Open AccessArticle

Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials

Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan

Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada

⁴

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

⁵

Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

⁶

Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy

⁷

Department of Mathematics, College of Sciences and Arts Onaizah, Qassim University, P.O. Box 6640, Buraydah 51452, Saudi Arabia

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2023, 12(1), 52; https://doi.org/10.3390/axioms12010052

Submission received: 20 November 2022 / Accepted: 27 December 2022 / Published: 3 January 2023

(This article belongs to the Special Issue Mathematical Analysis and Applications III)

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Abstract

In this paper, we introduce two new subclasses of bi-univalent functions using the q-Hermite polynomials. Furthermore, we establish the bounds of the initial coefficients

υ_{2}

υ_{3}

, and

υ_{4}

of the Taylor–Maclaurin series and that of the Fekete–Szegö functional associated with the new classes, and we give the many consequences of our findings.

Keywords:

q-convolution operator; coefficient estimates; q-Hermite; bi-univalent functions; Babalola operator

MSC:

05A30; 30C45; 11B65; 47B38

1. Introduction and Preliminaries

Let

M (U)

denote the class of analytic functions in the open unit disk

U = \{z : z \in C and |z| < 1\} .

Let

A

be the subclass of

M (U)

whose functions f satisfy the normalization condition given by

f (0) = f^{'} (0) - 1 = 0,

that is, each function f in

A

can be represented by the following Taylor–Maclaurin series expansion:

f (z) = z + \sum_{k = 2}^{\infty} υ_{k} z^{k} (z \in U) .

(1)

Moreover, let

S

be the subclass of

A

whose functions are univalent in

U

. The Koebe one-quarter theorem ensures that the image of

U

under every

f \in 𝒮

contains a disk of radius

1 / 4

It is known that every function

f \in S

has an inverse

f^{- 1}

defined by

f^{- 1} (f (z)) = z (z \in U),

and

f^{- 1} (f (ω)) = ω (|ω| < r_{0} (f); r_{0} (f) \geq \frac{1}{4}),

where

f^{- 1} (ω) = g (ω) = ω - υ_{2} ω^{2} + (2 υ_{2}^{2} - υ_{3}) ω^{3} - (5 υ_{2}^{3} - 5 υ_{2} υ_{3} + υ_{4}) ω^{4} + \dots .

(2)

A function is said to be bi-univalent in

U

if both f and

f^{- 1}

are univalent in

U

. Let

Σ

denote the class of bi-univalent functions in

U

given by (1).

Lewin [1] investigated the class

Σ

and showed that

|v_{2}| < 1.51 .

Subsequently, Brannan and Clunie [2] conjectured that

|v_{2}| < \sqrt{2} .

Netanyahu [3], on the other hand, showed that

max_{f \in Σ} |v_{2}| = \frac{4}{3} .

The Taylor–Maclaurin coefficients

|v_{n}|

(n \geq 3, n \in N)

in (1) are still unknown, and it is an open problem.

Similar to the subclasses

S^{*} (ζ)

and

K (ζ)

of the starlike and convex functions of the order

ζ

(0 \leq ζ < 1),

respectively, that we are familiar with, Brannan and Taha [4] gave two subclasses of

Σ

, which are called

S_{Σ}^{*} (ζ)

and

K_{Σ} (ζ)

of the bi-starlike functions and bi-convex functions of the order

ζ

(0 \leq ζ < 1),

respectively. It should be remarked here that, in their pioneering work, Srivastava et al. [5] actually revived the study of analytic and bi-univalent functions in recent years.

Moreover, for two analytic functions

s_{1}

and

s_{2},

the function

s_{1}

is called subordinated to the function

s_{2},

denoted as

s_{1} (z) ≺ s_{2} (z) (z \in U),

if there is an analytic function w in

U

with

w (0) = 0 and |w (z)| < 1,

such that

s_{1} (z) = s_{2} (w (z)) .

If the function

s_{2}

\in S,

then

s_{1} (z) ≺ s_{2} (z) \Leftrightarrow s_{1} (0) = s_{2} (0) and s_{1} (U) \subset s_{2} (U) .

In 2008, Babalola [6] defined the operator

I_{n}^{σ} : A ⟶ A

I_{m}^{τ} f (z) = (ν_{τ} * ν_{τ, m}^{- 1} * f) (z)

(3)

where

ν_{τ, m} (z) = \frac{z}{{(1 - z)}^{τ - (m - 1)}}, τ - (m - 1) > 0, ν_{τ} = ν_{τ, 0}

and

ν_{τ, m}^{- 1}

is such that

(ν_{τ, m} * ν_{τ, m}^{- 1}) (z) = \frac{z}{1 - z} (τ, m \in N) .

Let

f \in A

, then (3) is equivalent to

I_{m}^{τ} f (z) = z + \sum_{j = 2}^{\infty}  [\frac{[τ + j - 1]!}{τ!} \frac{[τ - m]!}{[τ + j - m - 1]!}] υ_{j} z^{j} .

(4)

The q-derivative operator

D_{q}

of a function was introduced and researched by Jackson [7,8].

D_{q} f (z) = \frac{f (q z) - f (z)}{z (q - 1)} = z^{- 1} \{z + \sum_{k = 2}^{\infty} {[k]}_{q} υ_{k} z^{k}\}

(5)

and

D_{q} f (0) = f^{'} (0)

. In particular,

f (z) = z^{k}

for k is a positive integer, the q-derivative of

f (z)

is given by

D_{q} z^{k} = \frac{{(z q)}^{k} - z^{k}}{z (q - 1)} = {[k]}_{q} z^{k - 1},

(6)

lim_{q ⟶ 1} {[k]}_{q} = lim_{q ⟶ 1} \frac{q^{k} - 1}{q - 1} = k .

(7)

For function

f (z)

given by (1) and

g (z)

given by

g (z) = z + \sum_{k = 2}^{\infty} c_{k} z^{k}

the convolution of

f (z)

and

g (z)

is defined by

(f * g) (z) = z + \sum_{k = 2}^{\infty} υ_{k} c_{k} z^{k} = (g * f) (z) .

Let

D_{q} (z) = \frac{z}{(1 - q z) (1 - z)} = z + (1 + e_{1}) z^{2} + (1 + e_{1} + e_{2}) z^{3} + \dots = z + \sum_{k = 2}^{\infty} {[k]}_{e} z^{k}

(8)

where

{[k]}_{e} = 1 + e_{1} + e_{2} + \dots + e_{k - 1}, e_{k} = q^{k} .

(9)

See [9,10] for additional information on q-derivative theories.

Quantum (or q-) calculus is a strong instrument for investigating a wide range of analytic functions, and it has sparked new research in mathematics and other fields. The first time it was used in the context of univalent functions was by Srivastava [11]. Many academics have studied q-calculus and its many applications due to the usefulness of q-analysis in mathematics and other areas. With the help of certain higher-order q-derivative operators, Khan et al. [12] constructed and analyzed a number of subclasses of q-starlike functions. Shi et al. (see also [13]) created a novel subclass of multivalent q-starlike Janowski functions using the q-differential operator. A variety of adequate requirements as well as some other noteworthy characteristics were investigated in both articles [12,14].

Because of the large range of applications and the usefulness of q-operators above fundamental operators, many scholars have looked into q-calculus in depth. Furthermore, Srivastava’s recently published survey-cum-expository review study [15,16,17] is useful for academics and scholars studying these topics.

The q-Hermite polynomial was first introduced by Rogers [18] (see also [19,20]) and is usually defined by means of their generating function as follows

B_{k} (s | q) = \sum_{k = 0}^{\infty} H_{k} (x; q) \frac{t^{k}}{{(q; q)}_{k}} = \prod_{k = 0}^{\infty} \frac{1}{1 - 2 x t q^{k} + t^{2} q^{2 k}} (0 < q < 1) .

The q-derivative of the q-Hermite polynomial is

D_{q} \{B_{k + 1} (s | q)\} = {[k]}_{q} B_{k} (s | q) .

(10)

Moreover, Ismail et al. [18] were able to define the recursion relation as

t B_{k} (s | q) = B_{k + 1} (s | q) + {[k]}_{q} B_{k - 1} (s | q)

(11)

with

B_{0} (s | q) = 1 and B_{- 1} (s | q) = 0 .

Also from (11), we have

\begin{matrix} B_{1} (s | q) = s \\ B_{2} (s | q) = s^{2} - 1 \\ B_{3} (s | q) = s^{3} - (2 + q) s \\ B_{4} (s | q) = s^{4} - (3 + 2 q + q^{2}) s^{2} + (1 + q + q^{2}) . \end{matrix}

Remark 1.

It is clear that

B_{k} (s | q = 1) = B_{c_{k}} (s)

is the Hermite polynomials. Moreover, when

B_{k} (s | q = 0) = U_{k} (s / 2),

we have Chebyshev polynomials of the first kind, and they are defined by the recursion relation,

2 s U_{k} (s) = U_{k - 1} (s) + U_{k + 1} (s)

(12)

with

U_{0} (s) = 1 a n d U_{- 1} (s) = 0 .

Next, we define the q-Babalola convolution operator which will be used throughout this paper.

Definition 1.

Let

f \in A

. Denote by

I_{γ, q} f (z)

the q-Babalola convolution operator defined by

I_{γ, q} f (z) = (ν_{τ, q} * ν_{γ, q}^{(- 1)} * f) (z)

(13)

where

ν_{γ, q} = \frac{z}{{(1 - q z)}^{γ} (1 - z)}, γ > - 1 a n d ν_{γ, q}^{(- 1)}

is such that

(ν_{γ, q} * ν_{γ, q}^{(- 1)}) (z) = \frac{z}{1 - z} .

Hence,

I_{γ, q} f (z) = z + \sum_{k = 2}^{\infty} \frac{{[k]}_{e}^{σ}}{{[k]}_{e}^{γ}} υ_{k} z^{k} = z + \sum_{k = 2}^{\infty} {(k]}_{e}^{γ} υ_{k} z^{k} .

(14)

where

{(k]}_{e}^{γ} = \frac{1 + e_{1} (τ) + e_{2} (τ) \dots e_{k - 1} (τ)}{1 + e_{1} (γ) + e_{2} (γ) \dots e_{k - 1} (γ)}

and

e_{k - 1} (τ) = \frac{(τ + k - 2)!}{(τ - 1)!} \frac{q^{k - 1}}{(k - 1)!}, e_{k - 1} (γ) = \frac{(γ + k - 2)!}{(γ - 1)!} \frac{q^{k - 1}}{(k - 1)!} .

Remark 2.

It is easily seen that, upon setting

q ⟶ 1 -

, the extended Babalola convolution operator

I_{γ, q} f (z)

reduces to the Babalola convolution operator

I_{σ}^{m} f (z)

which was introduced and studied by Babalola [6]. For

m = τ = 1

, the extended Babalola convolution operator

I_{γ, q} f (z)

reduces to the q-derivative operator introduced and studied by Jackson [7,8]. Moreover, if

m = τ

and

q ⟶ 1 -

, we have the Ruscheweyh’s operator [21].

Consider the univalent normalized functions of the kind (1); the Fekete–Szegö functional

| υ_{3} - φ υ_{2}^{2} |

has a long history in geometric function theory. The authors in [22] disproved Paley’s conjecture and Littlewood’s that the coefficients of odd univalent functions are confined by unity in 1933. Since then, the functional has gotten much attention, especially in subclasses of the family of univalent functions. This problem appears to have piqued the interest of scholars in recent years (see, for example, [23,24]).

We know that the q-Hermite polynomials and q-convolution operators still have not been studied with bi-univalent functions. The main goal of this paper is to start looking at the properties of the bi-univalent functions that are connected to q-Hermite polynomials and the q-convolution operator. In this study, the initial coefficient estimates for the Fekete–Szegö problem of analytic and bi-univalent functions are determined using the q-Hermite polynomial expansions and the q-convolution operator.

In Definition 2, we describe a class of convex bi-univalent functions that are defined by the q-convolution operator and linked to the q-Hermite polynomial.

Definition 2.

Let

N (z, s, q)

be defined as follows:

N (z, s, q) = \sum_{k = 2}^{\infty} B_{k} (s | q) z^{k} .

(15)

A function

f \in Σ

given by (1) is said to be in the class

Γ_{Σ}^{q} (s, γ, τ)

, if the following conditions are satisfied:

1 + \frac{z D_{q}^{2} (I_{γ, q} f (z))}{D_{q} (I_{γ, q} f (z))} ≺ N (z, s, q)

(16)

and

1 + \frac{ω D_{q}^{2} (I_{γ, q} f^{- 1} (ω))}{D_{q} (I_{γ, q} f^{- 1} (ω))} ≺ N (ω, s, q) .

(17)

Where

s \in (\frac{1}{2}, 1), 0 < q < 1, z \in U, ω \in U, γ = τ - m > - 1

In Definition 3, we describe a class of starlike bi-univalent functions that are defined by the q-convolution operator and linked to the q-Hermite polynomial.

Definition 3.

Let

N (z, s, q)

be defined as follows:

N (z, s, q) = \sum_{k = 2}^{\infty} B_{k} (s | q) z^{k} .

(18)

A function

f \in Σ

given by (1) is said to be in the class

Π_{Σ}^{q} (s, γ, τ)

, if the following conditions are satisfied:

\frac{z D_{q} (I_{γ, q} f (z))}{I_{γ, q} f (z)} ≺ N (z, s, q)

(19)

and

\frac{ω D_{q} (I_{γ, q} f^{- 1} (ω))}{I_{γ, q} f^{- 1} (ω)} ≺ N (ω, s, q) .

(20)

We must recall the following lemma in order to arrive at our primary conclusions.

As is usually the case, we let

P

be the family of functions

p (z) = 1 + p_{1} z + p_{2} z^{2} + . . .

regular with positive real part, for

z \in U

Lemma 1

([25]). Let

φ (z) \in P

, then

| p_{j} | \leq 2 (j \in N) .

2. Coefficient Estimates for the Class $Γ_{Σ}^{q} (s, γ, τ)$

The initial coefficient bounds of the class

Γ_{Σ}^{q} (s, γ, τ)

of bi-univalent functions are investigated in this section.

Theorem 1.

Let

f \in Γ_{Σ}^{q} (s, γ, τ)

. Then,

| υ_{2} | \leq \sqrt{Ψ_{1} (s, q, γ)},

(21)

| υ_{3} | \leq \frac{s^{2}}{{[2]}_{b}^{2} {({(2]}_{e}^{γ})}^{2}} + \frac{s}{4 {[2]}_{b} {[3]}_{b} {(3]}_{e}^{γ}},

(22)

and

\begin{matrix} | υ_{4} | & \leq \frac{s^{3} (2 (1 + e_{1}) {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2})}{{[2]}_{e}^{3} {[4]}_{e} {(4]}_{e}^{γ} {({(2]}_{e}^{γ})}^{2}} - \frac{s^{2} (2 e_{1} {[2]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} + 5 {[4]}_{e} {(4]}_{e}^{γ})}{2 {[2]}_{e}^{3} {[4]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} {(4]}_{e}^{γ}} \\ + \frac{s^{3} - 2 s - 2 q s - 4}{{[2]}_{e} {[4]}_{e} {(4]}_{e}^{γ}} \end{matrix}

where

Ψ_{1} (s, q, γ) = \frac{s^{3}}{| s^{2} ({[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}) - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2} (s^{2} - s - 1) |} .

(23)

Proof.

Let

f \in Σ

be given by (1) be in the class

Γ_{Σ}^{q} (s, γ, τ)

. Then,

1 + \frac{z D_{q}^{2} (I_{γ, q} f (z))}{D_{q} (I_{γ, q} f (z))} = N (d (z), s, q)

(24)

and

1 + \frac{z D_{q}^{2} (I_{γ, q} f^{- 1} (ω))}{D_{q} (I_{γ, q} f^{- 1} (ω))} = N (ϖ (ω), s, q),

(25)

Let

ϱ, η \in P

be defined as

ϱ (z) = \frac{1 + d (z)}{1 - d (z)} = 1 + ϱ_{1} z + ϱ_{2} z^{2} + ϱ_{3} z^{3} + \dots \Rightarrow d (z) = \frac{ϱ (z) - 1}{ϱ (z) + 1}, (z \in U)

(26)

and

η (ω) = \frac{1 + ϖ (ω)}{1 - ϖ (ω)} = 1 + η_{1} ω + η_{2} ω^{2} + η_{3} ω^{3} + \dots \Rightarrow ϖ (ω) = \frac{η (ω) - 1}{η (ω) + 1}, (ω \in U) .

(27)

It follows that from (26) and (27) that

d (z) = \frac{1}{2}  [ϱ_{1} z + (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) z^{2} + (ϱ_{3} - ϱ_{1} ϱ_{2} + \frac{ϱ_{1}^{3}}{4}) z^{3} + \dots]

(28)

and

ϖ (ω) = \frac{1}{2}  [η_{1} ω + (η_{2} - \frac{η_{1}^{2}}{2}) ω^{2} + (η_{3} - η_{1} η_{2} + \frac{η_{1}^{3}}{4}) ω^{3} + \dots] .

(29)

From (28) and (29), applying

N (z, s, q)

as given in (18), we see that

\begin{matrix} N (d (z), s, q) = 1 + \frac{B_{1} (s | q)}{2} ϱ_{1} z +  [\frac{B_{1} (s | q)}{2} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} ϱ_{1}^{2}] z^{2} \\ +  [\frac{B_{1} (s | q)}{2} (ϱ_{3} - ϱ_{1} ϱ_{2} + \frac{ϱ_{1}^{3}}{4}) + \frac{B_{2} (s | q)}{2} ϱ_{1} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} ϱ_{1}^{3}] z^{3} + \dots \end{matrix}

and

\begin{matrix} N (ϖ (ω), s, q) & = 1 + \frac{B_{1} (s | q)}{2} η_{1} ω +  [\frac{B_{1} (s | q)}{2} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} η_{1}^{2}] ω^{2} \\ +  [\frac{B_{1} (s | q)}{2} (η_{3} - η_{1} η_{2} + \frac{η_{1}^{3}}{4}) + \frac{B_{2} (s | q)}{2} η_{1} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} η_{1}^{3}] ω^{3} + \dots . \end{matrix}

(30)

It follows from (24), (30), and (25), we have

{[2]}_{e} {(2]}_{e}^{γ} υ_{2} = \frac{B_{1} (s | q)}{2} ϱ_{1},

(31)

{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} υ_{3} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2} υ_{2}^{2} = \frac{B_{1} (s | q)}{2} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} ϱ_{1}^{2},

(32)

\begin{matrix} {[3]}_{e} {[4]}_{e} {(4]}_{e}^{γ} υ_{4} & - {[2]}_{e} {[3]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} e_{1} υ_{2} υ_{3} + {[2]}_{e}^{3} {({(2]}_{e}^{γ})}^{3} υ_{2}^{3} \\ = \frac{B_{1} (s | q)}{2} (ϱ_{3} - ϱ_{1} ϱ_{2} + \frac{ϱ_{1}^{3}}{4}) + \frac{B_{2} (s | q)}{2} ϱ_{1} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} ϱ_{1}^{3}, \end{matrix}

(33)

- {[2]}_{e} {(2]}_{e}^{γ} υ_{2} = \frac{B_{1} (s | q)}{2} η_{1},

(34)

(2 {[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}) υ_{2}^{2} - {[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} υ_{3} = \frac{B_{1} (s | q)}{2} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} η_{1}^{2},

(35)

\begin{matrix} (2 (2 + e_{1}) {[2]}_{e} {[3]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} - 5 {[3]}_{e} {[4]}_{e} {(4]}_{e}^{γ} - {[2]}_{e}^{3} {({(2]}_{e}^{γ})}^{2}) υ_{2}^{3} + (5 {[3]}_{e} {[4]}_{e} {(4]}_{e}^{γ} \\ + {[2]}_{e} {[3]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} e_{1}) υ_{2} υ_{3} - {[3]}_{e} {[4]}_{e} {(4]}_{e}^{γ} υ_{4} = \frac{B_{1} (s | q)}{2} (η_{3} - η_{1} η_{2} + \frac{η_{1}^{3}}{4}) \\ + \frac{B_{2} (s | q)}{2} η_{1} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} η_{1}^{3} . \end{matrix}

(36)

Adding (31) and (34), we have

ϱ_{1} = - η_{1}, ϱ_{1}^{2} = η_{1}^{2} and ϱ_{1}^{3} = - η_{1}^{3}

(37)

and

υ_{2}^{2} = \frac{{(B_{1} (s | q))}^{2} (ϱ_{1}^{2} + η_{1}^{2})}{8 {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}} .

(38)

Moreover, adding (32) and (35) and applying (37) yields

4 υ_{2}^{2} [{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{b}^{2} {({(2]}_{e}^{γ})}^{2}] = B_{1} (s | q) (ϱ_{2} + η_{2}) - η_{1}^{2} (B_{1} (s | q) - B_{2} (s | q)) .

(39)

Applying (37) in (38) gives

η_{1}^{2} = \frac{4 {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2} υ_{2}^{2}}{{(B_{1} (s | q))}^{2}} .

(40)

Putting (40) into (39) and with some calculations, we have

| υ_{2} |^{2} = |\frac{{(B_{1} (s | q))}^{3} (ϱ_{2} + η_{2})}{4 [{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{b}^{2} {({(2]}_{e}^{γ})}^{2}] {(B_{1} (s | q))}^{2} + 4 {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2} (B_{1} (s | q) - B_{2} (s | q))}| .

Applying triangular inequality and Lemma 1, we have

| υ_{2} | \leq \sqrt{Ψ_{1} (s, q, γ)} .

(41)

Subtracting (35) from (32) and with some calculations, we have

υ_{3} = υ_{2}^{2} + \frac{B_{1} (s | q) [ϱ_{2} - η_{2}]}{4 {[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}}

(42)

υ_{3} = \frac{{(B_{1} (s | q))}^{2} ϱ_{1}^{2}}{4 {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}} + \frac{B_{1} (s | q) [ϱ_{2} - η_{2}]}{4 {[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}} .

(43)

Applying triangular inequality and Lemma 1, we have

| υ_{3} | \leq \frac{s^{2}}{{[2]}_{b}^{2} {({(2]}_{e}^{γ})}^{2}} + \frac{s}{4 {[2]}_{b} {[3]}_{b} {(3]}_{e}^{γ}} .

(44)

Subtracting (36) from (33), we have

\begin{matrix} 2 {[2]}_{e} {[4]}_{e} {(4]}_{e}^{γ} υ_{4} & = \frac{(2 (1 + e_{1}) {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}) {(B_{1} (s | q))}^{3} ϱ_{1}^{3}}{4 {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}} \\ - \frac{(2 e_{1} {[2]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} + 5 {[4]}_{e} {(4]}_{e}^{γ}) {(B_{1} (s | q))}^{2} ϱ_{1} (ϱ_{2} - η_{2})}{8 {[2]}_{e}^{2} {(2]}_{e}^{γ} {(3]}_{e}^{γ}} \\ + \frac{B_{1} (s | q) (ϱ_{3} - η_{3})}{2} + \frac{[B_{2} (s | q) - B_{1} (s | q)] ϱ_{1} (ϱ_{2} + η_{2})}{2} \\ + \frac{(B_{1} (s | q) - 2 B_{2} (s | q) + B_{3} (s | q)) ϱ_{1}^{3}}{4} . \end{matrix}

(45)

Applying triangular inequality and Lemma 1, we have

\begin{matrix} | υ_{4} | & \leq \frac{s^{3} (2 (1 + e_{1}) {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2})}{{[2]}_{e}^{3} {[4]}_{e} {(4]}_{e}^{γ} {({(2]}_{e}^{γ})}^{2}} - \frac{s^{2} (2 e_{1} {[2]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} + 5 {[4]}_{e} {(4]}_{e}^{γ})}{2 {[2]}_{e}^{3} {[4]}_{e} {(2]}_{e}^{γ} {(3]}_{e}^{γ} {(4]}_{e}^{γ}} \\ + \frac{s^{3} - 2 s - 2 q s - 4}{{[2]}_{e} {[4]}_{e} {(4]}_{e}^{γ}} . \end{matrix}

□

3. Coefficient Estimates for the Class $Π_{Σ}^{q} (s, γ, τ)$

The initial coefficient bounds of the class

Π_{Σ}^{q} (s, γ, τ)

of bi-univalent functions are investigated in this section.

Theorem 2.

Let

f \in Π_{Σ}^{q} (s, γ, τ)

. Then,

| υ_{2} | \leq \sqrt{X_{1} (s, q, γ)},

(46)

| υ_{3} | \leq \frac{s^{2}}{e_{1}^{2} {({(2]}_{e}^{γ})}^{2}} + \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}},

(47)

and

\begin{matrix} | υ_{4} | & \leq \frac{s^{3} ({(2]}_{e}^{γ} {(3]}_{e}^{γ} (4 e_{1} + 2 e_{2}) - 2 {({(2]}_{e}^{γ})}^{3} e_{1} - 10 (e_{1} + e_{2} + e_{3}) {(4]}_{e}^{γ})}{2 (e_{1} + e_{2} + e_{3}) {({(2]}_{e}^{γ})}^{3} {(4]}_{e}^{γ} e_{1}^{3}} \\ - \frac{5 s^{2}}{2 {(2]}_{e}^{γ} {(3]}_{e}^{γ} e_{1} (e_{1} + e_{2})} + \frac{s^{3} - 2 s - 2 q s - 4}{(e_{1} + e_{2} + e_{3}) {(4]}_{e}^{γ}} \end{matrix}

where

X_{1} (s, q, γ) = \frac{2 s^{3}}{| s^{2} {2 (e_{1} + e_{2}) {(3]}_{e}^{γ} + {(2]}_{e}^{γ} - {({(2]}_{e}^{γ})}^{2} (1 + 2 e_{1})} - 2 {({(2]}_{e}^{γ})}^{2} e_{1}^{2} (s^{2} - s - 1) |} .

(48)

Proof.

Let

f \in Σ

be given by (1) be in the class

Π_{Σ}^{q} (s, γ, τ)

. Then,

\frac{z D_{q} (I_{γ, q} f (z))}{I_{γ, q} f (z)} = N (d (z), s, q)

(49)

and

\frac{ω D_{q} (I_{γ, q} f^{- 1} (ω))}{I_{γ, q} f^{- 1} (ω)} = N (ϖ (ω), s, q) .

(50)

Let

ϱ, η \in P

be defined by

ϱ (z) = \frac{1 + d (z)}{1 - d (z)} = 1 + ϱ_{1} (z) + ϱ_{2} z^{2} + ϱ_{3} z^{3} + \dots \Rightarrow d (z) = \frac{ϱ (z) - 1}{ϱ (z) + 1}, (z \in U)

(51)

and

η (ω) = \frac{1 + ϖ (ω)}{1 - ϖ (ω)} = 1 + η_{1} (ω) + η_{2} ω^{2} + η_{3} ω^{3} + \dots \Rightarrow ϖ (ω) = \frac{η (ω) - 1}{η (ω) + 1}, (ω \in U) .

(52)

It follows that from (51) and (52) that

d (z) = \frac{1}{2}  [ϱ_{1} z + (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) z^{2} + (ϱ_{3} - ϱ_{1} ϱ_{2} + \frac{ϱ_{1}^{3}}{4}) z^{3} + \dots]

(53)

and

ϖ (ω) = \frac{1}{2}  [η_{1} ω + (η_{2} - \frac{η_{1}^{2}}{2}) ω^{2} + (η_{3} - η_{1} η_{2} + \frac{η_{1}^{3}}{4}) ω^{3} + \dots] .

(54)

From (53) and (54), applying

N (z, s, q)

as given in (18), we see that

\begin{matrix} N (d (z), s, q) = 1 + \frac{B_{1} (s | q)}{2} ϱ_{1} z +  [\frac{B_{1} (s | q)}{2} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} ϱ_{1}^{2}] z^{2} \\ +  [\frac{B_{1} (s | q)}{2} (ϱ_{3} - ϱ_{1} ϱ_{2} + \frac{ϱ_{1}^{3}}{4}) + \frac{B_{2} (s | q)}{2} ϱ_{1} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} ϱ_{1}^{3}] z^{3} + \dots \end{matrix}

and

\begin{matrix} N (ϖ (ω), s, q) & = 1 + \frac{B_{1} (s | q)}{2} η_{1} ω +  [\frac{B_{1} (s | q)}{2} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} η_{1}^{2}] ω^{2} \\ +  [\frac{B_{1} (s | q)}{2} (η_{3} - η_{1} η_{2} + \frac{η_{1}^{3}}{4}) + \frac{B_{2} (s | q)}{2} η_{1} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} η_{1}^{3}] ω^{3} + \dots . \end{matrix}

(55)

It follows from (49), (55), and (50), we have

{(2]}_{e}^{γ} e_{1} υ_{2} = \frac{B_{1} (s | q)}{2} ϱ_{1},

(56)

(e_{1} + e_{2}) {(3]}_{e}^{γ} υ_{3} - {({(2]}_{e}^{γ})}^{2} e_{1} υ_{2}^{2} = \frac{B_{1} (s | q)}{2} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} ϱ_{1}^{2},

(57)

\begin{matrix} (e_{1} + e_{2} + e_{3}) {(4]}_{e}^{γ} υ_{4} & - (2 e_{1} + e_{2}) {(2]}_{e}^{γ} {(3]}_{e}^{γ} υ_{2} υ_{3} + e_{1} {({(2]}_{e}^{γ})}^{3} υ_{2}^{3} \\ = \frac{B_{1} (s | q)}{2} (ϱ_{3} - ϱ_{1} ϱ_{2} + \frac{ϱ_{1}^{3}}{4}) + \frac{B_{2} (s | q)}{2} ϱ_{1} (ϱ_{2} - \frac{ϱ_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} ϱ_{1}^{3}, \end{matrix}

(58)

- {(2]}_{e}^{γ} e_{1} υ_{2} = \frac{B_{1} (s | q)}{2} η_{1},

(59)

2 {(3]}_{e}^{γ} (e_{1} + e_{2}) υ_{2}^{2} - e_{1} {({(2]}_{e}^{γ})}^{2} υ_{2}^{2} - (e_{1} + e_{2}) {(3]}_{e}^{γ} υ_{3} = \frac{B_{1} (s | q)}{2} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{2} (s | q)}{4} η_{1}^{2},

(60)

\begin{matrix} ({(2]}_{e}^{γ} {(3]}_{e}^{γ} (4 e_{1} + 2 e_{2}) - 5 {(4]}_{e}^{γ} (e_{1} + e_{2} + e_{3}) - {({(2]}_{e}^{γ})}^{2} e_{1}) υ_{2}^{3} - ({(2]}_{e}^{γ} {(3]}_{e}^{γ} (2 e_{1} + e_{2}) \\ + 5 {(4]}_{e}^{γ} (e_{1} + e_{2} + e_{3})) υ_{2} υ_{3} - {(4]}_{e}^{γ} (e_{1} + e_{2} + e_{3}) υ_{4} = \frac{B_{1} (s | q)}{2} (η_{3} - η_{1} η_{2} + \frac{η_{1}^{3}}{4}) \\ + \frac{B_{2} (s | q)}{2} η_{1} (η_{2} - \frac{η_{1}^{2}}{2}) + \frac{B_{3} (s | q)}{8} η_{1}^{3} . \end{matrix}

(61)

Adding (56) and (59), we have

ϱ_{1} = - η_{1}, ϱ_{1}^{2} = η_{1}^{2} and ϱ_{1}^{3} = - η_{1}^{3}

(62)

and

υ_{2}^{2} = \frac{{(B_{1} (s | q))}^{2} (ϱ_{1}^{2} + η_{1}^{2})}{8 {({(2]}_{e}^{γ})}^{2} e_{1}^{2}} .

(63)

Moreover, adding (57) and (60) and applying (62) yields

2 υ_{2}^{2} {2 (e_{1} + e_{2}) {(3]}_{e}^{γ} + {(2]}_{e}^{γ} - {({(2]}_{e}^{γ})}^{2} (1 + 2 e_{1})} = B_{1} (s | q) (ϱ_{2} + η_{2}) - η_{1}^{2} (B_{1} (s | q) - B_{2} (s | q)) .

(64)

Applying (62) in (63) gives

η_{1}^{2} = \frac{4 {({(2]}_{e}^{γ})}^{2} e_{1}^{2} υ_{2}^{2}}{{(B_{1} (s | q))}^{2}} .

(65)

Putting (65) into (64) and with some calculations, we have

| υ_{2} |^{2} = |\frac{{(B_{1} (s | q))}^{3} (ϱ_{2} + η_{2})}{2 [2 (e_{1} + e_{2}) {(3]}_{e}^{γ} + {(2]}_{e}^{γ} - {({(2]}_{e}^{γ})}^{2} (1 + 2 e_{1})] {(B_{1} (s | q))}^{2} + 4 {({(2]}_{e}^{γ})}^{2} e_{1}^{2} (B_{1} (s | q) - B_{2} (s | q))}| .

Applying triangular inequality and Lemma 1, we have

| υ_{2} | \leq \sqrt{X_{1} (s, q, γ)} .

(66)

Subtracting (60) from (57) and with some calculations, we have

υ_{3} = υ_{2}^{2} + \frac{B_{1} (s | q) [ϱ_{2} - η_{2}]}{4 (e_{1} + e_{2}) {(3]}_{e}^{γ}}

(67)

υ_{3} = \frac{{(B_{1} (s | q))}^{2} ϱ_{1}^{2}}{4 e_{1}^{2} {({(2]}_{e}^{γ})}^{2}} + \frac{B_{1} (s | q) [ϱ_{2} - η_{2}]}{4 (e_{1} + e_{2}) {(3]}_{e}^{γ}} .

(68)

Applying triangular inequality and Lemma 1, we have

| υ_{3} | \leq \frac{s^{2}}{e_{1}^{2} {({(2]}_{e}^{γ})}^{2}} + \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}} .

(69)

Subtracting (61) from (58), we have

\begin{matrix} 2 (e_{1} + e_{2} + e_{3}) {(4]}_{e}^{γ} υ_{4} & = \frac{{(3]}_{e}^{γ} (4 e_{1} + 2 e_{2}) {(B_{1} (s | q))}^{3} ϱ_{1}^{3}}{8 e_{1}^{2} {({(2]}_{e}^{γ})}^{2}} - \frac{{(B_{1} (s | q))}^{3} ϱ_{1}^{3}}{4 e_{1}^{2}} \\ - \frac{5 {(4]}_{e}^{γ} (e_{1} + e_{2} + e_{3}) {(B_{1} (s | q))}^{3} ϱ_{1}^{3}}{4 {({(2]}_{e}^{γ})}^{3} e_{1}^{3}} \\ - \frac{5 {(4]}_{e}^{γ} (e_{1} + e_{2} + e_{3}) {(B_{1} (s | q))}^{2} ϱ_{1} (ϱ_{2} - η_{2})}{8 {(2]}_{e}^{γ} {(3]}_{e}^{γ} e_{1} (e_{1} + e_{2})} \\ + \frac{B_{1} (s | q) (ϱ_{3} - η_{3})}{2} + \frac{[B_{2} (s | q) - B_{1} (s | q)] ϱ_{1} (ϱ_{2} + η_{2})}{2} \end{matrix}

(70)

\begin{matrix} + \frac{(B_{1} (s | q) - 2 B_{2} (s | q) + B_{3} (s | q)) ϱ_{1}^{3}}{4} . \end{matrix}

(71)

Applying triangular inequality and Lemma 1, we have

\begin{matrix} | υ_{4} | & \leq \frac{s^{3} ({(2]}_{e}^{γ} {(3]}_{e}^{γ} (4 e_{1} + 2 e_{2}) - 2 {({(2]}_{e}^{γ})}^{3} e_{1} - 10 (e_{1} + e_{2} + e_{3}) {(4]}_{e}^{γ})}{2 (e_{1} + e_{2} + e_{3}) {({(2]}_{e}^{γ})}^{3} {(4]}_{e}^{γ} e_{1}^{3}} \\ - \frac{5 s^{2}}{2 {(2]}_{e}^{γ} {(3]}_{e}^{γ} e_{1} (e_{1} + e_{2})} + \frac{s^{3} - 2 s - 2 q s - 4}{(e_{1} + e_{2} + e_{3}) {(4]}_{e}^{γ}} . \end{matrix}

□

4. Fekete–Szego Inequalities for the Function Class $Γ_{Σ}^{q} (s, γ, τ)$

Theorem 3.

Let

f \in Γ_{Σ}^{q} (s, γ, τ)

. Then, for some

φ \in R

|υ_{3} - φ υ_{2}^{2}| \leq \{\begin{matrix} 2 | 1 - φ | Ψ_{1} (s, q, γ) & (| 1 - φ | \geq \frac{s}{{[2]}_{b} {[3]}_{b} {(3]}_{e}^{γ} Ψ_{1} (s, q, γ)}) \\ \frac{2 s}{{[2]}_{b} {[3]}_{b} {(3]}_{e}^{γ}} & (| 1 - φ | \leq \frac{s}{{[2]}_{b} {[3]}_{b} {(3]}_{e}^{γ} Ψ_{1} (s, q, γ)}), \end{matrix}

where

Ψ_{1} (s, q, γ) = \frac{s^{3}}{| s^{2} ({[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2}) - {[2]}_{e}^{2} {({(2]}_{e}^{γ})}^{2} (s^{2} - s - 1) |} .

(72)

Proof.

From (42), we have

\begin{matrix} υ_{3} - φ υ_{2}^{2} & = υ_{2}^{2} + \frac{B_{1} (s | q) [ϱ_{2} - η_{2}]}{4 {[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}} - φ υ_{2}^{2} . \end{matrix}

By triangular inequality, we have

| υ_{3} - φ υ_{2}^{2} | \leq \frac{s}{{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}} + | 1 - φ | Ψ_{1} (s, q, γ) .

(73)

Suppose

| 1 - φ | Ψ_{1} (s, q, γ) \geq \frac{s}{{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}}

then we have

| υ_{3} - φ υ_{2}^{2} | \leq 2 | 1 - φ | Ψ_{1} (s, q, γ)

(74)

where

| 1 - φ | \geq \frac{s}{{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} Ψ_{1} (s, q, γ)}

and suppose

| 1 - φ | Ψ_{1} (s, q, γ) \leq \frac{s}{{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}}

then we have

| υ_{3} - δ υ_{2}^{2} | \leq \frac{2 s}{{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ}}

where

| 1 - φ | \leq \frac{s}{{[2]}_{e} {[3]}_{e} {(3]}_{e}^{γ} Ψ_{1} (s, q, γ)}

and

Ψ_{1} (s, q, γ)

is given in (72). □

5. Fekete–Szego Inequalities for the Function Class $Π_{Σ}^{q} (s, γ, τ)$

Theorem 4.

Let

f \in Π_{Σ}^{q} (s, γ, τ)

. Then, for some

φ \in R

|υ_{3} - φ υ_{2}^{2}| \leq \{\begin{matrix} 2 | 1 - φ | X_{1} (s, q, γ) & (| 1 - φ | \geq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ} X_{1} (s, q, γ)}) \\ \frac{2 s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}} & (| 1 - φ | \leq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ} X_{1} (s, q, γ)}), \end{matrix}

where

X_{1} (s, q, γ) = \frac{2 s^{3}}{| s^{2} {2 (e_{1} + e_{2}) {(3]}_{e}^{γ} + {(2]}_{e}^{γ} - {({(2]}_{e}^{γ})}^{2} (1 + 2 e_{1})} - 2 {({(2]}_{e}^{γ})}^{2} e_{1}^{2} (s^{2} - s - 1) |} .

(75)

Proof.

From (67), we have

\begin{matrix} υ_{3} - φ υ_{2}^{2} & = υ_{2}^{2} + \frac{B_{1} (s | q) [ϱ_{2} - η_{2}]}{4 (e_{1} + e_{2}) {(3]}_{e}^{γ}} - φ υ_{2}^{2} . \end{matrix}

By triangular inequality, we have

| υ_{3} - φ υ_{2}^{2} | \leq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}} + | 1 - φ | X_{1} (s, q, γ) .

(76)

Suppose

| 1 - φ | X_{1} (s, q, γ) \geq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}}

then we have

| υ_{3} - φ υ_{2}^{2} | \leq 2 | 1 - φ | X_{1} (s, q, γ)

(77)

where

| 1 - φ | \geq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ} X_{1} (s, q, γ)}

and suppose

| 1 - φ | Ψ_{1} (s, q, γ) \leq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}}

then we have

| υ_{3} - δ υ_{2}^{2} | \leq \frac{2 s}{(e_{1} + e_{2}) {(3]}_{e}^{γ}}

where

| 1 - φ | \leq \frac{s}{(e_{1} + e_{2}) {(3]}_{e}^{γ} X_{1} (s, q, γ)}

and

X_{1} (s, q, γ)

is given in (75). □

6. Conclusions

As we mentioned earlier, q-calculus is a vital tool for understanding a large class of analytic functions and its applications. Several useful results related to the q-version of the starlike function and the q-derivative, bi-univalent functions, for instance, were provided in [26,27,28,29,30,31]. In recent decades, the orthogonal polynomials and special functions have played an essential role in mathematics, physics, engineering, and other research disciplines. In our current analysis, we used q-Hermite polynomials and q-convolution operators and systematically defined two new subclasses of bi-univalent functions, which was primarily prompted by the recent research cited in this paper. We then obtained several significant findings, such as bonds for the initial coefficients of

υ_{2}

υ_{3}

, and

υ_{4}

of the Taylor–Maclaurin series and the Fekete–Szegö functional results for our established function classes.

Moreover, to have more new theorems under the present examinations, new generalizations and applications can be explored with some positive and novel outcomes in various fields of science, mainly in geometric function theory. These recent surveys will be presented in the future research work being processed by the authors of the present paper.

However, the purported trivial (

p, q

)-calculus extension was clearly demonstrated to be a relatively insignificant variation of the classical q-calculus, the extra parameter p being redundant or superfluous (see, for details, [17], p. 340, and [32], pp. 1511–1512). This observation by Srivastava (see [17,32]) will indeed also apply to any future attempt to produce the rather straightforward

(p, q)

-variants of the results which we have presented in this paper.

Author Contributions

Conceptualization, I.A.-S.; Formal analysis, A.C.; Investigation, I.A.-S.; Resources, H.M.S. and N.A.; Data curation, H.M.S.; Writing—original draft, A.C.; Writing—review & editing, N.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Acknowledgments

The last author thanks the Deanship of Scientific Research at Qassim University for supporting her work.

Conflicts of Interest

The authors declare no conflict of interest.

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Al-Shbeil, I.; Cătaş, A.; Srivastava, H.M.; Aloraini, N. Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials. Axioms 2023, 12, 52. https://doi.org/10.3390/axioms12010052

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Al-Shbeil I, Cătaş A, Srivastava HM, Aloraini N. Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials. Axioms. 2023; 12(1):52. https://doi.org/10.3390/axioms12010052

Chicago/Turabian Style

Al-Shbeil, Isra, Adriana Cătaş, Hari Mohan Srivastava, and Najla Aloraini. 2023. "Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials" Axioms 12, no. 1: 52. https://doi.org/10.3390/axioms12010052

APA Style

Al-Shbeil, I., Cătaş, A., Srivastava, H. M., & Aloraini, N. (2023). Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials. Axioms, 12(1), 52. https://doi.org/10.3390/axioms12010052

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Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials

Abstract

1. Introduction and Preliminaries

2. Coefficient Estimates for the Class $Γ_{Σ}^{q} (s, γ, τ)$

3. Coefficient Estimates for the Class $Π_{Σ}^{q} (s, γ, τ)$

4. Fekete–Szego Inequalities for the Function Class $Γ_{Σ}^{q} (s, γ, τ)$

5. Fekete–Szego Inequalities for the Function Class $Π_{Σ}^{q} (s, γ, τ)$

6. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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Coefficient Estimates of New Families of Analytic Functions Associated with q-Hermite Polynomials

Abstract

1. Introduction and Preliminaries

2. Coefficient Estimates for the Class Γ Σ q ( s , γ , τ )

3. Coefficient Estimates for the Class Π Σ q ( s , γ , τ )

4. Fekete–Szego Inequalities for the Function Class Γ Σ q ( s , γ , τ )

5. Fekete–Szego Inequalities for the Function Class Π Σ q ( s , γ , τ )

6. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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2. Coefficient Estimates for the Class $Γ_{Σ}^{q} (s, γ, τ)$

3. Coefficient Estimates for the Class $Π_{Σ}^{q} (s, γ, τ)$

4. Fekete–Szego Inequalities for the Function Class $Γ_{Σ}^{q} (s, γ, τ)$

5. Fekete–Szego Inequalities for the Function Class $Π_{Σ}^{q} (s, γ, τ)$