde{G} (x) d x]}^{γ} = [(A) \int_{p}^{q} G_{*} (x, γ) d x, (A) \int_{p}^{q} G^{*} (x, γ) d x] = (I A) \int_{p}^{q} G_{γ} (x) d x,

(6)

\forall

γ \in (0, 1] .

For all

γ \in (0, 1],

{F A}_{([p, q], γ)}

denotes the collection of all

F A

-integrable

F N V M

s over

[p, q]

Definition 2

([42]). Let

G : [ⱱ, ⱳ] \to R_{I}^{+}

I V M

and

G \in {I R}_{[ⱱ, ⱳ]}

. Then, interval Riemann–Liouville-type integrals of

G

are established as

I_{ⱱ^{+}}^{α} G (y) = \frac{1}{Γ (α)} \int_{ⱱ}^{y} {(y - t)}^{α - 1} G (t) d t (y > ⱱ),

(7)

I_{ⱳ^{-}}^{α} G (y) = \frac{1}{Γ (α)} \int_{y}^{ⱳ} {(t - y)}^{α - 1} G (t) d t (y < ⱳ),

(8)

where

α > 0

and

Γ

is the gamma function.

Recently, Allahviranloo et al. [43] introduced the fuzzy version of established fractional integral integrals for the following definition:

Definition 3.

Let

α > 0

and

L ([ⱱ, ⱳ], F_{0})

be the collection of all Lebesgue measurable

F N V M

s on

[ⱱ, ⱳ]

. Then, the fuzzy left and right Riemann–Liouville fractional integrals of

\tilde{G} \in

L ([ⱱ, ⱳ], F_{0})

with order

α > 0

are defined by

I_{ⱱ^{+}}^{α} \tilde{G} (y) = \frac{1}{Γ (α)} \int_{ⱱ}^{y} {(y - t)}^{α - 1} \tilde{G} (t) d t, (y > ⱱ),

(9)

and

I_{ⱳ^{-}}^{α} \tilde{G} (y) = \frac{1}{Γ (α)} \int_{y}^{ⱳ} {(t - y)}^{α - 1} \tilde{G} (t) d t, (y < ⱳ),

(10)

respectively, where

Γ (y) = \int_{0}^{\infty} t^{y - 1} e^{- t} d t

represents the Euler gamma function.

The fuzzy left and right Riemann–Liouville fractional integral

y

based on left and right end point functions can then be established, that is

\begin{array}{l} {[I_{ⱱ^{+}}^{α} \tilde{G} (y)]}^{γ} = \frac{1}{Γ (α)} \int_{ⱱ}^{y} {(y - t)}^{α - 1} G_{γ} (t) d t \\ = \frac{1}{Γ (α)} \int_{ⱱ}^{y} {(y - t)}^{α - 1} [G_{*} (t, γ), G^{*} (t, γ)] d t, (y > ⱱ), \end{array}

(11)

where

I_{ⱱ^{+}}^{α} G_{*} (y, γ) = \frac{1}{Γ (α)} \int_{ⱱ}^{y} {(y - t)}^{α - 1} G_{*} (t, γ) d t, (y > ⱱ),

and

I_{ⱱ^{+}}^{α} G^{*} (y, γ) = \frac{1}{Γ (α)} \int_{ⱱ}^{y} {(y - t)}^{α - 1} G^{*} (t, γ) d t, (y > ⱱ),

(12)

Similarly,

\begin{array}{l} {[I_{ⱳ^{-}}^{α} \tilde{G} (y)]}^{γ} = \frac{1}{Γ (α)} \int_{y}^{ⱳ} {(t - y)}^{α - 1} G_{γ} (t) d t \\ = \frac{1}{Γ (α)} \int_{y}^{ⱳ} {(t - y)}^{α - 1} [G_{*} (t, γ), G^{*} (t, γ)] d t, (y < ⱳ), \end{array}

(13)

where

I_{ⱳ^{-}}^{α} G_{*} (y, γ) = \frac{1}{Γ (α)} \int_{y}^{ⱳ} {(t - y)}^{α - 1} G_{*} (t, γ) d t, (y < ⱳ),

and

I_{ⱳ^{-}}^{α} G^{*} (y, γ) = \frac{1}{Γ (α)} \int_{y}^{ⱳ} {(t - y)}^{α - 1} G^{*} (t, γ) d t, (y < ⱳ) .

(14)

Lemma 2

([44]). Let

ỽ : [0, 1] \to R^{+}

and

\tilde{G} : [p, q] \to F_{0}^{+}

be convex

F N V M s

[p, q],

whose

γ

-cuts set up the sequence of

Ι V M

G_{γ} : [p, q] \subset R \to R_{I}^{+}

provided by

G_{γ} (y) = [G_{*} (y, γ), G^{*} (y, γ)]

for all

y \in [p, q]

and for all

γ \in [0, 1]

. If

\tilde{G} \in L ([p, q], F_{0})

, then

\frac{1}{α ỽ (\frac{1}{2})} \tilde{G} (\frac{p + q}{2}) \leq_{F} \frac{Γ (α)}{{(q - p)}^{α}} [I_{p^{+}}^{α} \tilde{G} (q) \oplus I_{q^{-}}^{α} \tilde{G} (p)] \leq_{F} [\tilde{G} (p) \oplus \tilde{G} (q)] \int_{0}^{1} t^{β - 1} [ỽ (t) + ỽ (1 - t)] d t .

(15)

Interval- and fuzzy Aumann-type integrals are established as follows for the coordinated

I V M

G (x, y)

and coordinated

F N V M

\tilde{G} (x, y)

Lemma 3

([46]). Let

\tilde{G} : ∆ [ⱱ, ⱳ] \times [p, q] \subset R^{2} \to F_{0}

be an

F N V M

on coordinates, whose

γ

-cuts set up the sequence of

Ι V M s

G_{γ} : ∆ \subset R^{2} \to R_{I}

provided by

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)]

for all

(x, y) \in ∆ = [ⱱ, ⱳ] \times [p, q]

and for all

γ \in [0, 1] .

Then,

\tilde{G}

is fuzzy double-integrable (

F D

-integrable) over

∆

if and only if

G_{*} (x, γ)

and

G^{*} (x, γ)

are both

D

-integrable over

∆ .

Moreover, if

\tilde{G}

F D

-integrable over

∆,

then

{[(F D) \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) d y d x]}^{\begin{matrix} γ \end{matrix}} = [(D) \int_{ⱱ}^{ⱳ} \int_{p}^{q} G_{*} ((x, y), γ) d y d x, (D) \int_{ⱱ}^{ⱳ} \int_{p}^{q} G^{*} ((x, y), γ) d y d x] = (I D) \int_{ⱱ}^{ⱳ} \int_{p}^{q} G_{γ} (x, y) d y d x,

(16)

for all

γ \in [0, 1] .

The family of all

F D

-integrable

F N V M

s over coordinates and

D

-integrable functions over coordinates are denoted by

{F O}_{∆}

and

O_{(∆, γ)}

for all

γ \in [0, 1] .

Here, the main definition of the fuzzy Riemann–Liouville fractional integral on the coordinates of the function

\tilde{G} (x, y)

is given by the following definition:

Definition 4

([45]). Let

\tilde{G} : ∆ \to F_{0}

and

\tilde{G} \in {F O}_{∆}

. The double fuzzy interval Riemann–Liouville-type integrals

I_{ⱱ^{+}, p^{+}}^{α, β}, I_{ⱱ^{+}, q^{-}}^{α, β}, I_{ⱳ^{-}, p^{+}}^{α, β}, a n d I_{ⱳ^{-}, q^{-}}^{α, β}

G

order

α, β > 0

are established by the following equations:

I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (x, y) = \frac{1}{Γ (α) Γ (β)} \int_{ⱱ}^{x} \int_{p}^{y} {{(x - t)}^{α - 1} (y - s)}^{β - 1} \tilde{G} (t, s) d s d t, (x > ⱱ, y > p),

(17)

I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (x, y) = \frac{1}{Γ (α) Γ (β)} \int_{ⱱ}^{x} \int_{y}^{q} {{(x - t)}^{α - 1} (s - y)}^{β - 1} \tilde{G} (t, s) d s d t, (x > ⱱ, y < q),

(18)

I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (x, y) = \frac{1}{Γ (α) Γ (β)} \int_{x}^{ⱳ} \int_{p}^{y} {{(t - x)}^{α - 1} (y - s)}^{β - 1} \tilde{G} (t, s) d s d t, (x < ⱳ, y > p),

(19)

I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (x, y) = \frac{1}{Γ (α) Γ (β)} \int_{x}^{ⱳ} \int_{y}^{q} {{(t - x)}^{α - 1} (s - y)}^{β - 1} \tilde{G} (t, s) d s d t, (x < ⱳ, y < q) .

(20)

Here, the newly established concept of coordinated

ỽ

-convexity over fuzzy number space in the codomain via fuzzy relation is provided by the following definition:

Definition 5.

The

F N V M

\tilde{G} : ∆ \to F_{0}

is referred to as a coordinated

ỽ

-convex

F N V M

∆

\begin{array}{l} \tilde{G} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ (t) ỽ (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ (t) ỽ (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ (1 - t) ỽ (ԟ) \tilde{G} (ⱳ, p) \oplus ỽ (1 - t) ỽ (1 - ԟ) \tilde{G} (ⱳ, q), \end{array}

(21)

for all

(ⱱ, ⱳ), (p, q) \in ∆

and

t, ԟ \in [0, 1],

where

\tilde{G} (x) \geq_{F} \tilde{0} .

If inequality (21) is reversed, then

\tilde{G}

is referred to as a coordinated

ỽ

-concave

F N V M

∆

Lemma 4.

Let

\tilde{G} : ∆ \to F_{0}

be a coordinated

F N V M

∆

. Then,

\tilde{G}

is a coordinated

ỽ

-convex

F N V M

∆

if and only if there are two coordinated

ỽ

-convex

F N V M

{\tilde{G}}_{x} : [p, q] \to F_{0}

{\tilde{G}}_{x} (w) = \tilde{G} (x, w)

and

{\tilde{G}}_{y} : [ⱱ, ⱳ] \to F_{0}

{\tilde{G}}_{y} (z) = \tilde{G} (z, y)

Theorem 1.

Let

\tilde{G} : ∆ \to F_{0}

be an

F N V M

∆

. Then, from

γ

-levels, we identify the collection of

Ι V M

G_{γ} : ∆ \to R_{I}^{+} \subset R_{I}

provided by

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)],

(22)

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. Then,

\tilde{G}

is coordinated

ỽ

-convex

F N V M

∆

if and only if, for all

γ \in [0, 1],

G_{*} ((x, y), γ)

and

G^{*} ((x, y), γ)

are both coordinated

ỽ

-convex.

Proof.

We assume that for each

γ \in [0, 1],

G_{*} (x, γ)

and

G^{*} (x, γ)

are coordinated

ỽ

-convex and

ỽ

-concave on

∆

, respectively. Then, for Equation (21), for all

(ⱱ, ⱳ), (p, q) \in ∆, t

and

ԟ \in [0, 1]

, we have

\begin{array}{l} G_{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ) \\ \leq ỽ (t) {ỽ (ԟ) G}_{*} ((ⱱ, p), γ) + ỽ (t) ỽ (1 - ԟ) G_{*} ((ⱱ, q), γ) + {ỽ (ԟ) ỽ (1 - t) G}_{*} ((ⱱ, p), γ) + ỽ (1 - t) ỽ (1 - ԟ) G_{*} ((ⱱ, q), γ), \end{array}

and

\begin{array}{l} G^{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ) \\ \leq ỽ (t) {ỽ (ԟ) G}_{*} ((ⱱ, p), γ) + ỽ (t) ỽ (1 - ԟ) G^{*} ((ⱱ, q), γ) + ỽ (ԟ) ỽ (1 - t) G^{*} ((ⱱ, p), γ) + ỽ (1 - t) ỽ (1 - ԟ) G^{*} ((ⱱ, q), γ), \end{array}

Then, by Equations (22), (3) and (5), we obtain

\begin{array}{l} G_{γ} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q)) \\ = [G_{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ), G^{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ)] \\ \leq_{I} ỽ (t) ỽ (ԟ) [G_{*} ((ⱱ, p), γ), G^{*} ((ⱱ, p), γ)] + ỽ (t) ỽ (1 - ԟ) [G_{*} ((ⱱ, q), γ), G^{\begin{matrix} \begin{matrix} * \end{matrix} \end{matrix}} ((ⱱ, q), γ)] \\ + ỽ (ԟ) ỽ (1 - t) [G_{*} ((ⱱ, p), γ), G^{*} ((ⱱ, p), γ)] + ỽ (1 - t) ỽ (1 - ԟ) [G_{*} ((ⱱ, q), γ), G^{*} ((ⱱ, q), γ)] \end{array}

that is

\begin{array}{l} \tilde{G} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ (t) ỽ (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ (t) ỽ (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ (1 - t) ỽ (1 - ԟ) \tilde{G} (ⱳ, p) \oplus ỽ (1 - t) ỽ (1 - ԟ) \tilde{G} (ⱳ, q), \end{array}

\tilde{G}

is coordinated

ỽ

-convex

F N V M

∆ .

Conversely, let

\tilde{G}

be coordinated

ỽ

-convex

F N V M

∆ .

Then, for all

(ⱱ, ⱳ), (p, q) \in ∆, t

and

ԟ \in [0, 1],

we have

\begin{array}{l} \tilde{G} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ (t) ỽ (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ (t) ỽ (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ (1 - t) ỽ (ԟ) \tilde{G} (ⱳ, p) \oplus ỽ (1 - t) ỽ (1 - ԟ) \tilde{G} (ⱳ, q) . \end{array}

Therefore, again from Equation (22), for each

γ \in [0, 1]

, we have

\begin{array}{l} G_{γ} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q)) \\ = [G_{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ), G^{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ)] . \end{array}

Again, using Equations (3) and (5), we obtain

\begin{array}{l} ỽ (t) ỽ (ԟ) G_{γ} (ⱱ, p) + ỽ (t) ỽ (1 - ԟ) G_{γ} (ⱱ, q) + ỽ (1 - t) ỽ (ԟ) G_{γ} (ⱳ, p) + ỽ (1 - t) ỽ (1 - ԟ) G_{γ} (ⱳ, q) \\ = ỽ (t) ỽ (ԟ) [G_{*} ((ⱱ, p), γ), G^{*} ((ⱱ, p), γ)] + ỽ (t) ỽ (1 - ԟ) [G_{*} ((ⱱ, q), γ), G^{*} ((ⱱ, q), γ)] \\ + ỽ (ԟ) ỽ (1 - t) [G_{*} ((ⱱ, p), γ), G^{*} ((ⱱ, p), γ)] + ỽ (1 - t) ỽ (1 - ԟ) [G_{*} ((ⱱ, q), γ), G^{*} ((ⱱ, q), γ)], \end{array}

for all

x, ω \in ∆

and

t \in [0, 1] .

Then, by the coordinated

ỽ

-convexity of

\tilde{G}

, we have, for all

x, ω \in ∆

and

t \in [0, 1]

, such that

\begin{array}{l} G_{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ) \\ \leq ỽ (t) ỽ (ԟ) G_{*} (ⱱ, p) + ỽ (t) ỽ (1 - ԟ) G_{*} (ⱱ, q) + ỽ (1 - t) ỽ (ԟ) G_{*} (ⱳ, p) + ỽ (1 - t) ỽ (1 - ԟ) G_{*} (ⱳ, q), \end{array}

and

\begin{array}{l} G^{*} ((t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q), γ) \\ \leq ỽ (t) ỽ (ԟ) G^{*} (ⱱ, p) + ỽ (t) ỽ (1 - ԟ) G^{*} (ⱱ, q) + ỽ (1 - t) ỽ (ԟ) G^{*} (ⱳ, p) + ỽ (1 - t) ỽ (1 - ԟ) G^{*} (ⱳ, q), \end{array}

for each

γ \in [0, 1] .

Hence, the result is as follows. □

Example 1.

We consider the

F N V M

\tilde{G} : [0, 1] \times [0, 1] \to F_{0}

established by

G (x, y) (σ) = \{\begin{matrix} \frac{σ - x y}{5 - x y}, & [x y, 5] \\ \frac{(6 + e^{x}) (6 + e^{y}) - σ}{(6 + e^{x}) (6 + e^{y}) - 5}, & σ \in (5, (6 + e^{x}) (6 + e^{y})] \\ 0, & o t h e r w i s e, \end{matrix}

(23)

Then, for each

γ \in [0, 1],

we have

G_{γ} (x) = [(1 - γ) x y + 5 γ, (1 - γ) (6 + e^{x}) (6 + e^{y}) + 5 γ]

. The endpoint functions

G_{*} ((x, y), γ),

G^{*} ((x, y), γ)

are coordinated

ỽ

-concave functions for each

γ \in [0, 1]

. Hence,

\tilde{G} (x, y)

is a coordinated

ỽ

-convex

F N V M

From Lemma 4 and Example 1, we can easily note that each

ỽ

-convex

F N V M

is a coordinated

ỽ

-convex

F N V M

. But, the opposite is not true.

Remark 1.

If one assumes that

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

with

γ = 1

, then

G

is referred to as a classical coordinated

ỽ

-convex function if

G

meets the inequality stated here:

\begin{array}{l} G (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq ỽ (t) {ỽ (ԟ) G}_{*} (ⱱ, p) + ỽ (t) ỽ (1 - ԟ) G_{*} (ⱱ, q) + {ỽ (ԟ) ỽ (1 - t) G}_{*} (ⱱ, p) + ỽ (1 - t) ỽ (1 - ԟ) G_{*} (ⱱ, q) . \end{array}

(24)

If one assumes that

ỽ (t) = t, ỽ (ԟ) = ԟ

and

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

with

γ = 1

, then

G

is referred to as a classical coordinated convex function if

G

meets the inequality stated here:

\begin{array}{l} G (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq t ԟ G (ⱱ, p) + t (1 - ԟ) G (ⱱ, q) + (1 - t) ԟ G (ⱳ, p) + (1 - t) (1 - ԟ) G (ⱳ, q) . \end{array}

(25)

Let us assume that

ỽ (t) = t, ỽ (ԟ) = ԟ

and

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

γ = 1

, as well as that

G_{*} ((x, y), γ)

is an affine function and

G^{*} ((x, y), γ)

is a concave function. If it meets the inequality stated here, see the following equation [63]:

\begin{array}{l} G (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \supseteq t ԟ G (ⱱ, p) + t (1 - ԟ) G (ⱱ, q) + (1 - t) ԟ G (ⱳ, p) + (1 - t) (1 - ԟ) G (ⱳ, q), \end{array}

(26)

is true.

Definition 6.

Let

\tilde{G} : ∆ \to F_{0}

be an

F N V M

∆

, provided using

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)],

(27)

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. If

G_{*} ((x, y), γ)

and

G^{*} ((x, y), γ)

are coordinated

ỽ

-convex (concave) and affine functions on

∆

for all

γ \in [0, 1],

respectively, then

\tilde{G}

is a coordinated left-

ỽ

-convex (concave)

F N V M

∆

Definition 7.

Let

\tilde{G} : ∆ \to F_{0}

be an

F N V M

∆

, established using

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)],

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. If

G_{*} ((x, y), γ)

and

G^{*} ((x, y), γ)

are coordinated

ỽ

-affine and

ỽ

- (concave) functions on

∆

, respectively, then

\tilde{G}

is a coordinated right-

ỽ

-convex (concave)

F N V M

∆

Theorem 2.

Let

∆

be a coordinated convex set, and let

\tilde{G} : ∆ \to F_{0}

be an

F N V M

. Then, from

γ

-levels, we obtain the collection of IVMs

G_{γ} : ∆ \to R_{I}^{+} \subset R_{I}

provided by

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)],

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. Then,

\tilde{G}

is a coordinated

ỽ

-concave

F N V M

∆

if and only if, for all

γ \in [0, 1],

G_{*} ((x, y), γ)

and

G^{*} ((x, y), γ)

are coordinated

ỽ

-concave and

ỽ

-convex functions, respectively.

Proof.

The demonstration of proof of Theorem 2 is similar to the demonstration proof of Theorem 1. □

In the next section, to avoid confusion, we will not include the symbols

(R)

(I R)

(F R)

(I D)

, and

(F D)

before the integral sign.

The main goal of this article is to develop a number of original fractional coordinated integral inequalities for the H-H types using a coordinated

ỽ

-concave

F N V M

. We acquired the most recent estimates for mappings, whose products were coordinated

ỽ

-concave

F N V M

s using the fuzzy fractional operators.

3. Main Results

Here is first result of the coordinated integral inequalities for the H-H type using the fuzzy fractional operators via the coordinated

ỽ

-concave

F N V M

Theorem 3.

Let

\tilde{G} : ∆ \to F_{0}^{+}

be a coordinated

ỽ

-convex

F N V M

∆

and let

ỽ : [0, 1] \to R^{+}

. Then, from

γ

-cuts, we set up the sequence of

Ι V M s

G_{γ} : ∆ \to R_{I}^{+}

are provided by

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)]

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. If

{\tilde{G} \in F O}_{∆}

, then the following inequalities hold:

\begin{array}{l} \frac{1}{ỽ^{2} (\frac{1}{2})} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq_{F} \frac{Γ (α + 1)}{{2 ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, \frac{p + q}{2}) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, \frac{p + q}{2})] \oplus \frac{Γ (β + 1)}{{2 ỽ (\frac{1}{2}) (q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, q) \oplus I_{q^{-}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, p)] \\ \leq_{F} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \oplus I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p)] \\ \leq_{F} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, p) \oplus I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, q) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, p) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ \\ \oplus \frac{α Γ (β + 1)}{{(q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (ⱱ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p) \oplus I_{p^{+}}^{β} \tilde{G} (ⱳ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p)] \times \int_{0}^{1} t^{α - 1} ỽ (t) + ỽ (1 - t) d t \\ \leq_{F} α β [\tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ \int_{0}^{1} t^{α - 1} [ỽ (t) + ỽ (1 - t)] d t . \end{array}

(28)

\tilde{G} (x, y)

is a coordinated

ỽ

-concave

F N V M

, then

\begin{array}{l} \frac{1}{ỽ^{2} (\frac{1}{2})} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \geq_{F} \frac{Γ (α + 1)}{{2 ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, \frac{p + q}{2}) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, \frac{p + q}{2})] \oplus \frac{Γ (β + 1)}{{2 ỽ (\frac{1}{2}) (q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, q) \oplus I_{q^{-}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, p)] \\ \geq_{F} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \oplus I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p)] \\ \geq_{F} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, p) \oplus I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, q) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, p) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ \\ \oplus \frac{α Γ (β + 1)}{{(q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (ⱱ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p) \oplus I_{p^{+}}^{β} \tilde{G} (ⱳ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p)] \times \int_{0}^{1} t^{α - 1} [ỽ (t) + ỽ (1 - t)] d t \\ \geq_{F} α β [\tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ \int_{0}^{1} t^{α - 1} [ỽ (t) + ỽ (1 - t)] d t . \end{array}

(29)

Proof.

Let

\tilde{G} : [ⱱ, ⱳ] \to F_{0}

be a coordinated

ỽ

-convex

F N V M

. Then, based on the hypothesis, we have

\frac{1}{ỽ^{2} (\frac{1}{2})} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \leq_{F} \tilde{G} (t ⱱ + (1 - t) ⱳ, t p + (1 - t) q) \oplus \tilde{G} ((1 - t) ⱱ + t ⱳ, (1 - t) p + t q) .

By using Theorem 1, for every

γ \in [0, 1]

, we have

\begin{matrix} \frac{1}{ỽ^{2} (\frac{1}{2})} G_{*} ((\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}), γ) \\ \leq G_{*} ((t ⱱ + (1 - t) ⱳ, t p + (1 - t) q), γ) + G_{*} (((1 - t) ⱱ + t ⱳ, (1 - t) p + t q), γ), \\ \frac{1}{ỽ^{2} (\frac{1}{2})} G^{*} ((\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}), γ) \\ \leq G^{*} ((t ⱱ + (1 - t) ⱳ, t p + (1 - t) q), γ) + G^{*} (((1 - t) ⱱ + t ⱳ, (1 - t) p + t q), γ) . \end{matrix}

By using Lemma 4, we have

\begin{matrix} \frac{1}{ỽ (\frac{1}{2})} G_{*} ((x, \frac{p + q}{2}), γ) \leq G_{*} ((x, t p + (1 - t) q), γ) + G_{*} ((x, (1 - t) p + t q), γ), \\ \frac{1}{ỽ (\frac{1}{2})} G^{*} ((x, \frac{p + q}{2}), γ) \leq G^{*} ((x, t p + (1 - t) q), γ) + G^{*} ((x, (1 - t) p + t q), γ), \end{matrix}

(30)

and

\begin{matrix} \frac{1}{ỽ (\frac{1}{2})} G_{*} ((\frac{ⱱ + ⱳ}{2}, y), γ) \leq G_{*} ((t ⱱ + (1 - t) ⱳ, y), γ) + G_{*} (((1 - t) ⱱ + t ⱳ, y), γ), \\ \frac{1}{ỽ (\frac{1}{2})} G^{*} ((\frac{ⱱ + ⱳ}{2}, y), γ) \leq G^{*} ((t ⱱ + (1 - t) ⱳ, y), γ) + G^{*} (((1 - t) ⱱ + t ⱳ, y), γ) . \end{matrix}

(31)

From (30) and (31), we have

\begin{array}{l} \frac{1}{ỽ (\frac{1}{2})} [G_{*} ((x, \frac{p + q}{2}), γ), G^{*} ((x, \frac{p + q}{2}), γ)] \\ \leq_{I} [G_{*} ((x, t p + (1 - t) q), γ), G^{*} ((x, t p + (1 - t) q), γ)] \\ + [G_{*} ((x, (1 - t) p + t q), γ), G^{*} ((x, (1 - t) p + t q), γ)], \end{array}

and

\begin{array}{l} \frac{1}{ỽ (\frac{1}{2})} [G_{*} ((\frac{ⱱ + ⱳ}{2}, y), γ), G^{*} ((\frac{ⱱ + ⱳ}{2}, y), γ)] \\ \leq_{I} [G_{*} ((t ⱱ + (1 - t) ⱳ, y), γ), G^{*} ((t ⱱ + (1 - t) ⱳ, y), γ)] \\ + [G_{*} ((t ⱱ + (1 - t) ⱳ, y), γ), G^{*} ((t ⱱ + (1 - t) ⱳ, y), γ)], \end{array}

It follows that

\frac{1}{ỽ (\frac{1}{2})} G_{γ} (x, \frac{p + q}{2}) \leq_{I} G_{γ} (x, t p + (1 - t) q) + G_{γ} (x, (1 - t) p + t q),

(32)

and

\frac{1}{ỽ (\frac{1}{2})} G_{γ} (\frac{ⱱ + ⱳ}{2}, y) \leq_{I} G_{γ} (t ⱱ + (1 - t) ⱳ, y) + G_{γ} (t ⱱ + (1 - t) ⱳ, y) .

(33)

For

G_{γ} (x, .)

and

G_{γ} (., y)

, both are coordinated

ỽ

-convex-

Ι V M

s; then, using inequality (15), for every

γ \in [0, 1]

, for inequalities (32) and (43), we have

\frac{1}{β ỽ (\frac{1}{2})} {G_{γ}}_{x} (\frac{p + q}{2}) \leq_{I} \frac{Γ (β)}{{(q - p)}^{β}} [I_{p^{+}}^{β} {G_{γ}}_{x} (q) + I_{q^{-}}^{β} {G_{γ}}_{x} (p)] \leq_{I} [{G_{γ}}_{x} (p) + {G_{γ}}_{x} (q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ

(34)

and

\frac{1}{α ỽ (\frac{1}{2})} {G_{γ}}_{y} (\frac{ⱱ + ⱳ}{2}) \leq_{I} \frac{Γ (α)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} {G_{γ}}_{y} (ⱳ) + I_{ⱳ^{-}}^{α} {G_{γ}}_{y} (ⱱ)] \leq_{I} [{G_{γ}}_{y} (ⱱ) + {G_{γ}}_{y} (ⱳ)] \int_{0}^{1} t^{α - 1} ỽ (t) + ỽ (1 - t) d t

(35)

Since

{G_{γ}}_{x} (w) = G_{γ} (x, w)

, then (34) can be written as

\frac{1}{β ỽ (\frac{1}{2})} G_{γ} (x, \frac{p + q}{2}) \leq_{I} \frac{Γ (β)}{{(q - p)}^{β}} [I_{p^{+}}^{α} G_{γ} (x, q) + I_{q^{-}}^{α} G_{γ} (x, p)] \leq_{I} [G_{γ} (x, p) + G_{γ} (x, q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ .

(36)

that is

\frac{1}{β ỽ (\frac{1}{2})} G_{γ} (x, \frac{p + q}{2}) \leq_{I} \frac{1}{{(q - p)}^{β}} [\int_{p}^{q} {(q - ԟ)}^{β - 1} G_{γ} (x, ԟ) d ԟ + \int_{p}^{q} {(ԟ - p)}^{β - 1} G_{γ} (x, ԟ) d ԟ] \leq_{I} [G_{γ} (x, p) + G_{γ} (x, q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ .

By multiplying double inequality (36) by

\frac{{(ⱳ - x)}^{α - 1}}{{(ⱳ - ⱱ)}^{α}}

and integrating with respect to

x

over

[ⱱ, ⱳ],

we have

\begin{array}{l} \frac{1}{{β (ⱳ - ⱱ)}^{α} ỽ (\frac{1}{2})} \int_{ⱱ}^{ⱳ} G_{γ} (x, \frac{p + q}{2}) {(ⱳ - x)}^{α - 1} d x \\ \leq_{I} \frac{1}{{{(ⱳ - ⱱ)}^{α} (q - p)}^{β}} \int_{ⱱ}^{ⱳ} \int_{p}^{q} {(ⱳ - x)}^{α - 1} {(q - ԟ)}^{β - 1} G_{γ} (x, ԟ) d ԟ d x + \int_{ⱱ}^{ⱳ} \int_{p}^{q} {(ⱳ - x)}^{α - 1} {(ԟ - p)}^{β - 1} G_{γ} (x, ԟ) d ԟ d x \\ \leq_{I} \frac{1}{{(ⱳ - ⱱ)}^{α}} [\int_{ⱱ}^{ⱳ} {(ⱳ - x)}^{α - 1} G_{γ} (x, p) d x + \int_{ⱱ}^{ⱳ} {(ⱳ - x)}^{α - 1} G_{γ} (x, q) d x] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ . \end{array}

(37)

Again, by multiplying double inequality (36) by

\frac{{(x - ⱱ)}^{α - 1}}{{(ⱳ - ⱱ)}^{α}}

and integrating with respect to

x

over

[ⱱ, ⱳ],

we have

\begin{array}{l} \frac{1}{{β (ⱳ - ⱱ)}^{α} ỽ (\frac{1}{2})} \int_{ⱱ}^{ⱳ} G_{γ} (x, \frac{p + q}{2}) {(x - ⱱ)}^{α - 1} d x \\ \leq_{I} \frac{1}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} \int_{ⱱ}^{ⱳ} \int_{p}^{q} {(x - ⱱ)}^{α - 1} {(q - ԟ)}^{β - 1} G_{γ} (x, ԟ) d ԟ d x \\ + \frac{1}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} \int_{ⱱ}^{ⱳ} \int_{p}^{q} {(x - ⱱ)}^{α - 1} {(ԟ - p)}^{β - 1} G_{γ} (x, ԟ) d ԟ d x \\ \leq_{I} \frac{1}{{(ⱳ - ⱱ)}^{α}} [\int_{ⱱ}^{ⱳ} {(x - ⱱ)}^{α - 1} G_{γ} (x, p) d x + \int_{ⱱ}^{ⱳ} {(x - ⱱ)}^{α - 1} G_{γ} (x, q) d x] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ . \end{array}

(38)

From (37), we have

\begin{array}{l} \frac{Γ (α + 1)}{{2 ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, \frac{p + q}{2})] \\ \leq_{I} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G_{γ} (ⱳ, q) + I_{ⱳ^{-}, p^{+}}^{α, β} G_{γ} (ⱳ, p)] \\ \leq_{I} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, p) + I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ . \end{array}

(39)

From (38), we have

\begin{array}{l} \frac{Γ (α + 1)}{{2 ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, \frac{p + q}{2})] \\ \leq_{I} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} G_{γ} (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G_{γ} (ⱱ, p)] \\ \leq_{I} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, p) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ . \end{array}

(40)

Since, from

γ

-cuts, we obtain the collection of

Ι V M

G_{γ} : ∆ \to R_{I}^{+}

, we then have

\begin{array}{l} \frac{Γ (α + 1)}{{2 ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, \frac{p + q}{2})] \\ \leq_{F} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \oplus I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱳ, p)] \\ \leq_{F} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, p) \oplus I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ . \end{array}

(41)

And

\begin{array}{l} \frac{Γ (α + 1)}{{2 ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, \frac{p + q}{2})] \\ \leq_{F} \frac{β Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p)] \\ \leq_{F} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, p) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, q)] \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ . \end{array}

(42)

Similarly, since

{\tilde{G}}_{y} (z) = \tilde{G} (z, y)

, from (35), (41) and (42), we then have

\begin{array}{l} \frac{Γ (β + 1)}{{2 ỽ (\frac{1}{2}) (q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, q)] \\ \leq_{F} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \oplus I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q)] \\ \leq_{F} \frac{α Γ (β + 1)}{{(q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (ⱱ, q) \oplus I_{p^{+}}^{β} \tilde{G} (ⱳ, q)] . \end{array}

(43)

and

\begin{array}{l} \frac{Γ (β + 1)}{{2 ỽ (\frac{1}{2}) (q - p)}^{α}} [I_{q^{-}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, p)] \\ \leq_{F} \frac{Γ (α + 1) Γ (β + 1)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p)] \\ \leq_{F} \frac{α Γ (β + 1)}{{(q - p)}^{β}} [I_{q^{-}}^{β} \tilde{G} (ⱱ, p) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p)] . \end{array}

(44)

The second, third and fourth inequalities of (28) will be the consequence of adding the inequalities of (41)–(44).

Now, for any

γ \in [0, 1]

, we have inequality (15)’s left portion:

\frac{1}{ỽ^{2} (\frac{1}{2})} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \leq_{I} \frac{Γ (β + 1)}{{ỽ (\frac{1}{2}) (q - p)}^{β}} [I_{p^{+}}^{β} G_{γ} (\frac{ⱱ + ⱳ}{2}, q) + I_{q^{-}}^{β} G_{γ} (\frac{ⱱ + ⱳ}{2}, p)]

(45)

and

\frac{1}{ỽ^{2} (\frac{1}{2})} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \leq_{I} \frac{Γ (α + 1)}{{ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, \frac{p + q}{2}) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, \frac{p + q}{2})] .

(46)

The following inequality is created by adding inequalities (45) and (46):

\begin{array}{l} \frac{1}{ỽ^{2} (\frac{1}{2})} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \leq_{I} \frac{Γ (α + 1)}{{ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, \frac{p + q}{2}) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, \frac{p + q}{2})] \\ + \frac{Γ (β + 1)}{{ỽ (\frac{1}{2}) (q - p)}^{β}} [I_{p^{+}}^{β} G_{γ} (\frac{ⱱ + ⱳ}{2}, q) + I_{q^{-}}^{β} G_{γ} (\frac{ⱱ + ⱳ}{2}, p)] . \end{array}

(47)

Similarly, since we obtain the set of

Ι V M

G_{γ} : ∆ \to R_{I}^{+}

for

γ \in [0, 1]

, the inequality can be expressed as follows:

\begin{array}{l} \frac{1}{ỽ^{2} (\frac{1}{2})} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq_{F} \frac{Γ (α + 1)}{{ỽ (\frac{1}{2}) (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, \frac{p + q}{2}) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, \frac{p + q}{2})] \oplus \frac{Γ (β + 1)}{{ỽ (\frac{1}{2}) (q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, q) \oplus I_{q^{-}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, p)] . \end{array}

(48)

The first inequality of (28) is this one.

Now, for any

γ \in [0, 1]

, we have inequality (15)’s right portion:

\frac{Γ (β)}{{(q - p)}^{β}} [I_{p^{+}}^{β} G_{γ} (ⱱ, q) + I_{q^{-}}^{β} G_{γ} (ⱱ, p)] \leq_{I} [G_{γ} (ⱱ, p) + G_{γ} (ⱱ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ

(49)

\frac{Γ (β)}{{(q - p)}^{β}} [I_{p^{+}}^{β} G_{γ} (ⱳ, q) + I_{q^{-}}^{β} G_{γ} (ⱳ, p)] \leq_{I} [G_{γ} (ⱳ, p) + G_{γ} (ⱳ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ

(50)

\frac{Γ (α)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, p) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, p)] \leq_{I} [G_{γ} (ⱱ, p) + G_{γ} (ⱳ, p)] \times \int_{0}^{1} t^{α - 1} ỽ (t) + ỽ (1 - t) d t

(51)

\frac{Γ (α)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, q) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, q)] \leq_{I} [G_{γ} (ⱱ, q) + G_{γ} (ⱳ, q)] \times \int_{0}^{1} t^{α - 1} ỽ (t) + ỽ (1 - t) d t

(52)

By summing inequalities (49)–(52) and then taking the multiplication of the resultant with

α β

, we have

\begin{array}{l} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, p) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, p) + I_{ⱱ^{+}}^{α} G_{γ} (ⱳ, q) + I_{ⱳ^{-}}^{α} G_{γ} (ⱱ, q)] \\ + \frac{α Γ (β + 1)}{{(q - p)}^{β}} [I_{p^{+}}^{β} G_{γ} (ⱱ, q) + I_{q^{-}}^{β} G_{γ} (ⱱ, p) + I_{p^{+}}^{β} G_{γ} (ⱳ, q) + I_{q^{-}}^{β} G_{γ} (ⱳ, p)] \\ \leq_{I} [G_{γ} (ⱱ, p) + G_{γ} (ⱱ, q) + G_{γ} (ⱳ, p) + G_{γ} (ⱳ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ \int_{0}^{1} t^{α - 1} ỽ (t) + ỽ (1 - t) d t . \end{array}

Since we receive the collection of

Ι V M

G_{γ} : ∆ \to R_{I}^{+}

from

γ

-cuts, we have

\begin{array}{l} \frac{β Γ (α + 1)}{{(ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, p) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, p) \oplus I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, q) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, q)] \\ \oplus \frac{α Γ (β + 1)}{{(q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (ⱱ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱱ, p) \oplus I_{p^{+}}^{β} \tilde{G} (ⱳ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p)] \\ \leq_{F} [\tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱳ, q)] \times \int_{0}^{1} ԟ^{β - 1} [ỽ (ԟ) + ỽ (1 - ԟ)] d ԟ \int_{0}^{1} t^{α - 1} ỽ (t) + ỽ (1 - t) d t . \end{array}

(53)

This is the final inequality of (28), and the conclusion has been established. □

Now, we acquire the exceptional cases that can be obtained from Theorem 3 by applying some restrictions over the parameters.

Remark 2.

If one assumes that

α = 1,

β = 1

and

ỽ (t) = t, ỽ (ԟ) = ԟ

, then using (28), as a result, there will be inequity [55]:

\begin{array}{l} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \supseteq_{F} \frac{\begin{matrix} 1 \end{matrix}}{2} [\frac{1}{ⱳ - ⱱ} \int_{ⱱ}^{ⱳ} \tilde{G} (x, \frac{p + q}{2}) d x \oplus \frac{1}{q - p} \int_{p}^{q} \tilde{G} (\frac{ⱱ + ⱳ}{2}, y) d y] \supseteq_{F} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) d y d x \\ \supseteq_{F} \frac{1}{4 (ⱳ - ⱱ)} [\int_{ⱱ}^{ⱳ} \tilde{G} (x, p) d x \oplus \int_{ⱱ}^{ⱳ} \tilde{G} (x, q) d x] \oplus \frac{\begin{matrix} 1 \end{matrix}}{4 (q - p)} [\int_{p}^{q} \tilde{G} (ⱱ, y) d y \oplus \int_{p}^{q} \tilde{G} (ⱳ, y) d y] \\ \supseteq_{F} \frac{\tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, q)}{4} . \end{array}

(54)

If one assumes that

α = 1

and

β = 1

, while

ỽ (t) = t, ỽ (ԟ) = ԟ

and

\tilde{G}

is the coordinated left-

U D

ỽ

-convex, then using (28), as a result, there will be inequity [46]:

\begin{array}{l} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq_{F} \frac{\begin{matrix} 1 \end{matrix}}{2} [\frac{1}{ⱳ - ⱱ} \int_{ⱱ}^{ⱳ} \tilde{G} (x, \frac{p + q}{2}) d x \oplus \frac{1}{q - p} \int_{p}^{q} \tilde{G} (\frac{ⱱ + ⱳ}{2}, y) d y] \leq_{F} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) d y d x \\ \leq_{F} \frac{1}{4 (ⱳ - ⱱ)} [\int_{ⱱ}^{ⱳ} \tilde{G} (x, p) d x \oplus \int_{ⱱ}^{ⱳ} \tilde{G} (x, q) d x] \oplus \frac{\begin{matrix} 1 \end{matrix}}{4 (q - p)} [\int_{p}^{q} \tilde{G} (ⱱ, y) d y \oplus \int_{p}^{q} \tilde{G} (ⱳ, y) d y] \\ \leq_{F} \frac{\tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, q)}{4} . \end{array}

(55)

ỽ (t) = t, ỽ (ԟ) = ԟ

and

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

γ = 1

, then using (28), we succeed in bringing about the upcoming inequity [35]:

\begin{array}{l} G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \supseteq \frac{Γ (α + 1)}{{4 (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G (ⱳ, \frac{p + q}{2}) + I_{ⱳ^{-}}^{α} G (ⱱ, \frac{p + q}{2})] + \frac{Γ (β + 1)}{{4 (q - p)}^{β}} [I_{p^{+}}^{β} G (\frac{ⱱ + ⱳ}{2}, q) + I_{q^{-}}^{β} G (\frac{ⱱ + ⱳ}{2}, p)] \\ \supseteq \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G (ⱳ, p) + I_{ⱳ^{-}, p^{+}}^{α, β} G (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G (ⱱ, p)] \\ \supseteq \frac{Γ (α + 1)}{{8 (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G (ⱳ, p) + I_{ⱱ^{+}}^{α} G (ⱳ, q) + I_{ⱳ^{-}}^{α} G (ⱱ, p) + I_{ⱳ^{-}}^{α} G (ⱱ, q)] . \\ + \frac{Γ (β + 1)}{{8 (q - p)}^{β}} [I_{p^{+}}^{β} G (ⱱ, q) + I_{q^{-}}^{β} G (ⱱ, p) + I_{p^{+}}^{β} G (ⱳ, q) + I_{q^{-}}^{β} G (ⱳ, p)] \\ \supseteq \frac{G (ⱱ, p) + G (ⱳ, p) + G (ⱱ, q) + G (ⱳ, q)}{4} . \end{array}

(56)

ỽ (t) = t, ỽ (ԟ) = ԟ

and

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

γ = 1

, then using (28), we succeed in bringing about the upcoming inequity [63]:

\begin{array}{l} G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \supseteq \frac{\begin{matrix} 1 \end{matrix}}{2} [\frac{1}{ⱳ - ⱱ} \int_{ⱱ}^{ⱳ} G (x, \frac{p + q}{2}) d x + \frac{1}{q - p} \int_{p}^{q} G (\frac{ⱱ + ⱳ}{2}, y) d y] \supseteq \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} G (x, y) d y d x \\ \supseteq \frac{1}{4 (ⱳ - ⱱ)} [\int_{ⱱ}^{ⱳ} G (x, p) d x + \int_{ⱱ}^{ⱳ} G (x, q) d x] + \frac{\begin{matrix} 1 \end{matrix}}{4 (q - p)} [\int_{p}^{q} G (ⱱ, y) d y + \int_{p}^{q} G (ⱳ, y) d y] \\ \supseteq \frac{G (ⱱ, p) + G (ⱳ, p) + G (ⱱ, q) + G (ⱳ, q)}{4} . \end{array}

(57)

\tilde{G}

is the coordinated right-

U D

ỽ

-convex with

ỽ (t) = t, ỽ (ԟ) = ԟ

and

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

with

γ = 1

, then using (28), we succeed in bringing about the upcoming inequity [48]:

\begin{array}{l} G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq \frac{Γ (α + 1)}{{4 (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G (ⱳ, \frac{p + q}{2}) + I_{ⱳ^{-}}^{α} G (ⱱ, \frac{p + q}{2})] + \frac{Γ (β + 1)}{{4 (q - p)}^{β}} [I_{p^{+}}^{β} G (\frac{ⱱ + ⱳ}{2}, q) + I_{q^{-}}^{β} G (\frac{ⱱ + ⱳ}{2}, p)] \\ \leq \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G (ⱳ, p) + I_{ⱳ^{-}, p^{+}}^{α, β} G (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G (ⱱ, p)] \\ \leq \frac{Γ (α + 1)}{{8 (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} G (ⱳ, p) G I_{ⱱ^{+}}^{α} G (ⱳ, q) + I_{ⱳ^{-}}^{α} G (ⱱ, p) + I_{ⱳ^{-}}^{α} G (ⱱ, q)] \\ + \frac{Γ (β + 1)}{{8 (q - p)}^{β}} [I_{p^{+}}^{β} G (ⱱ, q) \tilde{+} I_{q^{-}}^{β} G (ⱱ, p) + I_{p^{+}}^{β} G (ⱳ, q) + I_{q^{-}}^{β} G (ⱳ, p)] \\ \leq \frac{G (ⱱ, p) + G (ⱳ, p) + G (ⱱ, q) + G (ⱳ, q)}{4} . \end{array}

(58)

In the next outcomes, we are going to find very interesting outcomes that will be obtained over the product of two coordinated

ỽ

-convex

F N V M

s. These inequalities are known as Pachpatte inequalities.

Theorem 4.

Let

\tilde{G}, \tilde{G} : ∆ \to F_{0}^{+}

be two coordinated

ỽ

-convex

F N V M

s on

∆

and let

ỽ_{1}, ỽ_{2} : [0, 1] \to R^{+}

. Then, from

γ

-cuts, we set up the sequence of

Ι V M

G_{γ}, G_{γ} : ∆ \to R_{I}^{+}

provided by

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)]

and

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)]

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. If

{\tilde{G} \otimes \tilde{G} \in F O}_{∆}

, then the following inequalities hold:

\begin{array}{l} \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, p)] \\ \oplus \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, p)] \\ \leq_{F} \tilde{E} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ \oplus \tilde{Ѵ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ \\ \oplus \tilde{Ϗ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ . \end{array}

(59)

\tilde{G}

and

\tilde{G}

are both coordinated

ỽ

-concave

F N V M

s on

∆

, then the inequality above can be expressed as follows:

\begin{array}{l} \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, p)] \\ \oplus \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, p)] \\ \geq_{F} \tilde{E} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ \oplus \tilde{Ѵ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ \\ \oplus \tilde{Ϗ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ . \end{array}

(60)

where

\begin{array}{l} \tilde{E} (ⱱ, ⱳ, p, q) = \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, q), \\ \tilde{Ѵ} (ⱱ, ⱳ, p, q) = \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱳ, q) \oplus \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱱ, q), \\ \tilde{Ѡ} (ⱱ, ⱳ, p, q) = \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, q) \oplus \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, p) \oplus \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, p), \\ \tilde{Ϗ} (ⱱ, ⱳ, p, q) = \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱳ, q) \oplus \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱱ, q) \oplus \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱳ, p) \oplus \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱱ, p), \end{array}

and for each

γ \in [0, 1],

\tilde{E} (ⱱ, ⱳ, p, q)

\tilde{Ѵ} (ⱱ, ⱳ, p, q)

\tilde{Ѡ} (ⱱ, ⱳ, p, q)

and

\tilde{Ϗ} (ⱱ, ⱳ, p, q)

are established as follows:

\begin{array}{l} E_{γ} (ⱱ, ⱳ, p, q) = [E_{*} ((ⱱ, ⱳ, p, q), γ), E^{*} ((ⱱ, ⱳ, p, q), γ)], \\ Ѵ_{γ} (ⱱ, ⱳ, p, q) = [Ѵ_{*} ((ⱱ, ⱳ, p, q), γ), Ѵ^{*} ((ⱱ, ⱳ, p, q), γ)], \\ Ѡ_{γ} (ⱱ, ⱳ, p, q) = [Ѡ_{*} ((ⱱ, ⱳ, p, q), γ), Ѡ^{*} ((ⱱ, ⱳ, p, q), γ)], \\ Ϗ_{γ} (ⱱ, ⱳ, p, q) = [Ϗ_{*} ((ⱱ, ⱳ, p, q), γ), Ϗ^{*} ((ⱱ, ⱳ, p, q), γ)] . \end{array}

Proof.

Let

\tilde{G}

and

\tilde{G}

be two coordinated

ỽ_{1}

and

ỽ_{2}

-convex

F N V M

s on

[ⱱ, ⱳ] \times [p, q]

, respectively. Then,

\begin{array}{l} \tilde{G} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ_{1} (t) ỽ_{1} (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{1} (t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{1} (1 - t) ỽ_{1} (ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱳ, q), \\ \tilde{G} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \\ \leq_{F} ỽ_{1} (t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{1} (t) ỽ_{1} (ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{1} (1 - t) ỽ_{1} (ԟ) \tilde{G} (ⱳ, q), \\ \tilde{G} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ_{1} (1 - t) ỽ_{1} (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{1} (t) ỽ_{1} (ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{1} (t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱳ, q), \\ \tilde{G} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \\ \leq_{F} ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{1} (1 - t) ỽ_{1} (ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{1} (t) ỽ_{1} (1 - ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{1} (t) ỽ_{1} (ԟ) \tilde{G} (ⱳ, q), \end{array}

and

\begin{array}{l} \tilde{G} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ_{2} (t) ỽ_{2} (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{2} (t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{2} (1 - t) ỽ_{2} (ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱳ, q), \\ \tilde{G} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \\ \leq_{F} ỽ_{2} (t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{2} (t) ỽ_{2} (ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{2} (1 - t) ỽ_{2} (ԟ) \tilde{G} (ⱳ, q), \\ \tilde{G} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ \leq_{F} ỽ_{2} (1 - t) ỽ_{2} (ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{2} (t) ỽ_{2} (ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{2} (t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱳ, q), \\ \tilde{G} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \\ \leq_{F} ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱱ, p) \oplus ỽ_{2} (1 - t) ỽ_{2} (ԟ) \tilde{G} (ⱱ, q) \oplus ỽ_{2} (t) ỽ_{2} (1 - ԟ) \tilde{G} (ⱳ, p) \oplus ỽ_{2} (t) ỽ_{2} (ԟ) \tilde{G} (ⱳ, q), \end{array}

Since

\tilde{G}

and

\tilde{G}

both are coordinated

ỽ_{1}

and

ỽ_{2}

-convex

F N V M

s on

[ⱱ, ⱳ] \times [p, q]

, respectively, then, for any

γ \in [0, 1]

, we have

\begin{array}{l} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \\ + G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \\ \leq_{I} E_{γ} (ⱱ, ⱳ, p, q) [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] \\ + Ѵ_{γ} (ⱱ, ⱳ, p, q) [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] \\ + Ѡ_{γ} (ⱱ, ⱳ, p, q) [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] \\ + Ϗ_{γ} (ⱱ, ⱳ, p, q) [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] . \end{array}

By taking the multiplication of above fuzzy inclusion with

t^{α - 1} ԟ^{β - 1}

and then taking the double integration of the result over

[0, 1] \times [0, 1]

with respect to (

t, ԟ

), we find that

\begin{array}{l} \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) d t d ԟ \\ + \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) d t d ԟ \\ + \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) d t d ԟ \\ + \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) d t d ԟ \\ \leq_{I} E_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ + Ѵ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ + Ѡ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ \\ + Ϗ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ \end{array}

(61)

From the right hand side of (61), we have

\begin{array}{l} \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) d t d ԟ \\ + \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) d t d ԟ \\ + \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) d t d ԟ \\ + \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) d t d ԟ \\ = \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G_{γ} (ⱳ, q) \times G_{γ} (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G_{γ} (ⱳ, p) \times G_{γ} (ⱳ, p)] . \end{array}

(62)

Combining (61) and (62), for each

γ \in [0, 1]

, we have

\begin{array}{l} \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G_{γ} (ⱳ, q) \times G_{γ} (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G_{γ} (ⱳ, p) \times G_{γ} (ⱳ, p)] \\ \leq_{I} E_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ + Ѵ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ + Ѡ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ \\ + Ϗ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ . \end{array}

Moreover, we have

\begin{array}{l} \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, p)] \\ \oplus \frac{Γ (α) Γ (β)}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, p)] \\ \leq_{F} \tilde{E} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ \oplus \tilde{Ѵ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (ԟ)] d t d ԟ \\ \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (1 - t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ \\ \oplus \tilde{Ϗ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (1 - ԟ) ỽ_{2} (ԟ) + ỽ_{1} (1 - t) ỽ_{2} (t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ) + ỽ_{1} (t) ỽ_{2} (1 - t) ỽ_{1} (ԟ) ỽ_{2} (1 - ԟ)] d t d ԟ . \end{array}

Hence, the required result is found. □

In Remark 3, some of the classical results are discussed as special cases of the Theorem 4.

Remark 3.

If one assumes that

ỽ (t) = t, ỽ (ԟ) = ԟ

α = 1

and

β = 1

, then using (59), as a result, there will be inequity [55]:

\begin{array}{l} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) \otimes \tilde{G} (x, y) d y d x \\ \supseteq_{F} \frac{1}{9} \tilde{E} (ⱱ, ⱳ, p, q) \oplus \frac{1}{18} [\tilde{Ѵ} (ⱱ, ⱳ, p, q) \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q)] \oplus \frac{1}{36} \tilde{Ϗ} (ⱱ, ⱳ, p, q) . \end{array}

(63)

\tilde{G}

is a coordinated left-

U D

ỽ

-convex with

ỽ (t) = t, ỽ (ԟ) = ԟ

and one assumes that

α = 1

and

β = 1

, then using (59), as a result, there will be inequity [46]:

\begin{array}{l} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) \otimes \tilde{G} (x, y) d y d x \\ \leq_{F} \frac{1}{9} \tilde{E} (ⱱ, ⱳ, p, q) \oplus \frac{1}{18} [\tilde{Ѵ} (ⱱ, ⱳ, p, q) \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q)] \oplus \frac{1}{36} \tilde{Ϗ} (ⱱ, ⱳ, p, q) . \end{array}

(64)

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

γ = 1

and

ỽ (t) = t, ỽ (ԟ) = ԟ

then, using (59), we succeed in bringing about the upcoming inequity [64]:

\begin{array}{l} \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [\begin{matrix} I_{ⱱ^{+}, p^{+}}^{α, β} G (ⱳ, q) \times G (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G (ⱳ, p) \times G (ⱳ, p) \end{matrix}] \\ + \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [\begin{matrix} I_{ⱳ^{-}, p^{+}}^{α, β} G (ⱱ, q) \times G (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G (ⱱ, p) \times G (ⱱ, p) \end{matrix}] \\ \supseteq (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)}) (\frac{1}{2} - \frac{β}{(β + 1) (β + 2)}) E (ⱱ, ⱳ, p, q) + \frac{α}{(α + 1) (α + 2)} (\frac{1}{2} - \frac{β}{(β + 1) (β + 2)}) Ѵ (ⱱ, ⱳ, p, q) \\ + (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)}) \frac{β}{(β + 1) (β + 2)} Ѡ (ⱱ, ⱳ, p, q) + \frac{β}{(β + 1) (β + 2)} \frac{α}{(α + 1) (α + 2)} Ϗ (ⱱ, ⱳ, p, q) . \end{array}

(65)

ỽ (t) = t, ỽ (ԟ) = ԟ

and

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

γ = 1

, then using (59), we succeed in bringing about the upcoming inequity [63]:

\begin{array}{l} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} G (x, y) \times G (x, y) d y d x \\ \supseteq \frac{1}{9} E (ⱱ, ⱳ, p, q) + \frac{1}{18} [Ѵ (ⱱ, ⱳ, p, q) + Ѡ (ⱱ, ⱳ, p, q)] + \frac{1}{36} Ϗ (ⱱ, ⱳ, p, q) . \end{array}

(66)

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

and

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

with

γ = 1

and

ỽ (t) = t, ỽ (ԟ) = ԟ

, then using (59), we succeed in bringing about the upcoming inequity [47]:

\begin{array}{l} \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [\begin{matrix} I_{ⱱ^{+}, p^{+}}^{α, β} G (ⱳ, q) \times G (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G (ⱳ, p) \times G (ⱳ, p) \end{matrix}] \\ + \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [\begin{matrix} + I_{ⱳ^{-}, p^{+}}^{α, β} G (ⱱ, q) \times G (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G (ⱱ, p) \times G (ⱱ, p) \end{matrix}] \\ \leq (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)}) (\frac{1}{2} - \frac{β}{(β + 1) (β + 2)}) E (ⱱ, ⱳ, p, q) + \frac{α}{(α + 1) (α + 2)} (\frac{1}{2} - \frac{β}{(β + 1) (β + 2)}) Ѵ (ⱱ, ⱳ, p, q) \\ + (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)}) \frac{β}{(β + 1) (β + 2)} Ѡ (ⱱ, ⱳ, p, q) + \frac{β}{(β + 1) (β + 2)} \frac{α}{(α + 1) (α + 2)} Ϗ (ⱱ, ⱳ, p, q) . \end{array}

(67)

Theorem 5.

Let

\tilde{G}, \tilde{G} : ∆ \to F_{0}^{+}

be a coordinated

ỽ

-convex

F N V M

∆

and let

ỽ : [0, 1] \to R^{+}

. Then, from

γ

-cuts, we set up the sequence of

Ι V M

G_{γ}, G_{γ} : ∆ \to R_{I}^{+}

provided by

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)]

and

G_{γ} (x, y) = [G_{*} ((x, y), γ), G^{*} ((x, y), γ)]

for all

(x, y) \in ∆

and for all

γ \in [0, 1]

. If

{\tilde{G} \otimes \tilde{G} \in F O}_{∆}

, then the following inequalities hold:

\begin{array}{l} \frac{1}{2 α β {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2})} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \otimes \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq_{F} \frac{Γ (α) Γ (β)}{{2 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, p)] \\ \oplus \frac{Γ (α) Γ (β)}{2 {(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, p)] \\ \oplus \tilde{E} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)]] d t d ԟ \\ \oplus \tilde{Ѵ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)]] d t d ԟ \\ \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)]] d t d ԟ \\ \oplus \tilde{Ϗ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)]] d t d ԟ . \end{array}

(68)

\tilde{G}

and

\tilde{G}

both are coordinated

ỽ

-concave

F N V M

s on

∆

, then the inequality above can be expressed as follows:

\begin{array}{l} \frac{1}{2 α β {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2})} \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \otimes \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \geq_{F} \frac{Γ (α) Γ (β)}{{2 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} \tilde{G} (ⱳ, q) \otimes \tilde{G} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} \tilde{G} (ⱳ, p) \otimes \tilde{G} (ⱳ, p)] \\ \oplus \frac{Γ (α) Γ (β)}{2 {(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} \tilde{G} (ⱱ, q) \otimes \tilde{G} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} \tilde{G} (ⱱ, p) \otimes \tilde{G} (ⱱ, p)] \\ \oplus \tilde{E} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)]] d t d ԟ \\ \oplus \tilde{Ѵ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)]] d t d ԟ \\ \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)]] d t d ԟ \\ \oplus \tilde{Ϗ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)]] d t d ԟ . \end{array}

(69)

where

\tilde{E} (ⱱ, ⱳ, p, q)

\tilde{Ѵ} (ⱱ, ⱳ, p, q)

\tilde{Ѡ} (ⱱ, ⱳ, p, q)

and

\tilde{Ϗ} (ⱱ, ⱳ, p, q)

are provided in Theorem 5.

Proof.

Since

\tilde{G}, \tilde{G} : ∆ \to F_{0}

for two

ỽ

-convex

F N V M

s, then using inequality (17) and for each

γ \in [0, 1],

we have

\begin{array}{l} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \times G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ = G_{γ} (\frac{t ⱱ + (1 - t) ⱳ}{2} + \frac{(1 - t) ⱱ + t ⱳ}{2}, \frac{ԟ p + (1 - ԟ) q}{2} + \frac{p + q}{2}) \times G_{γ} (\frac{t ⱱ + (1 - t) ⱳ}{2} + \frac{(1 - t) ⱱ + t ⱳ}{2}, \frac{ԟ p + (1 - ԟ) q}{2} + \frac{(1 - ԟ) p + ԟ q}{2}) \\ \leq_{I} {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \times [\begin{matrix} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) + G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) + G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \end{matrix}] \\ \times [\begin{matrix} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) + G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) + G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \end{matrix}] \\ \leq_{I} {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \times [\begin{matrix} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \\ + G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \end{matrix}] \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \times [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \\ \begin{matrix} + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \end{matrix} \end{matrix}] E_{γ} (ⱱ, ⱳ, p, q) \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \\ \times [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \end{matrix}] Ѵ_{γ} (ⱱ, ⱳ, p, q) \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \\ \times [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \end{matrix}] Ѡ_{γ} (ⱱ, ⱳ, p, q) \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \\ \times [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \\ \begin{matrix} + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \end{matrix} \end{matrix}] Ϗ_{γ} (ⱱ, ⱳ, p, q) . \end{array}

By taking the multiplication of above fuzzy inclusion with

t^{α - 1} ԟ^{β - 1}

and then taking the double integration of the result over

[0, 1] \times [0, 1]

with respect to (

t, ԟ

), we have

\begin{array}{l} \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \times G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) d t d ԟ \\ \leq_{I} {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) \times \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [\begin{matrix} G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, ԟ p + (1 - ԟ) q) \\ + G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} (t ⱱ + (1 - t) ⱳ, (1 - ԟ) p + ԟ q) \\ + G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \times G_{γ} ((1 - t) ⱱ + t ⱳ, (1 - ԟ) p + ԟ q) \end{matrix}] d t d ԟ \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) E_{γ} (ⱱ, ⱳ, p, q) \\ \times \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \\ \begin{matrix} + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \end{matrix} \end{matrix}] d t d ԟ \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) Ѵ_{γ} (ⱱ, ⱳ, p, q) \times \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \end{matrix}] d t d ԟ \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) Ѡ_{γ} (ⱱ, ⱳ, p, q) \\ \times \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (1 - t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \end{matrix}] d t d ԟ \\ + {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) Ϗ_{γ} (ⱱ, ⱳ, p, q) \\ \times \int_{0}^{1} \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [\begin{matrix} ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] \\ \begin{matrix} + ỽ_{1} (1 - t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \\ + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] \end{matrix} \end{matrix}] d t d ԟ, \end{array}

which implies that

\begin{array}{l} \frac{1}{α β} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \times G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq_{F} \frac{Γ (α) Γ (β) {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2})}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G_{γ} (ⱳ, q) \times G_{γ} (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G_{γ} (ⱳ, p) \times G_{γ} (ⱳ, p)] \\ + \frac{Γ (α) Γ (β) {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2})}{{(ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱳ^{-}, p^{+}}^{α, β} G_{γ} (ⱱ, q) \times G_{γ} (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G_{γ} (ⱱ, p) \times G_{γ} (ⱱ, p)] \\ + 2 {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) E_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)]] d t d ԟ \\ + 2 {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) Ѵ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)]] d t d ԟ \\ + 2 {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) Ѡ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ)]] d t d ԟ \\ + 2 {ỽ_{1}}^{2} (\frac{1}{2}) {ỽ_{2}}^{2} (\frac{1}{2}) Ϗ_{γ} (ⱱ, ⱳ, p, q) \int_{0}^{1} t^{α - 1} ԟ^{β - 1} [ỽ_{1} (t) ỽ_{1} (ԟ) [ỽ_{2} (1 - t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ)] + ỽ_{1} (t) ỽ_{1} (1 - ԟ) [ỽ_{2} (1 - t) ỽ_{2} (1 - ԟ) + ỽ_{2} (t) ỽ_{2} (ԟ) + ỽ_{2} (t) ỽ_{2} (1 - ԟ)]] d t d ԟ, \end{array}

Since

γ \in [0, 1]

, after simplification, we reached the required conclusion. □

In Remark 4, some classical results have been obtained. It is clear that the results can be obtained using Theorem 5 instead of using other classical results.

Remark 4.

If one assumes that

ỽ (t) = t, ỽ (ԟ) = ԟ

α = 1

and

β = 1

, then using (68), as a result, there will be inequity [55]:

\begin{array}{l} 4 \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \otimes \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \supseteq_{F} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) \otimes \tilde{G} (x, y) d y d x \oplus \frac{5}{36} \tilde{E} (ⱱ, ⱳ, p, q) \\ \oplus \frac{7}{36} [\tilde{Ѵ} (ⱱ, ⱳ, p, q) \tilde{+} \tilde{Ѡ} (ⱱ, ⱳ, p, q)] \oplus \frac{2}{9} \tilde{Ϗ} (ⱱ, ⱳ, p, q) . \end{array}

(70)

\tilde{G}

is coordinated left-

U D

ỽ

-convex with

ỽ (t) = t, ỽ (ԟ) = ԟ

and one assumes that

α = 1

and

β = 1

, then using (68), as a result, there will be inequity [46]:

\begin{array}{l} 4 \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \otimes \tilde{G} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq_{F} \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} \tilde{G} (x, y) \otimes \tilde{G} (x, y) d y d x \oplus \frac{5}{36} \tilde{E} (ⱱ, ⱳ, p, q) \\ \oplus \frac{7}{36} [\tilde{Ѵ} (ⱱ, ⱳ, p, q) \oplus \tilde{Ѡ} (ⱱ, ⱳ, p, q)] \oplus \frac{2}{9} \tilde{Ϗ} (ⱱ, ⱳ, p, q) . \end{array}

(71)

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

ỽ (t) = t, ỽ (ԟ) = ԟ

and

γ = 1

, then using (68), we succeed in bringing about the upcoming inequity [63]:

\begin{array}{l} 4 G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \times G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \supseteq \frac{1}{(ⱳ - ⱱ) (q - p)} \int_{ⱱ}^{ⱳ} \int_{p}^{q} G (x, y) \times G (x, y) d y d x + \frac{5}{36} E (ⱱ, ⱳ, p, q) \\ + \frac{7}{36} [Ѵ (ⱱ, ⱳ, p, q) + Ѡ (ⱱ, ⱳ, p, q)] + \frac{2}{9} Ϗ (ⱱ, ⱳ, p, q) . \end{array}

(72)

G_{*} ((x, y), γ) \neq G^{*} ((x, y), γ)

with

γ = 1

and

ỽ (t) = t, ỽ (ԟ) = ԟ

, then using (68), we succeed in bringing about the upcoming inequity [64]:

\begin{array}{l} 4 G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \times G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \supseteq \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [\begin{matrix} I_{ⱱ^{+}, p^{+}}^{α, β} G (ⱳ, q) \times G (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G (ⱳ, p) \times G (ⱳ, p) \\ + I_{ⱳ^{-}, p^{+}}^{α, β} G (ⱱ, q) \times G (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G (ⱱ, p) \times G (ⱱ, p) \end{matrix}] \\ + [\frac{α}{2 (α + 1) (α + 2)} + \frac{β}{(β + 1) (β + 2)} (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)})] E (ⱱ, ⱳ, p, q) \\ + [\frac{1}{2} (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)}) + \frac{α}{(α + 1) (α + 2)} \frac{β}{(β + 1) (β + 2)}] Ѵ (ⱱ, ⱳ, p, q) \\ + [\frac{1}{2} (\frac{1}{2} - \frac{β}{(β + 1) (β + 2)}) + \frac{α}{(α + 1) (α + 2)} \frac{β}{(β + 1) (β + 2)}] Ѡ (ⱱ, ⱳ, p, q) \\ + [\frac{1}{4} - \frac{α}{(α + 1) (α + 2)} \frac{β}{(β + 1) (β + 2)}] Ϗ (ⱱ, ⱳ, p, q) . \end{array}

(73)

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

and

G_{*} ((x, y), γ) = G^{*} ((x, y), γ)

with

γ = 1

and

ỽ (t) = t, ỽ (ԟ) = ԟ

, then using (68), we succeed in bringing about the upcoming inequity [47]:

\begin{array}{l} 4 G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \times G (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) \\ \leq \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [\begin{matrix} I_{ⱱ^{+}, p^{+}}^{α, β} G (ⱳ, q) \times G (ⱳ, q) + I_{ⱱ^{+}, q^{-}}^{α, β} G (ⱳ, p) \times G (ⱳ, p) \\ + I_{ⱳ^{-}, p^{+}}^{α, β} G (ⱱ, q) \times G (ⱱ, q) + I_{ⱳ^{-}, q^{-}}^{α, β} G (ⱱ, p) \times G (ⱱ, p) \end{matrix}] \\ + [\frac{α}{2 (α + 1) (α + 2)} + \frac{β}{(β + 1) (β + 2)} (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)})] E (ⱱ, ⱳ, p, q) \\ + [\frac{1}{2} (\frac{1}{2} - \frac{α}{(α + 1) (α + 2)}) + \frac{α}{(α + 1) (α + 2)} \frac{β}{(β + 1) (β + 2)}] Ѵ (ⱱ, ⱳ, p, q) \\ + [\frac{1}{2} (\frac{1}{2} - \frac{β}{(β + 1) (β + 2)}) + \frac{α}{(α + 1) (α + 2)} \frac{β}{(β + 1) (β + 2)}] Ѡ (ⱱ, ⱳ, p, q) \\ + [\frac{1}{4} - \frac{α}{(α + 1) (α + 2)} \frac{β}{(β + 1) (β + 2)}] Ϗ (ⱱ, ⱳ, p, q) . \end{array}

(74)

4. Examples

In Example 2, we discuss

ỽ

-concave

F N V M s

over coordinates.

Example 2.

We consider the

F N V M

\tilde{G} : [0, 1] \times [0, 1] \to F_{0}

established by

\tilde{G} (x, y) (σ) = \{\begin{matrix} \frac{σ - (6 - e^{x}) (6 - e^{y})}{(6 - e^{x}) (6 - e^{y}) - 25}, σ \in [(6 - e^{x}) (6 - e^{y}), 25] \\ \frac{35 x y - σ}{35 x y - 25}, σ \in (25, 35 x y] \\ 0, o t h e r w i s e . \end{matrix}

(75)

Then, for each

γ \in [0, 1],

we have

G_{γ} (x, y) = [(1 - γ) (6 - e^{x}) (6 - e^{y}) + 25 γ, 35 (1 - γ) x y + 25 γ]

. The endpoint functions

G_{*} ((x, y), γ),

G^{*} ((x, y), γ)

are coordinated

ỽ

-concave and

ỽ

-convex functions for each

γ \in [0, 1]

. Hence,

\tilde{G} (x, y)

is a coordinated

ỽ

-concave

F N V M

Example 3.

We assume that the

F N V M

\tilde{G} : [0, 2] \times [0, 2] \to F_{0}

is established by

G (x, y) (σ) = \{\begin{matrix} \frac{σ}{(2 - \sqrt{x}) (2 - \sqrt{y})}, σ \in [0, (2 - \sqrt{x}) (2 - \sqrt{y})] \\ \frac{2 (2 - \sqrt{x}) (2 - \sqrt{y}) - σ}{(2 - \sqrt{x}) (2 - \sqrt{y})}, σ \in ((2 - \sqrt{x}) (2 - \sqrt{y}), 2 (2 - \sqrt{x}) (2 - \sqrt{y})] \\ 0, o t h e r w i s e, \end{matrix}

(76)

Then, for each

γ \in [0, 1],

we have

G_{γ} (x, y) = [γ (2 - \sqrt{x}) (2 - \sqrt{y}), (2 - γ) (2 - \sqrt{x}) (2 - \sqrt{y})]

. The end point functions

G_{*} ((x, y), γ),

G^{*} ((x, y), γ)

are coordinated

ỽ

-convex functions for each

γ \in [0, 1]

, respectively. Hence,

\tilde{G} (x, y)

is a

ỽ

-coordinated convex

F N V M

\begin{array}{l} G_{γ} (\frac{ⱱ + ⱳ}{2}, \frac{p + q}{2}) = [γ, (2 - γ)] \\ \frac{Γ (α + 1)}{{4 (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, \frac{p + q}{2}) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, \frac{p + q}{2})] \oplus \frac{Γ (β + 1)}{{4 (q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, q) \oplus I_{q^{-}}^{β} \tilde{G} (\frac{ⱱ + ⱳ}{2}, p)] \\ = [γ (2 - \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{8} π), (2 - γ) (2 - \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{8} π)] \\ \frac{Γ (α + 1) Γ (β + 1)}{{4 (ⱳ - ⱱ)}^{α} {(q - p)}^{β}} [I_{ⱱ^{+}, p^{+}}^{α, β} G_{γ} (ⱳ, q) \oplus I_{ⱱ^{+}, q^{-}}^{α, β} G_{γ} (ⱳ, p) \oplus I_{ⱳ^{-}, p^{+}}^{α, β} G_{γ} (ⱱ, q) \oplus I_{ⱳ^{-}, q^{-}}^{α, β} G_{γ} (ⱱ, p)] \\ = [γ (\frac{33}{8} - \sqrt{2} - \frac{\sqrt{2}}{2} π + \frac{π}{8} + \frac{π^{2}}{32}), (2 - γ) (\frac{33}{8} - \sqrt{2} - \frac{\sqrt{2}}{2} π + \frac{π}{8} + \frac{π^{2}}{32})] \\ \frac{Γ (α + 1)}{{8 (ⱳ - ⱱ)}^{α}} [I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, p) \oplus I_{ⱱ^{+}}^{α} \tilde{G} (ⱳ, q) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, p) \oplus I_{ⱳ^{-}}^{α} \tilde{G} (ⱱ, q)] \\ \oplus \frac{Γ (β + 1)}{{8 (q - p)}^{β}} [I_{p^{+}}^{β} \tilde{G} (ⱱ, q) \oplus I_{p^{+}}^{β} \tilde{G} (ⱳ, q) \oplus I_{q^{-}}^{β} \tilde{G} (ⱱ, p) \oplus I_{q^{-}}^{β} \tilde{G} (ⱳ, p)] \\ = [\frac{34 \sqrt{2} + (\sqrt{2} - 4) π - 24}{8 \sqrt{2}} γ, \frac{34 \sqrt{2} + (\sqrt{2} - 4) π - 24}{8 \sqrt{2}} (2 - γ)] \\ \frac{G_{γ} (p, ⱳ) + G_{γ} (σ, ⱳ) + G_{γ} (p, q) + G_{γ} (σ, q)}{4} = [γ (\frac{9}{2} - 2 \sqrt{2}), (2 - γ) (\frac{9}{2} - 2 \sqrt{2})] . \end{array}

That is

\begin{array}{l} [(1 - γ) + 4 γ, 9 (1 - γ) + 4 γ] \leq_{I} [(1 - γ) (2 - \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{8} π) + 4 γ, (1 - γ) (2 + \frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{8} π) + 4 γ] \\ \leq_{I} [(1 - γ) (\frac{33}{8} - \sqrt{2} - \frac{\sqrt{2}}{2} π + \frac{π}{8} + \frac{π^{2}}{32}) + 4 γ, (1 - γ) (\frac{33}{8} + \sqrt{2} + \frac{\sqrt{2}}{2} π + \frac{π}{8} + \frac{π^{2}}{32}) + 4 γ] \\ \leq_{I} [\frac{34 \sqrt{2} + (\sqrt{2} - 4) π - 24}{8 \sqrt{2}} (1 - γ) + 4 γ, \frac{34 \sqrt{2} + (\sqrt{2} + 4) π + 24}{8 \sqrt{2}} (1 - γ) + 4 γ] \\ \leq_{I} \frac{34 \sqrt{2} + (\sqrt{2} - 4) π - 24}{8 \sqrt{2}} (1 - γ) + 4 γ . \end{array}

Hence, Theorem 4 has been verified.

5. Conclusions and Future Plans

Different fields examine the theory of inequality from different angles. The establishment of novel, cohesive and improved versions of already established outcomes is one of the rigorous viewpoints. We present a new expanded notion of convex mappings in the current proceedings and analyze some implications of the proposed notion; the new class is known as coordinated

ỽ

-convex

F N V M s

. Additionally, we investigate the containments of the HH and Pachpatte types, effectively putting the recently developed concept of coordinated

ỽ

-convex convexity into practice along with fractional integral operators. The findings’ dependability is confirmed by simulations and instances. This class has the benefit of combining multiple other recognized convexity classes into one. As a result, the results obtained from this class likewise coincide with classical results. In future, this class will be extended to non-convex classes, and their applications will be found via various fractional integral operators by using the well-known “Condition C”.

Author Contributions

Conceptualization, Z.Z.; validation, Z.Z. and A.A.A.A.; formal analysis, A.A.L. and K.H.H.; investigation, A.A.L. and K.H.H.; resources, writing—original draft, Z.Z.; writing—review and editing, Z.Z. and A.A.A.A.; Visualization, Z.Z., A.A.A.A., A.A.L. and K.H.H.; supervision, Z.Z. and A.A.A.A.; project administration, Z.Z., A.A.A.A., A.A.L. and K.H.H. All authors have read and agreed to the published version of the manuscript.

Funding

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-56).

Data Availability Statement

There is no data availability statement to be declared.

Conflicts of Interest

The authors claim to have no conflicts of interest.

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Zhou, Z.; Al Ahmadi, A.A.; Lupas, A.A.; Hakami, K.H. A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain. Axioms 2024, 13, 666. https://doi.org/10.3390/axioms13100666

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Zhou Z, Al Ahmadi AA, Lupas AA, Hakami KH. A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain. Axioms. 2024; 13(10):666. https://doi.org/10.3390/axioms13100666

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Zhou, Zizhao, Ahmad Aziz Al Ahmadi, Alina Alb Lupas, and Khalil Hadi Hakami. 2024. "A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain" Axioms 13, no. 10: 666. https://doi.org/10.3390/axioms13100666

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A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Examples

5. Conclusions and Future Plans

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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