Open AccessArticle

Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications

Miguel Vivas Cortez

^1,†

Ali Althobaiti

^2,*,†

Abdulrahman F. Aljohani

^3,†

and

Saad Althobaiti

^4,†

Escuela de Ciencias F’ısicas y Matemáticas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador

Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 47512, Saudi Arabia

⁴

Department of Sciences and Technology, Ranyah University Collage, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2024, 13(7), 471; https://doi.org/10.3390/axioms13070471

Submission received: 20 May 2024 / Revised: 15 June 2024 / Accepted: 21 June 2024 / Published: 12 July 2024

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

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Abstract

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down (

U

D

) relations and over newly defined class

U

D

-ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of

U

D

-relations, recent findings have been developed using fuzzy Aumman’s, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields.

Keywords:

U·D-ħ–Godunova–Levin convex fuzzy-number mappings; Ostrowski’s inequality; fuzzy Hermite-Hadamard type inequalities

MSC:

26A33; 26A51; 26D10

1. Introduction

In recent years, numerous scholars in analysis and various branches of mathematics have shown a growing interest in inequality theory [1,2]. Many real-world problems can be viewed as integral equations, emphasizing the importance of generalizing integral inequalities to address such issues [3].

In Moore’s renowned book, the introductory chapter provides an interactive exploration of numerical data, serving as an initiation to interval analysis in numerical analysis (refer to [4]). Over the last five decades, numerous applications have emerged across various domains, including interval differential equations, computer graphics, aero elasticity, and optimization of neural networks. Recently, several authors have extensively investigated various integral inequalities within the context of interval-valued functions (refer to [5,6,7]).

It is widely acknowledged that the convexity of functions plays a pivotal role in numerous scientific disciplines, encompassing probability theory, economics, and optimal control theory. Moreover, various inequalities have been extensively documented in the literature (refer to [8,9]). The subsequent inequality is commonly known as the classical Hermite-Hadamard inequality regarding equality:

f : K \to N

is a convex function defined on the interval

K

of real numbers, and

θ, λ \in K

with

θ < λ

, then

f (\frac{θ + λ}{2}) \leq \frac{1}{λ - θ} \int_{θ}^{λ} f (б) d б \leq \frac{f (θ) + f (λ)}{2} .

(1)

Both inequalities are valid in the opposite direction if f exhibits concavity.

Varosanec initially introduced the concept of ℎ-convexity in 2007 (refer to [10]), exploring various generalizations and extensions of this inequality (refer to [11]). Several authors have subsequently developed more intricate Hermite-Hadamard inequalities involving ℎ-convex functions (refer to [12,13]). Moreover, Costa proposed a Jensen-type inequality for fuzzy interval-valued functions (refer to [14]). In the realm of interval-valued functions, Zhao et al. presented a novel Hermite-Hadamard inequality for ℎ-convex functions (refer to [15]).

Using the ℎ-Godunova–Levin function (refer to [16,17]), Almutairi and Kiliman demonstrated the following inequality in 2019.

f : K \to N

is a convex function defined on the interval

K

of real numbers, and

θ, λ \in K

with

θ < λ

, then

f (κ θ + (1 - κ) λ) \leq \frac{f (θ)}{ħ (κ)} + \frac{f (λ)}{ħ (1 - κ)},

(2)

where,

ħ : [0, 1] \subseteq K \to N^{+}

such that

ħ ≢ 0

Several mathematicians have expanded upon the Ostrowski inequality in various directions. Notably, several scientific articles have delved into this topic, exploring different forms of convexity. For instance, İşcan et al. [18] investigated the concept of a harmonically s-convex function. Set [19] introduced the fractional version of the Ostrowski-type inequality using Riemann–Liouville fractional operators. Liu [20] utilized the equality established by Set to devise new refinements of the Ostrowski-type inequality for an MT-convex function. Tunç [21] examined the Ostrowski-type inequality for an ℏ-convex function. Ozdemir et al. [22] derived a fresh version of the Ostrowski-type inequality for an (α, m)-convex function. Agarwal et al. [23] explored a more generalized Ostrowski-type inequality using a Raina fractional integral operator. Sarikaya et al. [24,25] employed local fractional integrals to establish new generalizations of the Ostrowski-type inequality. Gürbuz et al. [26,27] utilized a Katugampola fractional operator for a generalized version of the Ostrowski inequality. Ahmad et al. [28,29] introduced some innovative generalizations of the Ostrowski inequality via an Atangana–Baleanu fractional operator for differentiable convex functions and for harmonical convexity, see [30,31,32,33]. For further details on recent advancements in the Ostrowski-type inequality, readers are referred to the following references (refer to [34,35,36]). Budak et al. [37] derived innovative fractional inequalities of the Ostrowski type for interval-valued functions, drawing on the definitions of gH-derivatives. Basic concepts related to fuzzy and fuzzy Aumman’s integral are in the following literature (see [38,39] and the references therein). Nanda [40] introduced the concept of convexity in fuzzy environment. For interval-valued convex mapping, see [41]. Khan et al. [42] introduced log-h-convex fuzzy-interval-valued functions as a distinct class of convex fuzzy-interval-valued functions, employing a fuzzy order relation. This class facilitated the establishment of Jensen and Hermite-Hadamard inequalities (see [43,44,45,46,47] and the references therein).

Naturally, numerous researchers have extensively explored and examined Ostrowski’s and Hermite-Hadamard inequalities in a novel context via newly defined class of

U

D

ħ

–Godunova–Levin convex fuzzy-number mappings. Consequently, several extensions and improvements have been developed. For example, refer to [15,39,40,47] and the references therein. In this investigation, we propose some further adjustments to Fejér, Pachpatte, and Ostrowski’s integral inequalities via

U

D

ħ

–Godunova–Levin convexity and fuzzy Aumman’s integrals.

2. Preliminaries

Consider

E_{C}

as the set comprising all closed and bounded intervals of

N

, and let

V

belong to

E_{C}

, defined as:

V = [V_{*}, V^{*}] = \{б \in N | V_{*} \leq б \leq V^{*}\}, (V_{*}, V^{*} \in N)

(3)

It is named a positive interval

[V_{*}, V^{*}]

V_{*} \geq 0

. The Definition of

E_{C}^{+}

, which represents the set of all positive intervals, is

E_{C}^{+} = \{[V_{*}, V^{*}] : [V_{*}, V^{*}] \in E_{C} and V_{*} \geq 0\} .

(4)

Let

ı \in N

and

ı \cdot V

be defined by

ı \cdot V = \{\begin{matrix} [ı V_{*}, {ı V}^{*}] i f ı > 0, \\ \{0\} i f ı = 0, \\ [ı V^{*} {, ı V}_{*}] i f ı < 0 . \end{matrix}

(5)

Subsequently, the Minkowski difference

Ϗ - V

, addition

V + Ϗ

, and multiplication

V \times Ϗ

for

V, Ϗ

belong to

E_{C}

are delineated as follows:

[Ϗ_{*}, Ϗ^{*}] + [V_{*}, V^{*}] = [Ϗ_{*} + V_{*}, Ϗ^{*} + V^{*}],

(6)

[Ϗ_{*}, Ϗ^{*}] \times [V_{*}, V^{*}] = [m i n \{Ϗ_{*} V_{*}, Ϗ^{*} V_{*}, Ϗ_{*} V^{*}, Ϗ^{*} V^{*}\}, m a x \{Ϗ_{*} V_{*}, Ϗ^{*} V_{*}, Ϗ_{*} V^{*}, Ϗ^{*} V^{*}\}],

(7)

[Ϗ_{*}, Ϗ^{*}] - [V_{*}, V^{*}] = [Ϗ_{*} - V^{*}, Ϗ^{*} - V_{*}] .

(8)

Remark 1.

(i) For given

[Ϗ_{*}, Ϗ^{*}], [V_{*}, V^{*}] \in E_{C},

the relation

“ \supseteq_{I} ”

defined on

E_{C}

[V_{*}, V^{*}] \supseteq_{I} [Ϗ_{*}, Ϗ^{*}]

if and only if

V_{*} \leq Ϗ_{*}, Ϗ^{*} {\leq V}^{*}

for all

[Ϗ_{*}, Ϗ^{*}], [V_{*}, V^{*}] \in E_{C}

is a partial interval inclusion relation. The relation

[V_{*}, V^{*}] \supseteq_{I} [Ϗ_{*}, Ϗ^{*}]

is coincident to

[V_{*}, V^{*}] \supseteq [Ϗ_{*}, Ϗ^{*}]

E_{C} .

It can be easily seen that “

\supseteq_{I}

” looks like “up and down” on the real line

N,

so we call

“ \supseteq_{I} ”

“up and down” (or “

U D

” order, in short) [44]. For

[Ϗ_{*}, Ϗ^{*}], [V_{*}, V^{*}] \in E_{C},

the Hausdorff–Pompeiu distance between intervals

[Ϗ_{*}, Ϗ^{*}]

and

[V_{*}, V^{*}]

is defined by

d_{H} ([Ϗ_{*}, Ϗ^{*}], [V_{*}, V^{*}]) = m a x \{|Ϗ_{*} - V_{*}|, |Ϗ^{*} - V^{*}|\} .

(9)

It is a familiar fact that

(E_{C}, d_{H})

is a complete metric space [37,38,39].

We will briefly review some essential concepts regarding fuzzy sets and fuzzy numbers since we will rely on the standard definitions of these sets.

Please note that we refer to

F

and

F_{0}

as the set of all fuzzy subsets and fuzzy numbers of

N

Given

\tilde{Ϗ} \in F_{0}

, the level sets or cut sets are given by

{[\tilde{Ϗ}]}^{ı} = \{б \in N | \tilde{Ϗ} (б) > ı\}

for all

ı \in [0, 1]

and by

{[\tilde{Ϗ}]}^{0} = \{б \in N | \tilde{Ϗ} (б) > 0\}

. These sets are known as

ı

-level sets or

ı

-cut sets of

\tilde{Ϗ}

, see [37].

Proposition 1 ([44]).

Let

\tilde{Ϗ}, \tilde{V} \in F_{0}

. Then, relation

“ \leq_{F} ”

is given on

F_{0}

\tilde{Ϗ} \leq_{F} \tilde{V}

when and only when

{{[\tilde{Ϗ}]}^{ı} \leq}_{I} {[\tilde{V}]}^{ı}

, for every

ı \in [0, 1],

which are left- and right-order relations.

Proposition 2 ([44]).

Let

\tilde{Ϗ}, \tilde{V} \in F_{0}

. Then, relation

“ \supseteq_{F} ”

is given on

F_{0}

\tilde{Ϗ} \supseteq_{F} \tilde{V}

when and only when

{[\tilde{Ϗ}]}^{ı} \supseteq_{I} {[\tilde{V}]}^{ı}

for every

ı \in [0, 1],

which is the

U D -

order relation on

F_{0}

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

\tilde{Ϗ}, \tilde{V} \in F_{0}

and

ı \in N

, then, for every

ı \in [0, 1],

the arithmetic operations addition “

\oplus ”

, multiplication “

\otimes ”

, and scaler multiplication “

⊙ ”

are defined by

{[\tilde{Ϗ} \oplus \tilde{V}]}^{ı} = {[\tilde{Ϗ}]}^{ı} + {[\tilde{V}]}^{ı},

(10)

{[\tilde{Ϗ} \otimes \tilde{V}]}^{ı} = {[\tilde{Ϗ}]}^{ı} \times {[\tilde{V}]}^{ı},

(11)

{[t ⊙ \tilde{Ϗ}]}^{ı} = t {[\tilde{Ϗ}]}^{ı},

(12)

over

[θ, λ]

Aumann Integrals for Interval and Fuzzy Number Mappings

Now we define and discuss some properties of Aumann integrals for interval and

F

N

M

Definition 1 ([37]).

f : [θ, λ] \subset N \to E_{C}

is an interval-valued mapping (

I - V - M

) satisfying that

f (б) = [f_{*} (б), f^{*} (б)]

, then

f

is an Aumann integrable over

[θ, λ]

when and only when

f_{*} (б)

and

f^{*} (б)

both are lebesgue integrable over

[θ, λ],

such that

(I A) \int_{θ}^{λ} f (б) d б = [\int_{θ}^{\begin{matrix} λ \end{matrix}} f_{*} (б) d б, \int_{θ}^{λ} f^{*} (б) d б] .

(13)

The literature suggests the following conclusions, see [37,38,47]:

Definition 2 ([44]).

A fuzzy-interval-valued map

\tilde{f} : Λ \subset N \to F_{0}

is named

F

N

V

M

. For each

ı \in (0, 1],

its

I - V - M

s are classified according to their

ı

-levels

f_{ı} : Λ \subset N \to E_{C}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

for all

б \in Λ .

Here, for each

ı \in (0, 1],

the end point real mappings

f_{*} (., ı), f^{*} (., ı) : Λ \to N

are called lower and upper mappings of

\tilde{f} (б)

Definition 3.

Let

\tilde{f} : [θ, λ] \subset N \to F_{0}

be an

F

N

V

M

. Then, fuzzy integral of

\tilde{f}

over

[θ, λ],

denoted by

(F A) \int_{θ}^{λ} \tilde{f} (б) d б

, is given level-wise by

{[(F A) \int_{θ}^{λ} \tilde{f} (б) d б]}^{\begin{matrix} ı \end{matrix}} = (I A) \int_{θ}^{λ} f_{ı} (б) d б = \{\int_{θ}^{λ} f (б, ı) d б : f (б, ı) \in R_{([θ, λ], ı)}\},

(14)

for all

ı \in (0, 1],

where

R_{([θ, λ], ı)}

denotes the collection of Riemannian integrable mappings of

I - V - M

s. The

F

N

V

M

\tilde{f}

F A

-integrable over

[θ, λ]

(F A) \int_{θ}^{λ} \tilde{f} (б) d б \in F_{0} .

Note that, if

f_{*} (б, ı), f^{*} (б, ı)

are Lebesgue-integrable, then

f

is fuzzy Aumann-integrable mapping over

[θ, λ]

, see [44].

Theorem 1 ([39]).

Let

\tilde{f} : [θ, λ] \subset N \to F_{0}

be an

F

N

V

M

, it’s

I - V - M

s are classified according to their

ı

-levels

f_{ı} : [θ, λ] \subset N \to E_{C}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in (0, 1] .

Then,

\tilde{f}

F A

-integrable over

[θ, λ]

if and only if,

f_{*} (б, ı)

and

f^{*} (б, ı)

are both

A

-integrable over

[θ, λ]

. Moreover, if

\tilde{f}

F A

-integrable over

[θ, λ],

then

{[(F A) \int_{θ}^{λ} \tilde{f} (б) d б]}^{ı} = [(A) \int_{θ}^{λ} f_{*} (б, ı) d б, (A) \int_{θ}^{λ} f^{*} (б, ı) d б] = (I A) \int_{θ}^{λ} f_{ı} (б) d б,

(15)

for all

ı \in (0, 1] .

For all

ı \in (0, 1],

{F R}_{([θ, λ], ı)}

denotes the collection of all

F A

-integrable

F

N

V

M

The family of all

(F A)

-integrable

F

N

M

s over

[θ, λ]

are denoted by

{F A}_{([θ, λ], ı)} .

Breckner discussed the coming emerging idea of interval-valued convexity in [41].

I

V

M

f : I = [θ, λ] \to E_{I}

is called convex

I

V

M

f (κ б + (1 - κ) s) \supseteq κ f (б) + (1 - κ) f (s),

(16)

for all

б, y \in [θ, λ], κ \in [0, 1]

, where

E_{I}

is the collection of all real valued intervals. If (16) is reversed, then

f

is called concave.

Definition 4 ([40]).

The

F

N

M

f : [θ, λ] \to F_{0}

is called convex

F

N

M

[θ, λ]

\tilde{f} (κ б + (1 - κ) s) \leq_{F} κ ⊙ \tilde{f} (б) \oplus (1 - κ) ⊙ \tilde{f} (s),

(17)

for all

б, s \in [θ, λ], κ \in [0, 1],

where

\tilde{f} (б) \geq_{F} \tilde{0}

for all

б \in [θ, λ] .

If (17) is reversed then,

\tilde{f}

is called concave

F

N

M

[θ, λ]

\tilde{f}

is affine if and only if it is both convex and concave

F

N

M

3. Hermite-Hadamard Inequalities over $U$ · $D$ - $ħ$ -Godunova–Levin Convex $F$ · $N$ · $M$

In this section, we start with the main Definition of

U

D

ħ

-Godunova–Levin convexity over fuzzy domain that will be helpful for the upcoming results. The fuzzy valued Hermite-Hadamard inequalities for

U

D

ħ

-Godunova–Levin convex

F

N

M

s are established in this section. Additionally, several instances are provided to support the theory produced in this study’s application.

Definition 5.

Let

K

be convex set and

ħ : [0, 1] \subseteq K \to N^{+}

such that

ħ ≢ 0

. Then

F

N

M

\tilde{f} : K \to F_{0}

is said to be

U

D

ħ

-Godunova–Levin convex on

K

\tilde{f} (κ б + (1 - κ) s) \supseteq_{F} \frac{\tilde{f} (б)}{ħ (κ)} \oplus \frac{\tilde{f} (s)}{ħ (1 - κ)},

(18)

for all

б, s \in K, κ \in [0, 1],

where

\tilde{f} (б) \geq_{F} \tilde{0} .

The

F

N

M

\tilde{f} : K \to F_{0}

is said to be

U

D

ħ

-Godunova–Levin concave on

K

if inequality (21) is reversed. Moreover,

\tilde{f}

is known as

U

D

ħ

-Godunova–Levin affine

F

N

M

K

\tilde{f} (κ б + (1 - κ) s) = \frac{\tilde{f} (б)}{ħ (κ)} \oplus \frac{\tilde{f} (s)}{ħ (1 - κ)},

(19)

for all

б, s \in K, κ \in [0, 1],

where

\tilde{f} (б) \geq_{F} \tilde{0} .

Remark 2.

The

U

D

ħ

-Godunova–Levin convex

F

N

M

s have some very nice properties similar to convex

F

N

M

(1): if $\tilde{f}$ is $U$ · $D$ - $ħ$ -Godunova–Levin convex $F$ · $N$ · $M$ , then $α \tilde{f}$ is also $U$ · $D$ - $ħ$ -Godunova–Levin convex for $α \geq 0$ .
(2): if $\tilde{f}$ and $\tilde{T}$ both are $U$ · $D$ - $ħ$ -Godunova–Levin convex $F$ · $N$ · $M$ s, then $m a x (\tilde{f} (б), \tilde{T} (б))$ is also $U$ · $D$ - $ħ$ -Godunova–Levin convex $F$ · $N$ · $M$ .

Here, we will go through a few unique exceptional cases of

U

D

ħ

-Godunova–Levin convex

F

N

M

(i): If $ħ (κ) = κ^{s},$ then $U$ · $D$ - $ħ$ -Godunova–Levin convex $F$ · $N$ · $M$ becomes $U$ · $D$ - $s$ -Godunova–Levin convex $F$ · $N$ · $M$ , that is

$\tilde{f} (κ б + (1 - κ) s) \supseteq_{F} \frac{\tilde{f} (б)}{κ^{s}} \oplus \frac{\tilde{f} (s)}{{(1 - κ)}^{s}}, \forall б, s \in K, κ \in [0, 1] .$

(20)
(ii): If $ħ (κ) = κ,$ then $U$ · $D$ - $ħ$ -Godunova–Levin convex $F$ · $N$ · $M$ becomes $U$ · $D$ -Godunova–Levin convex $F$ · $N$ · $M$ , see [46], that is

$\tilde{f} (κ б + (1 - κ) s) \supseteq_{F} \frac{\tilde{f} (б)}{κ} \oplus \frac{\tilde{f} (s)}{1 - κ}, \forall б, s \in K, κ \in [0, 1] .$

(21)
(iii): If $ħ (κ) \equiv 1,$ then $U$ · $D$ - $ħ$ -Godunova–Levin convex $F$ · $N$ · $M$ becomes $U$ · $D$ -Godunova–Levin $P$ $F$ · $N$ · $M$ , that is

$\tilde{f} (κ б + (1 - κ) s) \supseteq_{F} \tilde{f} (б) \oplus \tilde{f} (s), \forall б, s \in K, κ \in [0, 1] .$

(22)

Note that, there are also new special cases (i) and (iii) as well.

Theorem 2.

Let

K

be convex set, non-negative real valued function

ħ : [0, 1] \subseteq K \to N

such that

ħ ≢ 0

and let

\tilde{f} : K \to F_{0}

be a

F

N

M

, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : K \subset N \to E_{I}^{+} \subset E_{I}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)],

(23)

for all

б \in K

and for all

ı \in [0, 1]

. Then

\tilde{f}

U

D

ħ

-Godunova–Levin convex on

K,

if and only if, for all

ı \in [0, 1],

f_{*} (б, ı)

ħ

-Godunova–Levin convex and

f^{*} (б, ı)

ħ

-Godunova–Levin concave.

Proof.

Assume that for each

ı \in [0, 1],

f_{*} (б, ı)

and

f^{*} (б, ı)

are

ħ

-Godunova–Levin convex and

ħ

-Godunova–Levin concave on

K

, respectively. Then, we have

f_{*} (κ б + (1 - κ) s, ı) \leq \frac{f_{*} (б, ı)}{ħ (κ)} + \frac{f_{*} (s, ı)}{ħ (1 - κ)}, \forall б, s \in K, κ \in [0, 1],

and

f^{*} (κ б + (1 - κ) s, ı) \geq \frac{f^{*} (б, ı)}{ħ (κ)} + \frac{f^{*} (s, ı)}{ħ (1 - κ)}, \forall б, s \in K, κ \in [0, 1] .

Then by (18), (5) and (6), we obtain

f_{ı} (κ б + (1 - κ) s) = [f_{*} (κ б + (1 - κ) s, ı), f^{*} (κ б + (1 - κ) s, ı)],

\supseteq_{I} [\frac{f_{*} (s, ı)}{ħ (κ)}, \frac{f^{*} (s, ı)}{ħ (κ)}] + [\frac{f_{*} (s, ı)}{ħ (1 - κ)}, \frac{f^{*} (s, ı)}{ħ (1 - κ)}],

that is

\tilde{f} (κ б + (1 - κ) s) \supseteq_{F} \frac{\tilde{f} (б)}{ħ (κ)} \oplus \frac{\tilde{f} (s)}{ħ (1 - κ)}, \forall б, s \in K, κ \in [0, 1] .

Hence,

\tilde{f}

U

D

ħ

-Godunova–Levin convex

F

N

M

K

Conversely, let

\tilde{f}

U

D

ħ

-Godunova–Levin convex

F

N

M

K .

Then, for all

б, s \in K

and

κ \in [0, 1],

we have

\tilde{f} (κ б + (1 - κ) s) \supseteq_{F} \frac{\tilde{f} (б)}{ħ (κ)} \oplus \frac{\tilde{f} (s)}{ħ (1 - κ)} .

Therefore, from (23), we have

f_{ı} (κ б + (1 - κ) s) = [f_{*} (κ б + (1 - κ) s, ı), f^{*} (κ б + (1 - κ) s, ı)] .

Again, from (18), (5) and (6), we obtain

\frac{f_{ı} (б)}{ħ (κ)} + \frac{f_{ı} (б)}{ħ (1 - κ)} = [\frac{f_{*} (s, ı)}{ħ (κ)}, \frac{f^{*} (s, ı)}{ħ (κ)}] + [\frac{f_{*} (s, ı)}{ħ (1 - κ)}, \frac{f^{*} (s, ı)}{ħ (1 - κ)}],

for all

б, s \in K

and

κ \in [0, 1] .

Then by

U

D

ħ

-Godunova–Levin convexity of

\tilde{f}

, we have for all

б, s \in K

and

κ \in [0, 1]

such that

f_{*} (κ б + (1 - κ) s, ı) \leq \frac{f_{*} (б, ı)}{ħ (κ)} + \frac{f_{*} (s, ı)}{ħ (1 - κ)},

and

f^{*} (κ б + (1 - κ) s, ı) \geq \frac{f^{*} (б, ı)}{ħ (κ)} + \frac{f^{*} (s, ı)}{ħ (1 - κ)},

for each

ı \in [0, 1] .

Hence, the result follows. □

Remark 3.

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1,

then

U

D

ħ

-Godunova–Levin convex

F

N

M

reduces to the

U

D

ħ

-Godunova–Levin convex function.

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

and

ħ (κ) = κ^{s}

with

s \in (0, 1)

, then

U

D

ħ

-Godunova–Levin convex

F

N

M

reduces to the

s

-Godunova–Levin convex function.

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

and

ħ (κ) = κ

, then

U

D

ħ

-Godunova–Levin convex

F

N

M

reduces to the

ħ

-Godunova–Levin convex function.

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

and

ħ (κ) = 1

, then

U

D

ħ

-Godunova–Levin convex

F

N

M

reduces to the

P

-convex function.

Example 1.

We consider

ħ (κ) = κ,

for

κ \in [0, 1]

and the

F

N

M

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

defined by

\tilde{f} (б) (σ) = \{\begin{matrix} \frac{σ}{{2 б}^{2}} σ \in [0, {2 б}^{2}] \\ \frac{{4 б}^{2} - σ}{{2 б}^{2}} σ \in ({2 б}^{2}, {4 б}^{2}] \\ 0 otherwise, \end{matrix}

(24)

Then, for each

ı \in [0, 1],

we have

f_{ı} (б) = [2 ı б^{2}, (4 - 2 ı) б^{2}]

. Since end point functions

f_{*} (б, ı),

f^{*} (б, ı)

are

ħ

-Godunova–Levin convex and

ħ

-Godunova–Levin concave functions for each

ı \in [0, 1]

, respectively. Hence

\tilde{f} (б)

U

D

ħ

-Godunova–Levin convex

F

N

M

Definition 6.

Let

\tilde{f} : [θ, λ] \to F_{0}

be a

F

N

M

, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : [θ, λ] \to E_{C}^{+} \subset E_{C}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)],

for all

ϰ \in [θ, λ]

and for all

ı \in [0, 1]

. Then,

\tilde{f}

is lower

ħ

-Godunova–Levin convex (upper

ħ

-Godunova–Levin concave)

F

N

M

o n

[θ, λ],

if and only if, for all

ı \in [0, 1],

f_{*} (ϰ, ı)

is a

ħ

-Godunova–Levin convex (

ħ

-Godunova–Levin concave) mapping and

f^{*} (ϰ, ı)

is a

ħ

-Godunova–Levin affine mapping.

Definition 7.

Let

\tilde{f} : [θ, λ] \to F_{0}

be a

F

N

M

, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : [θ, λ] \to E_{C}^{+} \subset E_{C}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)],

for all

ϰ \in [θ, λ]

and for all

ı \in [0, 1]

. Then,

\tilde{f}

is an upper

ħ

-Godunova–Levin convex (

ħ

-Godunova–Levin concave)

F

N

M

[θ, λ],

if and only if, for all

ı \in [0, 1],

f_{*} (ϰ, ı)

is an

ħ

-Godunova–Levin affine mapping and

f^{*} (ϰ, ı)

is a

ħ

-Godunova–Levin convex (

ħ

-Godunova–Levin concave) mapping.

Remark 4.

ħ (κ) = κ

, then both concepts “

U

D

ħ

-Godunova–Levin convex

F

N

M

” and “

ħ

-Godunova–Levin convex

F

N

M

”, are behave alike when

f

is lower

U

D

ħ

-Godunova–Levin convex

F

N

M

Both concepts “

U

D

ħ

-Godunova–Levin convex fuzzy number mapping”, and “

ħ

-Godunova–Levinconvex interval-valued mapping” are coincident when

f

is lower

ħ

-Godunova–Levin convex

F

N

M

with

ı = 1

The following result discuss the Hermite-Hadamard inequality over

U

D

ħ

-Godunova–Levin convex

F

N

M

Theorem 3.

Let

\tilde{f} : [θ, λ] \to F_{0}

be a

U

D

ħ

-Godunova–Levin convex

F

N

M

with non-negative real valued function

ħ : [0, 1] \to N^{+}

and

ħ (\frac{1}{2}) \neq 0

, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : [θ, λ] \subset N \to E_{I}^{+}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in [0, 1]

. If

\tilde{f} \in {F A}_{([θ, λ], ı)}

, then

\frac{ħ (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

(25)

\tilde{f}

ħ

-Godunova–Levin concave

F

N

M

, then (25) is reversed.

\frac{ħ (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \subseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \subseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

(26)

Proof.

Let

\tilde{f} : [θ, λ] \to F_{0}

be a

U

D

ħ

-Godunova–Levin convex

F

N

M

. Then, for

a, b \in [θ, λ]

, we have

\tilde{f} (κ a + (1 - κ) b) \supseteq_{F} \frac{\tilde{f} (a)}{ħ (κ)} \oplus \frac{\tilde{f} (b)}{ħ (1 - κ)}

κ = \frac{1}{2}

, then we have

\tilde{f} (\frac{a + b}{2}) \supseteq_{F} \frac{\tilde{f} (a)}{ħ (\frac{1}{2})} \oplus \frac{\tilde{f} (b)}{ħ (\frac{1}{2})} .

Let

a = κ θ + (1 - κ) λ

and

b = (1 - κ) θ + κ λ

. Then, above inequality we have

ħ (\frac{1}{2}) ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \tilde{f} (κ θ + (1 - κ) λ) \oplus \tilde{f} ((1 - κ) θ + κ λ) .

Therefore, for every

ı \in [0, 1]

, we have

\begin{matrix} ħ (\frac{1}{2}) f_{*} (\frac{θ + λ}{2}, ı) \leq f_{*} (κ θ + (1 - κ) λ, ı) + f_{*} ((1 - κ) θ + κ λ, ı), \\ ħ (\frac{1}{2}) f^{*} (\frac{θ + λ}{2}, ı) \geq f^{*} (κ θ + (1 - κ) λ, ı) + f^{*} ((1 - κ) θ + κ λ, ı) . \end{matrix}

Then

\begin{matrix} ħ (\frac{1}{2}) \int_{0}^{1} f_{*} (\frac{θ + λ}{2}, ı) d κ \leq \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) d κ + \int_{0}^{1} f_{*} ((1 - κ) θ + κ λ, ı) d κ, \\ ħ (\frac{1}{2}) \int_{0}^{1} f^{*} (\frac{θ + λ}{2}, ı) d κ \geq \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) d κ + \int_{0}^{1} f^{*} ((1 - κ) θ + κ λ, ı) d κ . \end{matrix}

It follows that

\begin{matrix} ħ (\frac{1}{2}) f_{*} (\frac{θ + λ}{2}, ı) \leq \frac{2}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) d б, \\ ħ (\frac{1}{2}) f^{*} (\frac{θ + λ}{2}, ı) \geq \frac{2}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) d б . \end{matrix}

That is

ħ (\frac{1}{2}) [f_{*} (\frac{θ + λ}{2}, ı), f^{*} (\frac{θ + λ}{2}, ı)] \supseteq_{I} \frac{2}{λ - θ} [\int_{θ}^{λ} f_{*} (б, ı) d б, \int_{θ}^{λ} f^{*} (б, ı) d б] .

Thus,

\frac{ħ (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б .

(27)

In a similar way as above, we have

\frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

(28)

Combining (27) and (28), we have

\frac{ħ (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

Hence, the required result.

Note that, by using same steps, the Formula (26) can be proved with the help of

ħ

-Godunova–Levin concave

F

N

M

. □

Remark 5.

ħ (κ) = \frac{1}{κ^{s}}

, then Theorem 3 simplifies to the outcome for

U

D

s

-convex

F

N

M

which is also new one:

2^{s - 1} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} \frac{1}{s + 1} ⊙ [\tilde{f} (θ) \oplus \tilde{f} (λ)] .

(29)

ħ (κ) = \frac{1}{κ}

, then Theorem 3 simplifies to the outcome for

U

D

-convex

F

N

M

which is also new one:

\tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} \frac{\tilde{f} (θ) \oplus \tilde{f} (λ)}{2} .

(30)

ħ (κ) \equiv 1

then Theorem 3 simplifies to the outcome for

U

D

P

F

N

M

which is also new one:

\frac{1}{2} \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} \tilde{f} (θ) \oplus \tilde{f} (λ) .

(31)

f_{*} (б, ı) \neq f^{*} (б, ı)

with

ı = 1

and

ħ (κ) = \frac{1}{κ}

, then Theorem 3 simplifies to the outcome for classical convex function, see [15]:

f (\frac{θ + λ}{2}) \supseteq \frac{1}{λ - θ} (I A) \int_{θ}^{λ} f (б) d б \supseteq \frac{f (θ) + f (λ)}{2} .

(32)

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

, then Theorem 3 simplifies to the outcome for classical

ħ

-convex function, see [46]:

\frac{ħ (\frac{1}{2})}{2} f (\frac{θ + λ}{2}) \leq \frac{1}{λ - θ} \int_{θ}^{λ} f (б) d б \leq [f (θ) + f (λ)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

(33)

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

and

ħ (κ) = \frac{1}{κ^{s}}

, then Theorem 3 simplifies to the outcome for classical

s

-convex function, see [46]:

2^{s - 1} f (\frac{θ + λ}{2}) \leq \frac{1}{λ - θ} \int_{θ}^{λ} f (б) d б \leq \frac{1}{s + 1} [f (θ) + f (λ)] .

(34)

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

and

ħ (κ) = \frac{1}{κ}

, then Theorem 3 simplifies to the outcome for classical convex function:

f (\frac{θ + λ}{2}) \leq \frac{1}{λ - θ} \int_{θ}^{λ} f (б) d б \leq \frac{f (θ) + f (λ)}{2} .

(35)

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

and

ħ (κ) \equiv 1

, then Theorem 3 simplifies to the outcome for classical

P

-convex function:

\frac{1}{2} f (\frac{θ + λ}{2}) \leq \frac{1}{λ - θ} \int_{θ}^{λ} f (б) d б \leq f (θ) + f (λ) .

(36)

Example 2.

We consider

ħ (κ) = \frac{1}{κ},

for

κ \in [0, 1]

, and the

F

N

M

\tilde{f} : [θ, λ] = [2, 3] \to F_{0}

defined by,

\tilde{f} (б) (θ) = \{\begin{matrix} \frac{θ - 2 + б^{\frac{1}{2}}}{1 - б^{\frac{1}{2}}} θ \in [2 - б^{\frac{1}{2}}, 3] \\ \frac{2 + б^{\frac{1}{2}} - θ}{б^{\frac{1}{2}} - 1} θ \in (3, 2 + б^{\frac{1}{2}}] \\ 0 otherwise . \end{matrix}

(37)

Then, for each

ı \in [0, 1],

we have

f_{ı} (б) = [(1 - ı) (2 - б^{\frac{1}{2}}) + 3 ı, (1 - ı) (2 + б^{\frac{1}{2}}) + 3 ı]

. Since left and right end point mappings

f_{*} (б, ı) = (1 - ı) (2 - б^{\frac{1}{2}}) + 3 ı,

f^{*} (б, ı) = (1 - ı) (2 + б^{\frac{1}{2}}) + 3 ı

, are

U

D

ħ

-Godunova–Levin convex mappings for each

ı \in [0, 1]

, then

\tilde{f} (б)

ħ

-Godunova–Levin convex

F

N

M

. We clearly see that

\tilde{f} \in L ([θ, λ], F_{0})

. Now computing the following

\frac{ħ (\frac{1}{2})}{2} f_{*} (\frac{θ + λ}{2}, ı) \leq \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) d б \leq [f_{*} (θ, ı) + f_{*} (λ, ı)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

\frac{ħ (\frac{1}{2})}{2} f_{*} (\frac{θ + λ}{2}, ı) = f_{*} (\frac{5}{2}, ı) = \frac{4 - \sqrt{10}}{2} (1 - ı) + 3 ı,

\frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) d б = \int_{2}^{3} ((1 - ı) (2 - б^{\frac{1}{2}}) + 3 ı) d б = \frac{(6 + 4 \sqrt{2} - 6 \sqrt{3})}{3} (1 - ı) + 3 ı,

[f_{*} (θ, ı) + f_{*} (λ, ı)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ = \frac{(4 - \sqrt{2} - \sqrt{3})}{2} (1 - ı) + 3 ı,

for all

ı \in [0, 1] .

That means

\frac{4 - \sqrt{10}}{2} (1 - ı) + 3 ı \leq \frac{(6 + 4 \sqrt{2} - 6 \sqrt{3})}{3} (1 - ı) + 3 ı \leq \frac{(4 - \sqrt{2} - \sqrt{3})}{2} (1 - ı) + 3 ı .

Similarly, it can be easily show that

\frac{ħ (\frac{1}{2})}{2} f^{*} (\frac{θ + λ}{2}, ı) \geq \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) d б \geq [f^{*} (θ, ı) + f^{*} (λ, ı)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

for all

ı \in [0, 1],

such that

\frac{ħ (\frac{1}{2})}{2} f^{*} (\frac{θ + λ}{2}, ı) = f^{*} (\frac{5}{2}, ı) = \frac{4 + \sqrt{10}}{2} (1 - ı) + 3 ı,

\frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) d б = \int_{2}^{3} ((1 - ı) (2 + б^{\frac{1}{2}}) + 3 ı) d б = \frac{(6 - 4 \sqrt{2} + 6 \sqrt{3})}{3} (1 - ı) + 3 ı,

[f^{*} (θ, ı) + f^{*} (λ, ı)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ = \frac{(4 + \sqrt{2} + \sqrt{3})}{2} (1 - ı) + 3 ı .

From which, we have

\frac{4 + \sqrt{10}}{2} (1 - ı) + 3 ı \geq \frac{(6 - 4 \sqrt{2} + 6 \sqrt{3})}{3} (1 - ı) + 3 ı \geq \frac{(4 + \sqrt{2} + \sqrt{3})}{2} (1 - ı) + 3 ı,

that is

[\frac{4 - \sqrt{10}}{2} (1 - ı) + 3 ı, \frac{4 + \sqrt{10}}{2} (1 - ı) + 3 ı] \supseteq_{I} [\frac{(6 + 4 \sqrt{2} - 6 \sqrt{3})}{3} (1 - ı) + 3 ı, \frac{(6 - 4 \sqrt{2} + 6 \sqrt{3})}{3} (1 - ı) + 3 ı]

\supseteq_{I} [\frac{(4 - \sqrt{2} - \sqrt{3})}{2} (1 - ı), \frac{(4 + \sqrt{2} + \sqrt{3})}{2} (1 - ı) + 3 ı]

for all

ı \in [0, 1] .

Hence,

\frac{ħ (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ \int_{0}^{1} \frac{1}{ħ (κ)} d κ .

Theorem 4.

Let

\tilde{f} : [θ, λ] \to F_{0}

be a

U

D

ħ

-Godunova–Levin convex

F

N

M

with non-negative real valued function

ħ : [0, 1] \to N^{+}

and

ħ (\frac{1}{2}) \neq 0,

it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : [θ, λ] \subset N \to E_{I}^{+}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in [0, 1]

. If

\tilde{f} \in {F A}_{([θ, λ], ı)}

, then

\frac{{[ħ (\frac{1}{2})]}^{2}}{4} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} ⪧_{2} \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} ⪧_{1} \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} ħ (κ) d κ,

(38)

\tilde{f}

is a

ħ

-Godunova–Levin concave

F

N

M

, then (38) is reversed.

\frac{{[ħ (\frac{1}{2})]}^{2}}{4} ⊙ \tilde{f} (\frac{θ + λ}{2}) \subseteq_{F} ⪧_{2} \subseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \subseteq_{F} ⪧_{1} \subseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} ħ (κ) d κ,

(39)

where

⪧_{1} = [\frac{\tilde{f} (θ) \oplus \tilde{f} (λ)}{2} \oplus \tilde{f} (\frac{θ + λ}{2})] ⊙ \int_{0}^{1} \frac{1}{ħ (κ)} d κ,

⪧_{2} = \frac{[ħ (\frac{1}{2})]}{4} ⊙ [\tilde{f} (\frac{3 θ + λ}{4}) \oplus \tilde{f} (\frac{θ + 3 λ}{4})],

and

⪧_{1} = [{⪧_{1}}_{*}, {⪧_{1}}^{*}]

⪧_{2} = [{⪧_{2}}_{*}, {⪧_{2}}^{*}] .

Proof.

Take

[θ, \frac{θ + λ}{2}],

we have

ħ (\frac{1}{2}) ⊙ \tilde{f} (\frac{κ θ + (1 - κ) \frac{θ + λ}{2}}{2} + \frac{(1 - κ) θ + κ \frac{θ + λ}{2}}{2}) \supseteq_{F} \tilde{f} (κ θ + (1 - κ) \frac{θ + λ}{2}) \oplus \tilde{f} ((1 - κ) θ + κ \frac{θ + λ}{2}) .

Therefore, for every

ı \in [0, 1]

, we have

\begin{matrix} ħ (\frac{1}{2}) f_{*} (\frac{κ θ + (1 - κ) \frac{θ + λ}{2}}{2} + \frac{(1 - κ) θ + κ \frac{θ + λ}{2}}{2}, ı) \\ \leq f_{*} (κ θ + (1 - κ) \frac{θ + λ}{2}, ı) + f_{*} ((1 - κ) θ + κ \frac{θ + λ}{2}, ı), \\ ħ (\frac{1}{2}) f^{*} (\frac{κ θ + (1 - κ) \frac{θ + λ}{2}}{2} + \frac{(1 - κ) θ + κ \frac{θ + λ}{2}}{2}, ı) \\ \geq f^{*} (κ θ + (1 - κ) \frac{θ + λ}{2}, ı) + f^{*} ((1 - κ) θ + κ \frac{θ + λ}{2}, ı) . \end{matrix}

In consequence, we obtain

\begin{matrix} \frac{ħ (\frac{1}{2})}{4} f_{*} (\frac{3 θ + λ}{4}, ı) \leq \frac{1}{λ - θ} \int_{θ}^{\frac{θ + λ}{2}} f_{*} (б, ı) d б, \\ \frac{ħ (\frac{1}{2})}{4} f^{*} (\frac{3 θ + λ}{4}, ı) \geq \frac{1}{λ - θ} \int_{θ}^{\frac{θ + λ}{2}} f^{*} (б, ı) d б . \end{matrix}

That is

\frac{ħ (\frac{1}{2})}{4} [f_{*} (\frac{3 θ + λ}{4}, ı), f^{*} (\frac{3 θ + λ}{4}, ı)] \supseteq_{I} \frac{1}{λ - θ} [\int_{θ}^{\frac{θ + λ}{2}} f_{*} (б, ı) d б, \int_{θ}^{\frac{θ + λ}{2}} f^{*} (б, ı) d б] .

It follows that

\frac{ħ (\frac{1}{2})}{4} ⊙ \tilde{f} (\frac{3 θ + λ}{4}) \supseteq_{F} \frac{1}{λ - θ} ⊙ \int_{θ}^{\frac{θ + λ}{2}} \tilde{f} (б) d б .

(40)

In a similar way as above, we have

\frac{ħ (\frac{1}{2})}{4} ⊙ \tilde{f} (\frac{θ + 3 λ}{4}) \supseteq_{F} \frac{1}{λ - θ} ⊙ \int_{\frac{θ + λ}{2}}^{λ} \tilde{f} (б) d б .

(41)

Combining (40) and (41), we have

\frac{ħ (\frac{1}{2})}{4} ⊙ [\tilde{f} (\frac{3 θ + λ}{4}) \oplus \tilde{f} (\frac{θ + 3 λ}{4})] \supseteq_{F} \frac{1}{λ - θ} ⊙ \int_{θ}^{λ} \tilde{f} (б) d б .

By using Theorem 3, we have

\frac{ħ (\frac{1}{2})}{4} ⊙ \tilde{f} (\frac{θ + λ}{2}) = \frac{ħ (\frac{1}{2})}{4} ⊙ \tilde{f} (\frac{1}{2} . \frac{3 θ + λ}{4} + \frac{1}{2} . \frac{θ + 3 λ}{4}) .

Therefore, for every

ı \in [0, 1]

, we have

\begin{matrix} \frac{{[ħ (\frac{1}{2})]}^{2}}{4} f_{*} (\frac{θ + λ}{2}, ı) = \frac{{[ħ (\frac{1}{2})]}^{2}}{4} f_{*} (\frac{1}{2} . \frac{3 θ + λ}{4} + \frac{1}{2} . \frac{θ + 3 λ}{4}, ı), \\ \frac{{[ħ (\frac{1}{2})]}^{2}}{4} f^{*} (\frac{θ + λ}{2}, ı) = \frac{{[ħ (\frac{1}{2})]}^{2}}{4} f^{*} (\frac{1}{2} . \frac{3 θ + λ}{4} + \frac{1}{2} . \frac{θ + 3 λ}{4}, ı), \end{matrix}

\begin{matrix} \leq \frac{{[ħ (\frac{1}{2})]}^{2}}{4} [\frac{f_{*} (\frac{3 θ + λ}{4}, ı)}{ħ (\frac{1}{2})} + \frac{f_{*} (\frac{θ + 3 λ}{4}, ı)}{ħ (\frac{1}{2})}], \\ \geq \frac{{[ħ (\frac{1}{2})]}^{2}}{4} [\frac{f^{*} (\frac{3 θ + λ}{4}, ı)}{ħ (\frac{1}{2})} + \frac{f^{*} (\frac{θ + 3 λ}{4}, ı)}{ħ (\frac{1}{2})}], \end{matrix}

\begin{matrix} = {⪧_{2}}_{*}, \\ = {⪧_{2}}^{*}, \end{matrix}

\begin{matrix} \leq \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) d б, \\ \geq \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) d б, \end{matrix}

\begin{matrix} \leq [\frac{f_{*} (θ, ı) + f_{*} (λ, ı)}{2} + f_{*} (\frac{θ + λ}{2}, ı)] \int_{0}^{1} ħ (κ) d κ, \\ \geq [\frac{f^{*} (θ, ı) + f^{*} (λ, ı)}{2} + f^{*} (\frac{θ + λ}{2}, ı)] \int_{0}^{1} ħ (κ) d κ, \end{matrix}

\begin{matrix} = {⪧_{1}}_{*}, \\ = {⪧_{1}}^{*}, \end{matrix}

\begin{matrix} \leq [\frac{f_{*} (θ, ı) + f_{*} (λ, ı)}{2} + \frac{1}{ħ (\frac{1}{2})} (f_{*} (θ, ı) + f_{*} (λ, ı))] \int_{0}^{1} \frac{1}{ħ (κ)} d κ, \\ \geq [\frac{f^{*} (θ, ı) + f^{*} (λ, ı)}{2} + \frac{1}{ħ (\frac{1}{2})} (f^{*} (θ, ı) + f^{*} (λ, ı))] \int_{0}^{1} \frac{1}{ħ (κ)} d κ, \end{matrix}

\begin{matrix} = [f_{*} (θ, ı) + f_{*} (λ, ı)] [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} \frac{1}{ħ (κ)} d κ, \\ = [f^{*} (θ, ı) + f^{*} (λ, ı)] [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} \frac{1}{ħ (κ)} d κ, \end{matrix}

that is

\frac{[ħ (\frac{1}{2})]}{4} ⊙ \tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} ⪧_{2} \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) d б \supseteq_{F} ⪧_{1} \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} \frac{1}{ħ (κ)} d κ,

hence, the result follows. □

Example 3.

We consider

ħ (κ) = κ,

for

κ \in [0, 1]

, and the

F

N

M

\tilde{f} : [θ, λ] = [2, 3] \to F_{0}

defined by,

f_{ı} (б) = [(1 - ı) (2 - б^{\frac{1}{2}}) + 3 ı, (1 - ı) (2 + б^{\frac{1}{2}}) + 3 ı],

as in Example 2, then

\tilde{f} (б)

U

D

ħ

-Godunova–Levin convex

F

N

M

. We have

f_{*} (б, ı) = (1 - ı) (2 - б^{\frac{1}{2}}) + 3 ı

and

f^{*} (б, ı) = (1 - ı) (2 + б^{\frac{1}{2}}) + 3 ı

. We now compute the following:

\begin{matrix} \frac{{[ħ (\frac{1}{2})]}^{2}}{4} f_{*} (\frac{θ + λ}{2}, ı) = f_{*} (\frac{5}{2}, ı) = \frac{4 - \sqrt{10}}{2} (1 - ı) + 3 ı, \\ \frac{{[ħ (\frac{1}{2})]}^{2}}{4} f^{*} (\frac{θ + λ}{2}, ı) = f^{*} (\frac{5}{2}, ı) = \frac{4 + \sqrt{10}}{2} (1 - ı) + 3 ı, \end{matrix}

\begin{matrix} {⪧_{2}}_{*} = \frac{ħ (\frac{1}{2})}{4} [f_{*} (\frac{3 θ + λ}{4}, ı) + f_{*} (\frac{θ + 3 λ}{4}, ı)] = \frac{5 - \sqrt{11}}{4} (1 - ı) + 3 ı, \\ {⪧_{2}}^{*} = \frac{ħ (\frac{1}{2})}{4} [f^{*} (\frac{3 θ + λ}{4}, ı) + f^{*} (\frac{θ + 3 λ}{4}, ı)] = \frac{7 + \sqrt{11}}{4} (1 - ı) + 3 ı . \end{matrix}

\begin{matrix} {⪧_{1}}_{*} = [\frac{f_{*} (θ, ı) + f_{*} (λ, ı)}{2} + f_{*} (\frac{θ + λ}{2}, ı)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ = \frac{(8 - \sqrt{2} - \sqrt{3} - \sqrt{10})}{4} (1 - ı) + 3 ı, \\ {⪧_{1}}^{*} = [\frac{f^{*} (θ, ı) + f^{*} (λ, ı)}{2} + f^{*} (\frac{θ + λ}{2}, ı)] \int_{0}^{1} \frac{1}{ħ (κ)} d κ = \frac{(8 + \sqrt{2} + \sqrt{3} + \sqrt{10})}{4} (1 - ı) + 3 ı, \end{matrix}

\begin{matrix} [f_{*} (θ, ı) + f_{*} (λ, ı)] [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} \frac{1}{ħ (κ)} d κ = \frac{(4 - \sqrt{2} - \sqrt{3})}{2} (1 - ı) + 3 ı, \\ [f^{*} (θ, ı) + f^{*} (λ, ı)] [\frac{1}{2} + \frac{1}{ħ (\frac{1}{2})}] \int_{0}^{1} \frac{1}{ħ (κ)} d κ = \frac{(4 + \sqrt{2} + \sqrt{3})}{2} (1 - ı) + 3 ı . \end{matrix}

Then we obtain that

\begin{matrix} (1 - ı) \frac{4 - \sqrt{10}}{2} + 3 ı \leq \frac{5 - \sqrt{11}}{4} (1 - ı) + 3 ı \leq \frac{(6 + 4 \sqrt{2} - 6 \sqrt{3})}{3} (1 - ı) + 3 ı \\ \leq \frac{(8 - \sqrt{2} - \sqrt{3} - \sqrt{10})}{4} (1 - ı) + 3 ı \leq \frac{(1 - ı) (4 - \sqrt{2} - \sqrt{3})}{2} + 3 ı \\ (1 - ı) \frac{4 + \sqrt{10}}{2} + 3 ı \geq \frac{7 + \sqrt{11}}{4} (1 - ı) + 3 ı \geq \frac{(6 - 4 \sqrt{2} + 6 \sqrt{3})}{3} (1 - ı) + 3 ı \\ \geq \frac{(8 + \sqrt{2} + \sqrt{3} + \sqrt{10})}{4} (1 - ı) + 3 ı \geq \frac{(1 - ı) (4 + \sqrt{2} + \sqrt{3})}{2} + 3 ı . \end{matrix}

Hence, Theorem 4 is verified.

The novel fuzzy Hermite-Hadamard inequalities for the product of two

U

D

ħ

-Godunova–Levin convex

F

N

M

s are found in the results.

Theorem 5.

Let

\tilde{f}, \tilde{J} : [θ, λ] \to F_{0}

be two

U

D

ħ

-Godunova–Levin convex

F

N

M

s with non-negative real valued functions

ħ_{1}, ħ_{2} : [0, 1] \to N^{+}

and

ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2}) \neq 0,

it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı}, J_{ı} : [θ, λ] \subset N \to E_{I}^{+}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

and

J_{ı} (б) = [J_{*} (б, ı), J^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in [0, 1]

. If

\tilde{f}, \tilde{J}

and

\tilde{f} \otimes \tilde{J} \in {F A}_{([θ, λ], ı)}

, then

\frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) \otimes \tilde{J} (б) d б \supseteq_{F} \tilde{M} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \oplus \tilde{N} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ,

(42)

\tilde{f}

ħ

-Godunova–Levin concave

F

N

M

, then (42) is reversed.

\frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) \otimes \tilde{J} (б) d б \subseteq_{F} \tilde{M} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \oplus \tilde{N} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ,

(43)

where

\tilde{M} (θ, λ) = \tilde{f} (θ) \otimes \tilde{J} (θ) \oplus \tilde{f} (λ) \otimes \tilde{J} (λ),

\tilde{N} (θ, λ) = \tilde{f} (θ) \otimes \tilde{J} (λ) \oplus \tilde{f} (λ) \otimes \tilde{J} (θ),

and

M_{ı} (θ, λ) = [M_{*} ((θ, λ), ı), M^{*} ((θ, λ), ı)]

and

N_{ı} (θ, λ) = [N_{*} ((θ, λ), ı), N^{*} ((θ, λ), ı)] .

Proof.

Let

\tilde{f}, \tilde{J} : [θ, λ] \to F_{0}

be two

U

D

ħ_{1}

-Godunova–Levin convex and

U

D

ħ_{2}

-Godunova–Levin convex

F

N

M

s. Then, we have

f_{*} (κ б + (1 - κ) s, ı) \leq \frac{f_{*} (б, ı)}{ħ (κ)} + \frac{f_{*} (s, ı)}{ħ (1 - κ)},

and

f^{*} (κ б + (1 - κ) s, ı) \geq \frac{f^{*} (б, ı)}{ħ (κ)} + \frac{f^{*} (s, ı)}{ħ (1 - κ)},

\begin{matrix} f_{*} (κ θ + (1 - κ) λ, ı) \leq \frac{f_{*} (б, ı)}{ħ_{1} (κ)} + \frac{f_{*} (s, ı)}{ħ_{1} (1 - κ)} \\ f^{*} (κ θ + (1 - κ) λ, ı) \geq \frac{f^{*} (б, ı)}{ħ_{1} (κ)} + \frac{f^{*} (s, ı)}{ħ_{1} (1 - κ)}, \end{matrix}

and

\begin{matrix} J_{*} (κ θ + (1 - κ) λ, ı) \leq \frac{J_{*} (б, ı)}{ħ_{2} (κ)} + \frac{J_{*} (s, ı)}{ħ_{2} (1 - κ)}, \\ J^{*} (κ θ + (1 - κ) λ, ı) \geq \frac{J^{*} (б, ı)}{ħ_{2} (κ)} + \frac{J^{*} (s, ı)}{ħ_{2} (1 - κ)} . \end{matrix}

From the Definition of

U

D

ħ

-Godunova–Levin convex

F

N

M

it follows that

\tilde{f} (x) {\begin{matrix} \begin{matrix} \geq \end{matrix} \end{matrix}}_{F} 0

and

\tilde{J} (x) {\begin{matrix} \begin{matrix} \geq \end{matrix} \end{matrix}}_{F} 0

, so

\begin{matrix} f_{*} (κ θ + (1 - κ) λ, ı) J_{*} (κ θ + (1 - κ) λ, ı) \\ \leq (\frac{f_{*} (б, ı)}{ħ_{1} (κ)} + \frac{f_{*} (s, ı)}{ħ_{1} (1 - κ)}) (\frac{J_{*} (б, ı)}{ħ_{2} (κ)} + \frac{J_{*} (s, ı)}{ħ_{2} (1 - κ)}) \\ = f_{*} (θ, ı) J_{*} (θ, ı) [\frac{1}{ħ_{1} (κ) ħ_{2} (κ)}] + f_{*} (λ, ı) J_{*} (λ, ı) [\frac{1}{ħ_{1} (1 - κ) ħ_{2} (1 - κ)}] \\ + f_{*} (θ, ı) J_{*} (λ, ı) [\frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)}] + f_{*} (λ, ı) J_{*} (θ, ı) [\frac{1}{ħ_{1} (1 - κ) ħ_{2} (κ)}], \\ f^{*} (κ θ + (1 - κ) λ, ı) J^{*} (κ θ + (1 - κ) λ, ı) \\ \geq (\frac{f^{*} (б, ı)}{ħ_{1} (κ)} + \frac{f^{*} (s, ı)}{ħ_{1} (1 - κ)}) (\begin{matrix} \frac{J^{*} (б, ı)}{ħ_{2} (κ)} + \frac{J^{*} (s, ı)}{ħ_{2} (1 - κ)} \end{matrix}) \\ = f^{*} (θ, ı) J^{*} (θ, ı) [\frac{1}{ħ_{1} (κ) ħ_{2} (κ)}] + f^{*} (λ, ı) J^{*} (λ, ı) [\frac{1}{ħ_{1} (1 - κ) ħ_{2} (1 - κ)}] \\ + f^{*} (θ, ı) J^{*} (λ, ı) [\frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)}] + f^{*} (λ, ı) J^{*} (θ, ı) [\frac{1}{ħ_{1} (1 - κ) ħ_{2} (κ)}] . \end{matrix}

Integrating both sides of above inequality over

[0, 1]

we get

\begin{matrix} \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) J_{*} (κ θ + (1 - κ) λ, ı) = \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (x, ı) J_{*} (x, ı) d x \\ \leq (f_{*} (θ, ı) J_{*} (θ, ı) + f_{*} (λ, ı) J_{*} (λ, ı)) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \\ + (f_{*} (θ, ı) J_{*} (λ, ı) + f_{*} (λ, ı) J_{*} (θ, ı)) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ, \\ \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) J^{*} (κ θ + (1 - κ) λ, ı) = \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (x, ı) J^{*} (x, ı) d x \\ \geq (f^{*} (θ, ı) J^{*} (θ, ı) + f^{*} (λ, ı) J^{*} (λ, ı)) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \\ + (f^{*} (θ, ı) J^{*} (λ, ı) + f^{*} (λ, ı) J^{*} (θ, ı)) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ . \end{matrix}

It follows that,

\begin{matrix} \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (x, ı) J_{*} (x, ı) d x \leq M_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \\ + N_{*} ((θ, λ)), ı \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ, \\ \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (x, ı) J^{*} (x, ı) d x \geq M^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \\ + N^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ, \end{matrix}

that is

\frac{1}{λ - θ} [\int_{θ}^{λ} f_{*} (x, ı) J_{*} (x, ı) d x, \int_{θ}^{λ} f^{*} (x, ı) J^{*} (x, ı) d x] \supseteq_{I} [M_{*} ((θ, λ), ı), M^{*} ((θ, λ), ı)] \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ + [N_{*} ((θ, λ), ı), N^{*} ((θ, λ), ı)] \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ .

Thus,

\frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (x) \otimes \tilde{J} (x) d x \supseteq_{F} \tilde{M} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ \oplus \tilde{N} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ,

and Theorem 5 has been established. □

Example 4.

We consider

ħ_{1} (κ) = \frac{1}{κ}, ħ_{2} (κ) = \frac{1}{κ},

for

κ \in [0, 1]

, and the

F

N

M

\tilde{f}, \tilde{J} : [θ, λ] = [0, 1] \to F_{0}

defined by,

\tilde{f} (б) (σ) = \{\begin{matrix} \frac{θ}{б} θ \in [0, б] \\ \frac{2 б - θ}{б} θ \in (б, 2 б] \\ 0 otherwise, \end{matrix}

\tilde{J} (б) (σ) = \{\begin{matrix} \frac{θ - б}{2 - б} θ \in [б, 2] \\ \frac{8 - e^{б} - θ}{8 - e^{б} - 2} θ \in (2, 8 - e^{б}] \\ 0 otherwise . \end{matrix}

Then, for each

ı \in [0, 1],

we have

f_{ı} (б) = [ı б, (2 - ı) б]

and

J_{ı} (б) = [(1 - ı) б + 2 ı, (1 - ı) (8 - e^{б}) + 2 ı] .

Since end point functions

f_{*} (б, ı) = ı б,

f^{*} (б, ı) = (2 - ı) б

and

J_{*} (б, ı) = (1 - ı) б + 2 ı

J^{*} (б, ı) = (1 - ı) (8 - e^{б}) + 2 ı

ħ_{1}, ħ_{2}

-Godunova–Levin convex functions for each

ı \in [0, 1]

. Hence

\tilde{f},

\tilde{J}

both are

U

D

ħ_{1}

U

D

ħ_{2}

-Godunova–Levin convex

F

N

M

s, respectively. We now computing the following

\begin{matrix} \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) \times J_{*} (б, ı) d б = \frac{1}{2} \int_{0}^{2} (ı (1 - ı) б^{2} + 2 ı^{2} б) d б = \frac{2}{3} ı (2 + ı) \\ \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) \times J^{*} (б, ı) d б = \frac{1}{2} \int_{0}^{2} ((1 - ı) (2 - ı) б (8 - e^{б}) + 2 ı (2 - ı) б) d б \approx \frac{(2 - ı)}{2} (\frac{1903}{250} - \frac{903}{250} ı), \end{matrix}

\begin{matrix} M_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ = \frac{4 ı}{3} \\ M^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ = \frac{2 (2 - ı) [(1 - ı) (8 - e^{2}) + 2 ı]}{3}, \end{matrix}

\begin{matrix} N_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ = \frac{2 ı^{2}}{3} \\ N^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ = \frac{(2 - ı) (7 - 5 ı)}{3}, \end{matrix}

for each

ı \in [0, 1],

that means

[\frac{2}{3} ı (1 + 2 ı), \frac{(2 - ı)}{2} (\frac{1903}{250} - \frac{903}{250} ı)] \supseteq_{I} \frac{1}{3} [2 ı (2 + ı), (2 - ı) [2 (1 - ı) (8 - e^{2}) - ı + 7]]

Hence, Theorem 5 is demonstrated.

Theorem 6.

Let

\tilde{f}, \tilde{J} : [θ, λ] \to F_{0}

be two

U

D

ħ_{1}

-Godunova–Levin convex and

U

D

ħ_{2}

- Godunova–Levin convex

F

N

M

s with non-negative real valued functions

ħ_{1}, ħ_{2} : [0, 1] \to N^{+}

, respectively and

ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2}) \neq 0,

respectively, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı}, J_{ı} : [θ, λ] \subset N \to E_{I}^{+}

are given, respectively, by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

and

J_{ı} (б) = [J_{*} (б, ı), J^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in [0, 1]

. If

\tilde{f} \otimes \tilde{J} \in {F A}_{([θ, λ], ı)}

, then

\frac{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \otimes \tilde{J} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) \otimes \tilde{J} (б) d б \oplus \tilde{M} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ \oplus \tilde{N} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ,

(44)

where

\tilde{M} (θ, λ) = \tilde{f} (θ) \otimes \tilde{J} (θ) \oplus \tilde{f} (λ) \otimes \tilde{J} (λ),

\tilde{N} (θ, λ) = \tilde{f} (θ) \otimes \tilde{J} (λ) \oplus \tilde{f} (λ) \otimes \tilde{J} (θ),

and

M_{ı} (θ, λ) = [M_{*} ((θ, λ), ı), M^{*} ((θ, λ), ı)]

and

N_{ı} (θ, λ) = [N_{*} ((θ, λ), ı), N^{*} ((θ, λ), ı)] .

Proof.

By hypothesis, for each

ı \in [0, 1],

we have

\begin{matrix} f_{*} (\frac{θ + λ}{2}, ı) \times J_{*} (\frac{θ + λ}{2}, ı) \\ f^{*} (\frac{θ + λ}{2}, ı) \times J^{*} (\frac{θ + λ}{2}, ı) \end{matrix}

\begin{matrix} \leq \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f_{*} (κ θ + (1 - κ) λ, ı) \times J_{*} (κ θ + (1 - κ) λ, ı) \\ + f_{*} (κ θ + (1 - κ) λ, ı) \times J_{*} ((1 - κ) θ + κ λ, ı) \end{matrix}] \\ + \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f_{*} ((1 - κ) θ + κ λ, ı) \times J_{*} (κ θ + (1 - κ) λ, ı) \\ + f_{*} ((1 - κ) θ + κ λ, ı) \times J_{*} ((1 - κ) θ + κ λ, ı) \end{matrix}], \\ \geq \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f^{*} (κ θ + (1 - κ) λ, ı) \times J^{*} (κ θ + (1 - κ) λ, ı) \\ + f^{*} (κ θ + (1 - κ) λ, ı) \times J^{*} ((1 - κ) θ + κ λ, ı) \end{matrix}] \\ + \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f^{*} ((1 - κ) θ + κ λ, ı) \times J^{*} (κ θ + (1 - κ) λ, ı) \\ + f^{*} ((1 - κ) θ + κ λ, ı) \times J^{*} ((1 - κ) θ + κ λ, ı) \end{matrix}], \end{matrix}

\begin{matrix} \leq \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f_{*} (κ θ + (1 - κ) λ, ı) {\times J}_{*} (κ θ + (1 - κ) λ, ı) \\ + f_{*} ((1 - κ) θ + κ λ, ı) \times J_{*} ((1 - κ) θ + κ λ, ı) \end{matrix}] \\ + \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} (ħ_{1} (κ) f_{*} (θ, ı) + ħ_{1} (1 - κ) f_{*} (λ, ı)) \\ \times (ħ_{2} (1 - κ) J_{*} (θ, ı) + ħ_{2} (κ) J_{*} (λ, ı)) \\ + ({ħ_{1} (1 - κ) f}_{*} (θ, ı) + ħ_{1} (κ) f_{*} (λ, ı)) \\ \times (ħ_{2} (κ) J_{*} (θ, ı) + ħ_{2} (1 - κ) J_{*} (λ, ı)) \end{matrix}], \\ \geq \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f^{*} (κ θ + (1 - κ) λ, ı) \times J^{*} (κ θ + (1 - κ) λ, ı) \\ + f^{*} ((1 - κ) θ + κ λ, ı) \times J^{*} ((1 - κ) θ + κ λ, ı) \end{matrix}] \\ + \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} (ħ_{1} (κ) f^{*} (θ, ı) + ħ_{1} (1 - κ) f^{*} (λ, ı)) \\ \times (ħ_{2} (1 - κ) J^{*} (θ, ı) + ħ_{2} (κ) J^{*} (λ, ı)) \\ + (ħ_{1} (1 - κ) f^{*} (θ, ı) + ħ_{1} (κ) f^{*} (λ, ı)) \\ \times (ħ_{2} (κ) J^{*} (θ, ı) + ħ_{2} (1 - κ) J^{*} (λ, ı)) \end{matrix}], \end{matrix}

\begin{matrix} = \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f_{*} (κ θ + (1 - κ) λ, ı) \times J_{*} (κ θ + (1 - κ) λ, ı) \\ + f_{*} ((1 - κ) θ + κ λ, ı) {\times J}_{*} ((1 - κ) θ + κ λ, ı) \end{matrix}] \\ + \frac{2}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} \{ħ_{1} (κ) ħ_{2} (κ) + ħ_{1} (1 - κ) ħ_{2} (1 - κ)\} N_{*} ((θ, λ), ı) \\ + \{ħ_{1} (κ) ħ_{2} (1 - κ) + ħ_{1} (1 - κ) ħ_{2} (κ)\} M_{*} ((θ, λ), ı) \end{matrix}], \\ = \frac{1}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} f^{*} (κ θ + (1 - κ) λ, ı) \times J^{*} (κ θ + (1 - κ) λ, ı) \\ + f^{*} ((1 - κ) θ + κ λ, ı) \times J^{*} ((1 - κ) θ + κ λ, ı) \end{matrix}] \\ + \frac{2}{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})} [\begin{matrix} \{ħ_{1} (κ) ħ_{2} (κ) + ħ_{1} (1 - κ) ħ_{2} (1 - κ)\} N^{*} ((θ, λ), ı) \\ + \{ħ_{1} (κ) ħ_{2} (1 - κ) + ħ_{1} (1 - κ) ħ_{2} (κ)\} M^{*} ((θ, λ), ı) \end{matrix}], \end{matrix}

Integrating over

[0, 1],

we have

\begin{matrix} \frac{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})}{2} f_{*} (\frac{θ + λ}{2}, ı) \times J_{*} (\frac{θ + λ}{2}, ı) \leq \frac{1}{λ - θ} (R) \int_{θ}^{λ} f_{*} (б, ı) \times J_{*} (б, ı) d б \\ + M_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ \\ + N_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ, \\ \frac{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})}{2} f^{*} (\frac{θ + λ}{2}, ı) \times J^{*} (\frac{θ + λ}{2}, ı) \geq \frac{1}{λ - θ} (R) \int_{θ}^{λ} f^{*} (б, ı) {\times J}^{*} (б, ı) d б \\ + M^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ \\ + N^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ, \end{matrix}

that is

\frac{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})}{2} ⊙ \tilde{f} (\frac{θ + λ}{2}) \otimes \tilde{J} (\frac{θ + λ}{2}) \supseteq_{F} \frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) \otimes \tilde{J} (б) d б

\oplus \tilde{M} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ \oplus \tilde{N} (θ, λ) ⊙ \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ,

Hence, the required result. □

Example 5.

We consider

ħ_{1} (κ) = \frac{1}{κ}, ħ_{2} (κ) = \frac{1}{κ},

for

κ \in [0, 1]

, and the

F

N

M

\tilde{f}, \tilde{J} : [θ, λ] = [0, 1] \to F_{0},

as in Example 4. Then, for each

ı \in [0, 1],

we have

f_{ı} (б) = [ı б, (2 - ı) б]

and

J_{ı} (б) = [(1 - ı) б + 2 ı, (1 - ı) (8 - e^{б}) + 2 ı]

and,

\tilde{f} (б), \tilde{J} (б)

are

U

D

ħ_{1}

-Godunova–Levin convex and

U

D

ħ_{2}

-Godunova–Levin convex

F

N

M

s, respectively. We have

f_{*} (б, ı) = ı б,

f^{*} (б, ı) = (2 - ı) б

and

J_{*} (б, ı) = (1 - ı) б + 2 ı

J^{*} (б, ı) = (1 - ı) (8 - e^{б}) + 2 ı

. We now computing the following

\begin{matrix} \frac{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})}{2} f_{*} (\frac{θ + λ}{2}, ı) \times J_{*} (\frac{θ + λ}{2}, ı) = 2 ı (1 + ı), \\ \frac{ħ_{1} (\frac{1}{2}) ħ_{2} (\frac{1}{2})}{2} f^{*} (\frac{θ + λ}{2}, ı) \times J^{*} (\frac{θ + λ}{2}, ı) = 2 [16 - 20 ı + 6 ı^{2} + (2 - 3 ı + ı^{2}) e] . \end{matrix}

\begin{matrix} \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) \times J_{*} (б, ı) d б = \frac{1}{2} \int_{0}^{2} (ı (1 - ı) б^{2} + 2 ı^{2} б) d б = \frac{4}{3} ı (3 - ı), \\ \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) \times J^{*} (б, ı) d б = \frac{1}{2} \int_{0}^{2} ((1 - ı) (2 - ı) б (8 - e^{б}) + 2 ı (2 - ı) б) d б \\ \approx \frac{(2 - ı)}{2} (\frac{1903}{250} - \frac{903}{250} ı) . \end{matrix}

\begin{matrix} M_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ = \frac{2 ı}{3}, \\ M^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (1 - κ)} d κ = \frac{(2 - ı) [(1 - ı) (8 - e^{2}) + 2 ı]}{3}, \end{matrix}

\begin{matrix} N_{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ = \frac{4 ı^{2}}{3}, \\ N^{*} ((θ, λ), ı) \int_{0}^{1} \frac{1}{ħ_{1} (κ) ħ_{2} (κ)} d κ = \frac{2 (2 - ı) (7 - 5 ı)}{3}, \end{matrix}

for each

ı \in [0, 1],

that means

2 [ı (1 + ı), [16 - 20 ı + 6 ı^{2} + (2 - 3 ı + ı^{2}) e]] \supseteq_{I} [\frac{2}{3} ı (2 + ı), \frac{(2 - ı)}{2} (\frac{1903}{250} - \frac{903}{250} ı)] + \frac{1}{3} [2 ı (1 + 2 ı), (2 - ı) [(1 - ı) (8 - e^{2}) - 8 ı + 14]],

hence, Theorem 6 is demonstrated.

The H-H Fejér inequalities for

U

D

ħ

-Godunova–Levin convex

F

N

M

s are now presented. The second H-H Fejér inequality for

U

D

ħ

-Godunova–Levin convex

F

N

M

is first obtained.

Theorem 7.

Let

\tilde{f} : [θ, λ] \to F_{0}

be an

U

D

ħ

-Godunova–Levin convex

F

N

M

with

ħ : [0, 1] \to N^{+}

, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : [θ, λ] \subset N \to E_{I}^{+}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in [0, 1]

. If

\tilde{f} \in {F R}_{([θ, λ], ı)}

and

B : [θ, λ] \to N, B (б) \geq 0,

symmetric with respect to

\frac{θ + λ}{2},

then

\frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} f (б) ⊙ B (б) d б \supseteq_{F} [f (θ) \oplus f (λ)] ⊙ \int_{0}^{1} \frac{B ((1 - κ) θ + κ λ)}{ħ (κ)} d κ .

(45)

Proof.

Let

\tilde{f}

be an

U

D

ħ

-Godunova–Levin convex

F

N

M

. Then, for each

ı \in [0, 1],

we have

\begin{matrix} f_{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) \\ \leq (\frac{f_{*} (θ, ı)}{ħ (κ)} + \frac{f_{*} (λ, ı)}{ħ (1 - κ)}) B (κ θ + (1 - κ) λ), \\ f^{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) \\ \geq (\frac{f^{*} (θ, ı)}{ħ (κ)} + \frac{f^{*} (λ, ı)}{ħ (1 - κ)}) B (κ θ + (1 - κ) λ) . \end{matrix}

(46)

And

\begin{matrix} f_{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) \\ \leq (\frac{f_{*} (θ, ı)}{ħ (1 - κ)} + \frac{f_{*} (λ, ı)}{ħ (κ)}) B ((1 - κ) θ + κ λ), \\ f^{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) \\ \geq (\frac{f^{*} (θ, ı)}{ħ (1 - κ)} + \frac{f^{*} (λ, ı)}{ħ (κ)}) B ((1 - κ) θ + κ λ) . \end{matrix}

(47)

After adding (46) and (47), and integrating over

[0, 1],

we get

\begin{matrix} \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ + \int_{0}^{1} f_{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ \leq \int_{0}^{1} [\begin{matrix} f_{*} (θ, ı) \{\frac{1}{ħ (κ)} B (κ θ + (1 - κ) λ) + \frac{1}{ħ (1 - κ)} B ((1 - κ) θ + κ λ)\} \\ + f_{*} (λ, ı) \{\frac{1}{ħ (1 - κ)} B (κ θ + (1 - κ) λ) + \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ)\} \end{matrix}] d κ, \\ \int_{0}^{1} f^{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ + \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ \geq \int_{0}^{1} [\begin{matrix} f^{*} (θ, ı) \{\frac{1}{ħ (κ)} B (κ θ + (1 - κ) λ) + \frac{1}{ħ (1 - κ)} B ((1 - κ) θ + κ λ)\} \\ + f^{*} (λ, ı) \{\frac{1}{ħ (1 - κ)} B (κ θ + (1 - κ) λ) + \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ)\} \end{matrix}] d κ, \end{matrix}

\begin{matrix} = 2 f_{*} (θ, ı) \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B (κ θ + (1 - κ) λ) \end{matrix} d κ + 2 f_{*} (λ, ı) \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ, \\ = 2 f^{*} (θ, ı) \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B (κ θ + (1 - κ) λ) \end{matrix} d κ + 2 f^{*} (λ, ı) \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ . \end{matrix}

Since

B

is symmetric, then

\begin{matrix} \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ + \int_{0}^{1} f_{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ \leq 2 [f_{*} (θ, ı) + f_{*} (λ, ı)] \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ, \\ \int_{0}^{1} f^{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ + \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ \geq 2 [f^{*} (θ, ı) + f^{*} (λ, ı)] \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ . \end{matrix}

(48)

Since

\begin{matrix} \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ = \int_{0}^{1} f_{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ = \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) B (б) d б \\ \int_{0}^{1} f^{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ = \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ = \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) B (б) d б \end{matrix}

(49)

Then from (49), (48) we have

\begin{matrix} \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) B (б) d б \leq [f_{*} (θ, ı) + f_{*} (λ, ı)] \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ, \\ \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) B (б) d б \geq [f^{*} (θ, ı) + f^{*} (λ, ı)] \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ, \end{matrix}

that is

[\frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) B (б) d б, \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) B (б) d б]

\supseteq_{I} [f_{*} (θ, ı) + f_{*} (λ, ı), f^{*} (θ, ı) + f^{*} (λ, ı)] \int_{0}^{1} \begin{matrix} \frac{1}{ħ (κ)} B ((1 - κ) θ + κ λ) \end{matrix} d κ

hence

\frac{1}{λ - θ} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) ⊙ B (б) d б \supseteq_{F} [\tilde{f} (θ) \oplus \tilde{f} (λ)] ⊙ \int_{0}^{1} \frac{B ((1 - κ) θ + κ λ)}{ħ (κ)} d κ .

Now, generalizing the first H-H Fejér inequalities for classical Godunova–Levin convex functions and we build the first H-H Fejér inequality for

U

D

ħ

-Godunova–Levin convex

F

N

M

. □

Theorem 8.

Let

\tilde{f} : [θ, λ] \to F_{0}

be an

U

D

ħ

-Godunova–Levin convex

F

N

M

with

ħ : [0, 1] \to N^{+}

, it’s

I - V - M

s are classified according to their

ı

-levels such that,

f_{ı} : [θ, λ] \subset N \to E_{I}^{+}

are given by

f_{ı} (б) = [f_{*} (б, ı), f^{*} (б, ı)]

for all

б \in [θ, λ]

and for all

ı \in [0, 1]

. If

\tilde{f} \in {F R}_{([θ, λ], ı)}

and

B : [θ, λ] \to N, B (б) \geq 0,

symmetric with respect to

\frac{θ + λ}{2},

and

\int_{θ}^{λ} B (б) d б > 0

, then

\tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) ⊙ B (б) d б .

(50)

Proof.

Since

\tilde{f}

is an

U

D

ħ

-Godunova–Levin convex, then for

ı \in [0, 1],

we have

\begin{matrix} f_{*} (\frac{θ + λ}{2}, ı) \leq \frac{1}{ħ (\frac{1}{2})} (f_{*} (κ θ + (1 - κ) λ, ı) + f_{*} ((1 - κ) θ + κ λ, ı)), \\ f^{*} (\frac{θ + λ}{2}, ı) \geq \frac{1}{ħ (\frac{1}{2})} (f^{*} (κ θ + (1 - κ) λ, ı) + f^{*} ((1 - κ) θ + κ λ, ı)), \end{matrix}

(51)

Since

B (κ θ + (1 - κ) λ) = B ((1 - κ) θ + κ λ)

, then by multiplying (51) by

B ((1 - κ) θ + κ λ)

and integrate it with respect to

κ

over

[0, 1],

we obtain

\begin{matrix} f_{*} (\frac{θ + λ}{2}, ı) \int_{0}^{1} B ((1 - κ) θ + κ λ) d κ \\ \leq \frac{1}{ħ (\frac{1}{2})} (\begin{matrix} \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ + \int_{0}^{1} f_{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \end{matrix}), \\ f^{*} (\frac{θ + λ}{2}, ı) \int_{0}^{1} B ((1 - κ) θ + κ λ) d κ \\ \geq \frac{1}{ħ (\frac{1}{2})} (\begin{matrix} \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ + \int_{0}^{1} f^{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \end{matrix}) . \end{matrix}

(52)

Since

\begin{matrix} \int_{0}^{1} f_{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ = \int_{0}^{1} f_{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ = \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (б, ı) B (б) d б \\ \int_{0}^{1} f^{*} ((1 - κ) θ + κ λ, ı) B ((1 - κ) θ + κ λ) d κ \\ = \int_{0}^{1} f^{*} (κ θ + (1 - κ) λ, ı) B (κ θ + (1 - κ) λ) d κ \\ = \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (б, ı) B (б) d б . \end{matrix}

(53)

And

\int_{0}^{1} B ((1 - κ) θ + κ λ) d κ = \frac{1}{λ - θ} \int_{θ}^{λ} B (б) d б

(54)

Then from (53) and (54), (52) we have

\begin{matrix} f_{*} (\frac{θ + λ}{2}, ı) \leq \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} \int_{θ}^{λ} f_{*} (б, ı) B (б) d б, \\ f^{*} (\frac{θ + λ}{2}, ı) \geq \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} \int_{θ}^{λ} f^{*} (б, ı) B (б) d б, \end{matrix}

from which, we have

\begin{matrix} [f_{*} (\frac{θ + λ}{2}, ı), f^{*} (\frac{θ + λ}{2}, ı)] \\ \supseteq_{I} \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} [\int_{θ}^{λ} f_{*} (б, ı) B (б) d б, \int_{θ}^{λ} f^{*} (б, ı) B (б) d б], \end{matrix}

that is

\tilde{f} (\frac{θ + λ}{2}) \supseteq_{F} \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) ⊙ B (б) d б .

This completes the proof. □

Remark 6.

From Theorem 7 and 8, we clearly see that:

B (б) = 1

, then we acquire the inequality (25).

Let

ı = 1

and

ħ (κ) = \frac{1}{κ}

. Then from (45) and (50), we acquire the following inequality, see [45]:

f (\frac{θ + λ}{2}) \supseteq \frac{1}{\int_{θ}^{λ} B (б) d б} (I A) \int_{θ}^{λ} f (б) B (б) d б \supseteq \frac{f (θ) + f (λ)}{2}

(55)

\tilde{f}

is lower Godunova–Levin convex

F

N

M

[θ, λ]

and

ħ (κ) = \frac{1}{κ},

then we derive the following subsequent inequality, see [33]:

\tilde{f} (\frac{θ + λ}{2}) \leq_{F} \frac{1}{\int_{θ}^{λ} B (б) d б} ⊙ (F A) \int_{θ}^{λ} \tilde{f} (б) ⊙ B (б) d б \leq_{F} \frac{\tilde{f} (θ) \oplus \tilde{f} (λ)}{2}

(56)

\tilde{f}

is lower Godunova–Levin convex

F

N

M

[θ, λ]

with

ı = 1

and

ħ (κ) = \frac{1}{κ},

then from (45) and (50) we derive the following subsequent inequality, see [15]:

f (\frac{θ + λ}{2}) \leq_{I} \frac{1}{λ - θ} (I A) \int_{θ}^{λ} f (б) d б \leq_{I} \frac{f (θ) + f (λ)}{2}

(57)

\tilde{f}

is lower Godunova–Levin convex

F

N

M

[θ, λ]

with

ı = 1

and

ħ (κ) = \frac{1}{κ}

, then from (45) and (50) we derive the following subsequent inequality, see [15]:

f (\frac{θ + λ}{2}) \leq_{I} \frac{1}{\int_{θ}^{λ} B (б) d б} (I A) \int_{θ}^{λ} f (б) B (б) d б \leq_{I} \frac{f (θ) + f (λ)}{2}

(58)

Let

ħ (κ) = \frac{1}{κ},

and

f_{*} (б, ı) = f^{*} (б, ı)

with

ı = 1

. Then from (45) and (50), we obtain following classical Fejér inequality.

f (\frac{θ + λ}{2}) \leq \frac{1}{\int_{θ}^{λ} B (б) d б} \int_{θ}^{λ} f (б) B (б) d б \leq \frac{f (θ) + f (λ)}{2}

(59)

Example 6.

We consider

ħ (κ) = \frac{1}{κ},

for

κ \in [0, 1]

, and the

F

N

M

f : [0, 2] \to F_{0}

defined by,

\tilde{f} (б) (θ) = \{\begin{matrix} \frac{θ - 2 + б^{\frac{1}{2}}}{\frac{3}{2} - 2 - б^{\frac{1}{2}}} θ \in [2 - б^{\frac{1}{2}}, \frac{3}{2}] \\ \frac{2 + б^{\frac{1}{2}} - θ}{2 + б^{\frac{1}{2}} - \frac{3}{2}} θ \in (\frac{3}{2}, 2 + б^{\frac{1}{2}}] \\ 0 otherwise, \end{matrix}

(60)

Then, for each

ı \in [0, 1],

we have

f_{ı} (б) = [(1 - ı) (2 - б^{\frac{1}{2}}) + \frac{3}{2} ı, (1 - ı) (2 + б^{\frac{1}{2}}) + \frac{3}{2} ı]

. Since end point mappings

f_{*} (б, ı),

f^{*} (б, ı)

are

ħ

-Godunova–Levin convex mappings for each

ı \in [0, 1]

, then

\tilde{f} (б)

U

D

ħ

-Godunova–Levin convex

F

N

M

. If

B (б) = \{\begin{matrix} \sqrt{б}, σ \in [0, 1], \\ \sqrt{2 - б}, σ \in (1, 2], \end{matrix}

then

B (2 - б) = B (б) \geq 0

, for all

б \in [0, 2]

. Since

f_{*} (б, ı) = (1 - ı) (2 - б^{\frac{1}{2}}) + \frac{3}{2} ı

and

f^{*} (б, ı) = (1 - ı) (2 + б^{\frac{1}{2}}) + \frac{3}{2} ı

. Now we compute the following:

\begin{matrix} \frac{1}{λ - θ} \int_{θ}^{λ} [f_{*} (б, ı)] B (б) d б = \frac{1}{2} \int_{0}^{2} [f_{*} (б, ı)] B (б) d б = \frac{1}{2} \int_{0}^{1} [f_{*} (б, ı)] B (б) d б + \frac{1}{2} \int_{1}^{2} f_{*} (б, ı) B (б) d б, \\ \frac{1}{λ - θ} \int_{θ}^{λ} [f^{*} (б, ı)] B (б) d б = \frac{1}{2} \int_{0}^{2} [f^{*} (б, ı)] B (б) d б = \frac{1}{2} \int_{0}^{1} [f^{*} (б, ı)] B (б) d б + \frac{1}{2} \int_{1}^{2} f^{*} (б, ı) B (б) d б, \end{matrix}

\begin{matrix} = \frac{1}{2} \int_{0}^{1} [(1 - ı) (2 - б^{\frac{1}{2}}) + \frac{3}{2} ı] (\sqrt{б}) d б + \frac{1}{2} \int_{1}^{2} [(1 - ı) (2 - б^{\frac{1}{2}}) + \frac{3}{2} ı] (\sqrt{2 - б}) d б \\ = \frac{ı + 5}{12} + \frac{1}{24} [3 π (ı - 1) - 4 (ı - 4)], \\ = \frac{1}{2} \int_{0}^{1} [(1 - ı) (2 + б^{\frac{1}{2}}) + \frac{3}{2} ı] (\sqrt{б}) d б + \frac{1}{2} \int_{1}^{2} [(1 - ı) (2 + б^{\frac{1}{2}}) + \frac{3}{2} ı] (\sqrt{2 - б}) d б \\ = \frac{1}{12} [11 - 5 ı] + \frac{1}{24} [- 3 π ı - 4 ı + 3 π + 16] . \end{matrix}

(61)

And

\begin{matrix} [f_{*} (θ, ı) + f_{*} (λ, ı)] \int_{0}^{1} \begin{matrix} \frac{B ((1 - κ) θ + κ λ)}{ħ (κ)} \end{matrix} d κ \\ = [4 (1 - ı) - \sqrt{2} (1 - ı) + 3 ı] [\int_{0}^{\frac{1}{2}} κ \sqrt{2 κ} d κ + \int_{\frac{1}{2}}^{1} κ \sqrt{2 (1 - κ)} d κ] \\ = \frac{1}{3} (4 (1 - ı) - \sqrt{2} (1 - ı) + 3 ı), \\ [f^{*} (θ, ı) + f^{*} (λ, ı)] \int_{0}^{1} \frac{B ((1 - κ) θ + κ λ)}{ħ (κ)} d κ \\ = [4 (1 - ı) + \sqrt{2} (1 - ı) + 3 ı] [\int_{0}^{\frac{1}{2}} κ \sqrt{2 κ} d κ + \int_{\frac{1}{2}}^{1} κ \sqrt{2 (1 - κ)} d κ] \\ = \frac{1}{3} (4 (1 - ı) + \sqrt{2} (1 - ı) + 3 ı) . \end{matrix}

(62)

From (61) and (62), we have

[\frac{ı + 5}{12} + \frac{1}{24} [3 π (ı - 1) - 4 (ı - 4)], \frac{1}{12} [11 - 5 ı] + \frac{1}{24} [- 3 π ı - 4 ı + 3 π + 16]] \supseteq_{I} [\frac{1}{3} (4 (1 - ı) - \sqrt{2} (1 - ı) + 3 ı), \frac{1}{3} (4 (1 - ı) + \sqrt{2} (1 - ı) + 3 ı)],

for all

ı \in [0, 1] .

Hence, Theorem 7 is verified.

For Theorem 8, we have

\begin{matrix} f_{*} (\frac{θ + λ}{2}, ı) = f_{*} (1, ı) = \frac{2 + ı}{2}, \\ f^{*} (\frac{θ + λ}{2}, ı) = f^{*} (1, ı) = \frac{3 (2 - ı)}{2}, \end{matrix}

(63)

\int_{θ}^{λ} B (б) d б = \int_{0}^{\begin{matrix} 1 \end{matrix}} \sqrt{б} d б + \int_{1}^{2} \sqrt{2 - б} d б = \frac{4}{3},

\begin{matrix} \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} \int_{θ}^{λ} f_{*} (б, ı) B (б) d б = \frac{3}{4} [\frac{13 - ı}{6} + \frac{π (ı - 1)}{4}], \\ \frac{2}{ħ (\frac{1}{2}) \int_{θ}^{λ} B (б) d б} \int_{θ}^{λ} f^{*} (б, ı) B (б) d б = \frac{3}{24} [[11 - 5 ı] + \frac{1}{2} [- 3 π ı - 4 ı + 3 π + 16]] . \end{matrix}

(64)

From (63) and (64), we have

[\frac{2 + ı}{2}, \frac{3 (2 - ı)}{2}] \supseteq_{I} [\frac{3}{4} [\frac{13 - ı}{6} + \frac{π (ı - 1)}{4}], \frac{3}{24} [[11 - 5 ı] + \frac{1}{2} [- 3 π ı - 4 ı + 3 π + 16]]] .

Hence, Theorem 8 has been verified.

4. Fuzzy Version of Ostrowski’s Type Inequality via $U$ · $D$ - $ħ$ -Godunova–Levin $F$ · $N$ · $M$ s

Here, an Ostrowski-type inequality was formulated in conjunction with several illustrations for Godunova-Levin functions within a broader category.

First, recalling some basic notations that will be helpful in this section such that:

Gamma and Beta functions are respectively characterized as

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

for

R (y) > 0

ß (y, ʑ) = \int_{0}^{1} κ^{y - 1} {(1 - κ)}^{ʑ - 1} d κ = \frac{Γ (y) Γ (ʑ)}{Γ (y + ʑ)}

for

R (y) > 0

R (ʑ) > 0

The integral representation of the hypergeometric function is

{}_{2}{F_{1}} (y, ʑ; c; б) = \frac{1}{ß (ʑ, c - ʑ)} \int_{0}^{1} κ^{ʑ - 1} {(1 - κ)}^{c - ʑ - 1} {(1 - x κ)}^{- y} d κ

for

|x| < 1

R (c) > 0

R (ʑ) > 0

The subsequent lemma aids in achieving our goal.

Lemma 1.

Let

f_{*} (., i), f^{*} (., i) : [θ, λ] \to N^{+}

be a differentiable function on

I_{0}

with

(θ, λ)

, where

i \in [0, 1]

. If

\tilde{f}

is integrable over

[θ, λ]

, then

[f_{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (φ, ı) d φ, f^{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (φ, ı) d φ] = \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} κ [{f_{*}}^{'} (κ б + (1 - κ) θ, ı), {f^{*}}^{'} (κ б + (1 - κ) θ, ı)] d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} κ [{f_{*}}^{'} (κ λ + (1 - κ) б, ı), {f^{*}}^{'} (κ λ + (1 - κ) б, ı)] d κ .

(65)

Proof.

Integration by parts finalizes the proof. □

Now, employing Lemma 1, we derive the principal outcomes.

Theorem 9.

Let

\tilde{f} : [θ, λ] \to F_{0}^{+}

be a differentiable function on

I_{0}

with

б \in (θ, λ)

, where

i \in [0, 1]

and let

\tilde{f}

be a integrable over

[θ, λ]

. If

|{\tilde{f}}^{'}|

ħ

-Godunova–Levin

F

N

M

s, with

|{\tilde{f}}^{'}| \geq_{F} \tilde{0}

. If

M = [m_{1}, m_{2}],

then

[|f_{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (φ, ı) d φ|, |f^{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (φ, ı) d φ|] \supseteq_{I} \frac{M [{(б - θ)}^{2} + {(λ - б)}^{2}]}{λ - θ} \int_{0}^{1} [\frac{1}{ƛ (κ) ħ (κ)} + \frac{1}{ƛ (κ) ħ (1 - κ)}] d κ,

(66)

for all

б \in [θ, λ],

where

m_{2} \geq |{f_{*}}^{'} (б, ı)|

and

|{f^{*}}^{'} (б, ı)| \geq m_{1}

Proof.

In accordance with Lemma 1 and

|f^{'}|

ħ

-Godunova–Levin

F

N

M

, for

ı \in [0, 1]

we have

|f_{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (φ, ı) d φ| = \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} κ {f_{*}}^{'} (κ б + (1 - κ) θ, ı) d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} κ {f_{*}}^{'} (κ λ + (1 - κ) б, ı) d κ

and

|f^{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (φ, ı) d φ| = \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} κ {f^{*}}^{'} (κ б + (1 - κ) θ, ı) d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} κ {f^{*}}^{'} (κ λ + (1 - κ) б, ı) d κ

From above equations, we have

|f_{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (φ, ı) d φ| \leq \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} κ [\frac{|{f_{*}}^{'} (б, ı)|}{ħ (κ)} + \frac{|{f_{*}}^{'} (θ, ı)|}{ħ (1 - κ)}] d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} κ [\frac{|{f_{*}}^{'} (λ, ı)|}{ħ (κ)} + \frac{|{f_{*}}^{'} (б, ı)|}{ħ (1 - κ)}] d κ,

and

|f^{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (φ, ı) d φ| \geq \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} κ [\frac{|{f^{*}}^{'} (б, ı)|}{ħ (κ)} + \frac{|{f^{*}}^{'} (θ, ı)|}{ħ (1 - κ)}] d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} κ [\frac{|{f^{*}}^{'} (λ, ı)|}{ħ (κ)} + \frac{|{f^{*}}^{'} (б, ı)|}{ħ (1 - κ)}] d κ .

As a result, we obtain

|f_{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (φ, ı) d φ| \leq \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} [\frac{|{f_{*}}^{'} (б, ı)|}{ƛ (κ) ħ (κ)} + \frac{|{f_{*}}^{'} (θ, ı)|}{ƛ (κ) ħ (1 - κ)}] d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} [\frac{|{f_{*}}^{'} (λ, ı)|}{ƛ (κ) ħ (κ)} + \frac{|{f_{*}}^{'} (б, ı)|}{ƛ (κ) ħ (1 - κ)}] d κ, \leq \frac{M {(б - θ)}^{2}}{λ - θ} \int_{0}^{1} [\frac{1}{ƛ (κ) ħ (κ)} + \frac{1}{ƛ (κ) ħ (1 - κ)}] d κ + \frac{M {(λ - б)}^{2}}{λ - θ} \int_{0}^{1} [\frac{1}{ƛ (κ) ħ (κ)} + \frac{1}{ƛ (κ) ħ (1 - κ)}] d κ,

and

|f^{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (φ, ı) d φ| \geq \frac{{(б - θ)}^{2}}{λ - θ} \int_{0}^{1} [\frac{|{f^{*}}^{'} (б, ı)|}{ƛ (κ) ħ (κ)} + \frac{|{f^{*}}^{'} (θ, ı)|}{ƛ (κ) ħ (1 - κ)}] d κ + \frac{{(λ - б)}^{2}}{λ - θ} \int_{0}^{1} [\frac{|{f^{*}}^{'} (λ, ı)|}{ƛ (κ) ħ (κ)} + \frac{|{f^{*}}^{'} (б, ı)|}{ƛ (κ) ħ (1 - κ)}] d κ, \geq \frac{M {(б - θ)}^{2}}{λ - θ} \int_{0}^{1} [\frac{1}{ƛ (κ) ħ (κ)} + \frac{1}{ƛ (κ) ħ (1 - κ)}] d κ + \frac{M {(λ - б)}^{2}}{λ - θ} \int_{0}^{1} [\frac{1}{ƛ (κ) ħ (κ)} + \frac{1}{ƛ (κ) ħ (1 - κ)}] d κ .

That is,

[|f_{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f_{*} (φ, ı) d φ|, |f^{*} (б, ı) - \frac{1}{λ - θ} \int_{θ}^{λ} f^{*} (φ, ı) d φ|] \supseteq_{I} \frac{M [{(б - θ)}^{2} + {(λ - б)}^{2}]}{λ - θ} \int_{0}^{1} [\frac{1}{ƛ (κ) ħ (κ)} + \frac{1}{ƛ (κ) ħ (1 - κ)}] d κ,

this completes the proof. □

5. Conclusions

Our principal objective entails applying classical integral operators and

U

D

-relations to recent findings elucidated by the authors of the referenced works [39,41,47]. Various forms of

U

D

-relations, and classical integral operators, along with fuzzy Aumman’s integral operators, have been applied to derive several novel outcomes. Our primary contributions encompass the establishment of fuzzy integral inequalities of the H.H., Fejér, Pachpatte and Ostrowski’s inequalities via newly defined class

U

D

ħ

-Godunova–Levin convex

F

N

M

s. Furthermore, we provided initial clarifications on concepts related to up and down, and pseudo order relations to underscore their differences and offer commentary on key revelations. The outcomes of this research are poised to significantly influence the realms of inequality and optimization theory.

Author Contributions

Conceptualization, A.A. and A.F.A.; validation, S.A. and A.F.A.; formal analysis, S.A. and A.F.A.; investigation, A.A. and M.V.C.; resources, A.A. and M.V.C.; writing—original draft, A.A. and M.V.C.; writing—review and editing, A.A., A.F.A. and S.A.; visualization, M.V.C. and A.A.; supervision, M.V.C. and A.A.; project administration, M.V.C. and A.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Taif University, Saudi Arabia, project No (TU-DSPP-2024-87).

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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Cortez, M.V.; Althobaiti, A.; Aljohani, A.F.; Althobaiti, S. Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications. Axioms 2024, 13, 471. https://doi.org/10.3390/axioms13070471

AMA Style

Cortez MV, Althobaiti A, Aljohani AF, Althobaiti S. Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications. Axioms. 2024; 13(7):471. https://doi.org/10.3390/axioms13070471

Chicago/Turabian Style

Cortez, Miguel Vivas, Ali Althobaiti, Abdulrahman F. Aljohani, and Saad Althobaiti. 2024. "Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications" Axioms 13, no. 7: 471. https://doi.org/10.3390/axioms13070471

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Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications

Abstract

1. Introduction

2. Preliminaries

Aumann Integrals for Interval and Fuzzy Number Mappings

3. Hermite-Hadamard Inequalities over $U$ · $D$ - $ħ$ -Godunova–Levin Convex $F$ · $N$ · $M$

4. Fuzzy Version of Ostrowski’s Type Inequality via $U$ · $D$ - $ħ$ -Godunova–Levin $F$ · $N$ · $M$ s

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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Article Menu

Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications

Abstract

1. Introduction

2. Preliminaries

Aumann Integrals for Interval and Fuzzy Number Mappings

3. Hermite-Hadamard Inequalities over U · D - ħ -Godunova–Levin Convex F · N · M

4. Fuzzy Version of Ostrowski’s Type Inequality via U · D - ħ -Godunova–Levin F · N · M s

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

3. Hermite-Hadamard Inequalities over $U$ · $D$ - $ħ$ -Godunova–Levin Convex $F$ · $N$ · $M$

4. Fuzzy Version of Ostrowski’s Type Inequality via $U$ · $D$ - $ħ$ -Godunova–Levin $F$ · $N$ · $M$ s