New Applications of Gegenbauer Polynomials on a New Family of Bi-Bazilevič Functions Governed by the q-Srivastava-Attiya Operator

Abstract

In the present paper, making use of Gegenbauer polynomials, we initiate and explore a new family ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ of holomorphic and bi-univalent functions which were defined in the unit disk $\mathbb{D}$ associated with the q-Srivastava–Attiya operator. We establish the bounds for $|a_2|$ and $|a_3|$, where $a_2$, $a_3$ are the initial Taylor–Maclaurin coefficients. For the new family of functions ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ we investigate the Fekete-Szegö inequality, special cases and consequences.

1. Introduction

We denote by $\mathcal{A}$ the family of holomorphic functions of the form

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,$$

(1)

in the open unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$.

We indicate by $\mathcal{S}$ the subfamily of $\mathcal{A}$ consisting of the functions which are univalent in $\mathbb{D}$.

A function $f\in \mathcal{A}$ is called a Bazilevič function in $\mathbb{D}$ if (see [1])

$$\Re \left(\frac{z^{1-\gamma }{f}^{\prime }\left(z\right)}{{\left(f\left(z\right)\right)}^{1-\gamma }}\right)>0,z\in \mathbb{D};\gamma \geqq 0.$$

The famous Koebe one-quarter theorem [2] ensures that the image of $\mathbb{D}$ under each univalent function $f\in \mathcal{A}$ contains a disk of radius $\frac{1}{4}$. Furthermore, each function $f\in \mathcal{S}$ has an inverse $f^{-1}$ defined by $f^{-1}\left(f\left(z\right)\right)=z$ and

$$f\left(f^{-1}\left(w\right)\right)=w,\left(\left|w\right|<r_0\left(f\right),r_0\left(f\right)\geqq \frac{1}{4}\right)$$

where the inverse of function f has the form

$$f^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\cdots .$$

(2)

We say that the function $f\in \mathcal{A}$ is bi-univalent in the unit disk $\mathbb{D}$ if the functions f and $f^{-1}$ are univalent in $\mathbb{D}$. The family of all bi-univalent functions in $\mathbb{D}$ is denoted by $\mathsf{\Sigma }$.

In fact, Srivastava et al. [3] have actually revived the study of analytic and bi-univalent functions in recent years. This was followed by such works as those by Ali et al. [4], Bulut et al. [5], Srivastava and et al. [6] and others (see, for example, [7,8,9,10,11,12]).

The examples of functions in the family $\mathsf{\Sigma }$ are:

$$\frac{z}{1-z},-log(1-z)\mathrm{and}\frac{1}{2}log\left(\frac{1+z}{1-z}\right);$$

see [3].

We notice that the family $\mathsf{\Sigma }$ is not empty. However, the Koebe function is not a member of $\mathsf{\Sigma }.$

The problem of obtaining the general coefficient bounds on the Taylor–Maclaurin coefficients

$$|a_n|(n\in \mathbb{N};n\geqq 3)$$

for functions $f\in \mathsf{\Sigma }$ is still not completely addressed for many of the subfamilies of $\mathsf{\Sigma }$. The Fekete–Szegö functional $\left|a_3-\eta a_2^2\right|$ for $f\in \mathcal{S}$ is well known for its rich history in the field of geometric function theory. Its origin was in the disproof by Fekete and Szegö [13] of the Littlewood–Paley conjecture that the coefficients of odd univalent functions are bounded by unity. In recent years, many authors have obtained Fekete–Szegö inequalities for different classes of functions (see [14,15,16,17]).

We say that if g is a univalent function in $\mathbb{D}$, then

$$f\prec g(z\in U)⟺f\left(0\right)=g\left(0\right)\mathrm{and}f\left(\mathbb{D}\right)\subseteq g\left(\mathbb{D}\right),$$

see [18].

The q-derivative operator ${\mathfrak{D}}_q$ for a function f is defined as follows:

$${\mathfrak{D}}_{q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z},0<q<1,z\ne 0.$$

For more details about this operator, see the papers [19,20,21,22,23], and for applications of $q-$ calculus associated with various families of analytic, univalent or multivalent functions, see [24,25,26,27].

For a function $f\in \mathcal{A}$, we deduce that

$${\mathfrak{D}}_{q}f\left(z\right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{z}^{n-1},$$

where ${\left[n\right]}_q$ is given by

$${\left[n\right]}_q=\frac{1-q^n}{1-q},forn\in \mathbb{N}.$$

${\left[n\right]}_q$ is called q-analogue of n. As $q⟶1^{-}$, then we have ${\left[n\right]}_q⟶n$ and ${\left[0\right]}_q=0$.

The q-analogue of the Hurwitz–Lerch zeta function found in [28] and defined by the series:

$${\varphi }_q(\mathfrak{s},\mathfrak{t};z)=\sum _{n=0}^{\infty }\frac{z^n}{{[n+\mathfrak{t}]}_q^{\mathfrak{s}}}.$$

(3)

where $\mathfrak{s}\in \mathbb{C},$$\left|z\right|<1,$$\mathfrak{t}\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-},$$\Re \left(\mathfrak{s}\right)>1$ when $\left|z\right|=1$, and the normalized form of the series given by the relation (3) is defined by:

$${\psi }_q(\mathfrak{s},\mathfrak{t};z)={[\mathfrak{t}+1]}_q^{\mathfrak{s}}\left\{{\varphi }_q(\mathfrak{s},\mathfrak{t};z)-{\left[\mathfrak{t}\right]}_q^{-\mathfrak{s}}\right\}=z+\sum _{n=2}^{\infty }{\left(\frac{{[1+\mathfrak{t}]}_q}{{[n+\mathfrak{t}]}_q}\right)}^{\mathfrak{s}}{z}^n.$$

(4)

In [28] the q-Srivastava–Attiya operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f:\mathcal{A}⟶\mathcal{A}$ is defined using the relations (1) and (4) as follows:

$${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)={\psi }_q(\mathfrak{s},\mathfrak{t};z)*f\left(z\right)=z+\sum _{n=2}^{\infty }{\left(\frac{{[\mathfrak{t}+1]}_q}{{[\mathfrak{t}+n]}_q}\right)}^{\mathfrak{s}}{a}_n{z}^n,$$

the symbol * stands for the Hadamard product.

Remark 1.

The q-Srivastava–Attiya operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ is a generalization of several known operators studied in earlier investigations, which are recalled below.

(1)

The operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}},$ for $q⟶1^{-},$ coincides with the Srivastava–Attiya operator; see [29] and for applications of this operator, see [24,30];

(2)

The operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ for $\mathfrak{s}=1$ reduces to the q-Bernardi operator; see [31];

(3)

The operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ for $\mathfrak{s}=\mathfrak{t}=1$ reduces to the q-Libera operator; see [31];

(4)

The operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ for $q⟶1^{-}$ and $\mathfrak{s}=1$ reduces to the Bernardi operator; see [32];

(5)

The operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ for $q⟶1^{-}$, $\mathfrak{s}=1$ and $\mathfrak{t}=0$ reduces to the Alexander operator; see [33].

In [34] the Gegenbauer polynomials $H_{\delta }\left(z,x\right)$ are studied, which are given by the following recurrence relation:

for a nonzero real constant $\delta$, a generating function of Gegenbauer polynomials is defined by

$$H_{\delta }\left(z,x\right)=\frac{1}{{\left(1-2xz+z^2\right)}^{\delta }},$$

where $x\in \left[-1,1\right]$ and $z\in \mathbb{D}$. For a fixed x, the function $H_{\delta }$ is holomorphic in $\mathbb{D}$, so it can be expanded in a Taylor series, with a note that if $x=cos\beta$, where $\beta \in (-\frac{\pi }{3},\frac{\pi }{3})$, then

$$H_{\delta }\left(z,x\right)=\frac{1}{{\left(1-2xz+z^2\right)}^{\delta }}=\sum _{n=0}^{\infty }{\mathcal{G}}_n^{\delta }\left(x\right)z^n,$$

(5)

where ${\mathcal{G}}_n^{\delta }\left(x\right)$ is a Gegenbauer polynomial of degree n.

Clearly, $H_{\delta }$ generates nothing when $\delta =0$. Thus, the generating function of the Gegenbauer polynomial is set to be

$$H_0\left(z,x\right)=1-log\left(1-2xz+z^2\right)=\sum _{n=0}^{\infty }{\mathcal{G}}_n^0\left(x\right)z^n.$$

Furthermore, it is worthwhile to mention that a normalization of $\delta$ to be greater than $-\frac{1}{2}$ is desirable; see [35,36,37,38]. We can also define the Gegenbauer polynomials using the relation of recurrence:

$${\mathcal{G}}_n^{\delta }\left(x\right)=\frac{1}{2}\left[2x\left(n+\delta -1\right){\mathcal{G}}_{n-1}^{\delta }\left(x\right)-\left(n+2\delta -2\right){\mathcal{G}}_{n-1}^{\delta }\left(x\right)\right],$$

with the initial values

$${\mathcal{G}}_0^{\delta }\left(x\right)=1,{\mathcal{G}}_1^{\delta }\left(x\right)=2\delta x\mathrm{and}{\mathcal{G}}_2^{\delta }\left(x\right)=2\delta \left(\delta +1\right)x^2-\delta .$$

(6)

Remark 2.

By choosing the particular values of δ, the Gegenbauer polynomial ${\mathcal{G}}_n^{\delta }\left(x\right)$ leads to well-known polynomials. These special cases are:

• The Chebyshev polynomials, taking $\delta =1$.
• The Legendre polynomials, taking $\delta =\frac{1}{2}$.

2. Main Results

We introduce the family of functions denoted by ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ and defined as follows:

Definition 1.

Assume that $0<\lambda \leqq 1$, $\gamma \geqq 0$ and h is analytic in $\mathbb{D}$, $h\left(0\right)=1$. The family ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ contains all the functions $f\in \mathsf{\Sigma }$ which satisfy the subordinations:

$$\frac{1}{2}\left(\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}+{\left(\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}\right)}^{\frac{1}{\lambda }}\right)\prec h\left(z\right)$$

and

$$\frac{1}{2}\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}+{\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}\right)}^{\frac{1}{\lambda }}\right)\prec h\left(w\right),$$

where the function $f^{-1}$ is given by (2).

Theorem 1.

Assume that $\gamma \geqq 0$ and $0<\lambda \leqq 1$. If the family ${\mathcal{J}}_{\mathsf{\Sigma }}(\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q;h)$ contains all the functions $f\in \mathsf{\Sigma }$ defined by the relation (1), with $h\left(z\right)=1+e_{1}z+e_2{z}^2+\cdots$. Then

$$\left|a_2\right|\leqq \frac{|2\lambda e_1{[2+\mathfrak{t}]}_q^{\mathfrak{s}}|}{(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}(\gamma +1)}$$

(7)

and

$$\begin{array}{c}\left|a_3\right|\leqq max\left\{\left|\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|,\left|\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_2}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}-\frac{2\lambda \mathsf{\Omega }(\lambda ,\gamma ){[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1^2}{\left(\gamma +2\right){\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^3{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|,\right.\hfill \\ \left|\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|,\left|\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_2}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right.\\ \hfill \left.\left.-\frac{\left(8{\lambda }^2(\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}+2\lambda \mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}\right)e_1^2}{{\left(\gamma +1\right)}^2\left(\gamma +2\right){\left[{1+\mathfrak{t}]}_q^{2\mathfrak{s}}(\lambda +1\right)}^3}\right|\right\},\end{array}$$

(8)

where

$$\mathsf{\Omega }(\lambda ,\gamma )=\lambda (\gamma -1)(\gamma +2)(\lambda +1)+(1-\lambda ){\left(\gamma +1\right)}^2.$$

(9)

Proof. 

Suppose that $f\in {\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;e_1;e_2)$. Then there exists the functions $\phi ,\chi :\mathbb{D}⟶\mathbb{D},$ holomorphically, where

$$\phi \left(z\right)=r_{1}z+r_2{z}^2+r_3{z}^3+\cdots (z\in \mathbb{D})$$

(10)

and

$$\chi \left(w\right)=s_{1}w+s_2{w}^2+s_3{w}^3+\cdots (w\in \mathbb{D}),$$

(11)

with $\varphi \left(0\right)=\psi \left(0\right)=0$, $\left|\phi \left(z\right)\right|<1$, $\left|\chi \left(w\right)\right|<1$, $z,w\in \mathbb{D}$ such that

$$\frac{1}{2}\left(\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}+{\left(\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}\right)}^{\frac{1}{\lambda }}\right)=1+e_1\phi \left(z\right)+e_2{\phi }^2\left(z\right)+\cdots$$

(12)

and

$$\frac{1}{2}\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}+{\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}\right)}^{\frac{1}{\lambda }}\right)=1+e_1\chi \left(w\right)+e_2{\chi }^2\left(w\right)+\cdots .$$

(13)

Combining (10)–(13), we yield

$$\frac{1}{2}\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}+{\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}\right)}^{\frac{1}{\lambda }}\right)=1+e_1{r}_{1}z+\left[e_1{r}_2+e_2{r}_1^2\right]z^2+\cdots$$

(14)

and

$$\frac{1}{2}\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}+{\left(\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}{f}^{-1}\left(w\right)\right)}^{1-\gamma }}\right)}^{\frac{1}{\lambda }}\right)=1+e_1{s}_{1}w+\left[e_1{s}_2+e_2{s}_1^2\right]w^2+\cdots .$$

(15)

If if the inequalities $\left|\phi \left(z\right)\right|<1$ and $\left|\chi \left(w\right)\right|<1,$$z,w\in \mathbb{D}$ are true, we know that

$$\left|r_j\right|\le 1,\left|s_j\right|\le 1,$$

(16)

for all $j\in \mathbb{N}$. For more details, see [2].

We obtain the next relation

$$\frac{(\lambda +1)(\gamma +1){[\mathfrak{t}+1]}_q^{\mathfrak{s}}}{2{[\mathfrak{t}+2]}_q^{\mathfrak{s}}\lambda }{a}_2=e_1{r}_1,$$

(17)

after simplifying in the relations (14) and (15)

$$\begin{array}{c}\frac{(\lambda +1)(\gamma +2){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{a}_3+\frac{\left[\lambda (\lambda +1)(\gamma -1)(\gamma +2)-(\lambda -1){\left(\gamma +1\right)}^2\right]{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}{4{\lambda }^2{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{a}_2^2\hfill \\ \hfill =e_1{r}_2+e_2{r}_1^2,\end{array}$$

(18)

$$-\frac{(1+\gamma ){[\mathfrak{t}+1]}_q^{\mathfrak{s}}(1+\lambda )}{2{[\mathfrak{t}+2]}_q^{\mathfrak{s}}\lambda }{a}_2=e_1{s}_1$$

(19)

and

$$\begin{array}{c}\frac{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\left(2a_2^2-a_3\right)+\frac{\left[\lambda (\gamma +2)(\gamma -1)(\lambda +1)+{\left(\gamma +1\right)}^2(1-\lambda )\right]{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}{4{\lambda }^2{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{a}_2^2\hfill \\ \hfill =e_1{s}_2+e_2{s}_1^2.\end{array}$$

(20)

Inequality (7) follows from (17) and (19). We deduce from the relations (17) and (18) that

$$\frac{(1+\lambda )(2+\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{a}_3=r_2+\left(\frac{e_2}{e_1}-\frac{\mathsf{\Omega }(\lambda ,\gamma )e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2}\right)r_1^2,$$

(21)

where $\mathsf{\Omega }(\lambda ,\gamma )$ is given by (9). By using the known sharp result (see in [18]):

$$|r_2-\mu r_1^2|\leqq max\left\{1,\left|\mu \right|\right\}$$

(22)

for all $\mu \in \mathbb{C}$, we obtain

$$\left|\frac{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}\right|\left|a_3\right|\leqq max\left\{1,\left|\frac{e_2}{e_1}-\frac{\mathsf{\Omega }(\lambda ,\gamma )e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2}\right|\right\}.$$

(23)

Following from (19) and (20), we deduce that

$$\begin{array}{c}\hfill -\frac{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{a}_3=s_2+\left(\frac{e_2}{e_1}-\frac{\left(4\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}+\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}\right)e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\right)s_1^2.\end{array}$$

(24)

Applying (22), we obtain

$$\begin{array}{cc}\left|\frac{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}\right|\left|a_3\right|\leqq \hfill & max\left\{1,\left|\frac{e_2}{e_1}\right.\right.\hfill \\ & \hfill \left.\left.-\frac{\left(4\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}+\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}\right)e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|\right\}.\end{array}$$

(25)

Inequality (8) follows from (23) and (25). □

If we take the generating function (5) of the Gegenbauer polynomials ${\mathcal{G}}_n^{\delta }\left(x\right)$ as $h\left(z\right)$, then from (6), we have $e_1=2\delta x$ and $e_2=2\delta \left(\delta +1\right)x^2-\delta$, and we obtain the next corollary.

Corollary 1.

If the class ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;H_{\delta }\left(z,x\right))$ contains all the functions $f\in \mathsf{\Sigma }$ given by (1), then

$$\left|a_2\right|\leqq \frac{{|4\lambda \delta x[2+\mathfrak{t}]}_q^{\mathfrak{s}}|}{(\gamma +1)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}$$

and

$$\begin{array}{c}\left|a_3\right|\leqq max\left\{\left|\frac{4\lambda \delta x{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|,\right.\hfill \\ \hfill \left|\frac{2\lambda \left(2\delta \left(\delta +1\right)x^2-\delta \right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}-\frac{8\lambda {\delta }^2{x}^2\mathsf{\Omega }(\lambda ,\gamma ){[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left(\gamma +2\right){\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^3{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|,\\ \hfill \left|\frac{4\lambda \delta x{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|,\left|\frac{2\left(2\delta \left(\delta +1\right)x^2-\delta \right)\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right.\\ \hfill \left.\left.-\frac{8\lambda {\delta }^2{x}^2\left(4\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}+\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}\right)}{{\left(1+\gamma \right)}^2\left(2+\gamma \right){\left(1+\lambda \right)}^3{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}\right|\right\},\end{array}$$

for all $\lambda ,\gamma ,\mathfrak{t},\mathfrak{s},q,x$ such that $0<\lambda \leqq 1$,$\mathfrak{t}\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$, $\gamma \geqq 0$, $\mathfrak{s}\in \mathbb{C}$,$x\in \mathbb{R}$, $0<q<1$, where $H_{\delta }\left(z,x\right)$ is given by (5).

The Fekete–Szegö inequality for the family functions ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ is given in the next theorem.

Theorem 2.

If the class ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ contains all the functions $f\in \mathsf{\Sigma }$ given by the relation (1), then

$$\begin{array}{c}\left|a_3-\eta a_2^2\right|\leqq \left|\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|\times \hfill \\ \hfill \times max\left\{1,\left|\frac{e_2}{e_1}-\frac{\left(\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}-2\eta \lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)e_1}{{\left(1+\lambda \right)}^2{\left(1+\gamma \right)}^2{[\mathfrak{t}+3]}_q^{\mathfrak{s}}{[\mathfrak{t}+1]}_q^{\mathfrak{s}}}\right|,\right.\\ \hfill \left.\left|\frac{e_2}{e_1}-\frac{\left(\mathsf{\Omega }(\lambda ,\gamma ){[\mathfrak{t}+1]}_q^{\mathfrak{s}}{[\mathfrak{t}+3]}_q^{\mathfrak{s}}+2(2-\eta )\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)e_1}{{\left(1+\gamma \right)}^2{\left(1+\lambda \right)}^2{[\mathfrak{t}+1]}_q^{\mathfrak{s}}{[\mathfrak{t}+3]}_q^{\mathfrak{s}}}\right|\right\},\end{array}$$

(26)

for all $\lambda ,\gamma ,\mathfrak{t},\mathfrak{s},q,x$ such that $0<\lambda \leqq 1$, $\gamma \geqq 0$, $\mathfrak{s}\in \mathbb{C}$, $\mathfrak{t}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$, $\eta \in \mathbb{C}$ and $0<q<1.$

Proof. 

We apply the notations from the proof of Theorem 1. From (17), (18) and (21), we have

$$\begin{array}{c}{a}_3-\eta a_2^2=\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\times \hfill \\ \hfill \times \left(r_2+\left(\frac{e_2}{e_1}-\frac{\left(\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}-2\eta \lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\right)r_1^2\right)\end{array}$$

and by using the known sharp result $|r_2-\mu r_1^2|\leqq max\left\{1,\left|\mu \right|\right\}$, we get

$$\begin{array}{cc}|a_3-\eta a_2^2|\leqq \hfill & |\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}|\times \\ & \hfill \times max\left\{1,\left|\frac{e_2}{e_1}-\frac{\left(\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}-2\eta \lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|\right\}.\end{array}$$

(27)

In the same way, from (19), (20) and (24), we conclude that

$$\begin{array}{c}{a}_3-\eta a_2^2=-\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\left(s_2+\left(\frac{e_2}{e_1}\right.\right.\hfill \\ \hfill \left.\left.-\frac{\left(\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}-2(\eta -2)\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)e_1}{{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\right)s_1^2\right)\end{array}$$

and by using $|s_2-\mu s_1^2|\leqq max\left\{1,\left|\mu \right|\right\}$, we get

$$\begin{array}{c}|a_3-\eta a_2^2|\leqq |\frac{2\lambda {[3+\mathfrak{t}]}_q^{\mathfrak{s}}{e}_1}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}|max\left\{1,\left|\frac{e_2}{e_1}\right.\right.\hfill \\ \hfill \left.\left.-\frac{\left(\mathsf{\Omega }(\lambda ,\gamma ){[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[3+\mathfrak{t}]}_q^{\mathfrak{s}}-2e_1(\eta -2)\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)}{{\left(1+\lambda \right)}^2{\left(1+\gamma \right)}^2{[\mathfrak{t}+1]}_q^{\mathfrak{s}}{[\mathfrak{t}+3]}_q^{\mathfrak{s}}}\right|\right\}.\end{array}$$

(28)

Inequality (26) follows from (27) and (28). □

Corollary 2.

If the class ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;H_{\delta }\left(z,x\right))$ contains all the functions $f\in \mathsf{\Sigma }$ given by the relations (1), then

$$\begin{array}{c}\left|a_3-\eta a_2^2\right|\leqq \left|\frac{4\lambda \delta x{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{(\gamma +2)(\lambda +1){[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right|\times \hfill \\ \times max\left\{1,\left|\frac{2\left(\delta +1\right)x^2-1}{2x}-\frac{2\delta x\left(\mathsf{\Omega }(\lambda ,\gamma ){[\mathfrak{t}+1]}_q^{\mathfrak{s}}{[\mathfrak{t}+3]}_q^{\mathfrak{s}}-2\eta \lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)}{{\left(1+\lambda \right)}^2{\left(1+\gamma \right)}^2{[\mathfrak{t}+1]}_q^{\mathfrak{s}}{[\mathfrak{t}+3]}_q^{\mathfrak{s}}}\right|,\right.\\ \hfill \left.\left|\frac{2\left(1+\delta \right)x^2-1}{2x}-\frac{2\delta x\left(\mathsf{\Omega }(\lambda ,\gamma ){[\mathfrak{t}+1]}_q^{\mathfrak{s}}{[\mathfrak{t}+3]}_q^{\mathfrak{s}}-2(\eta -2)\lambda (\gamma +2)(\lambda +1){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\right)}{{[\mathfrak{t}+1]}_q^{\mathfrak{s}}{\left(\gamma +1\right)}^2{\left(\lambda +1\right)}^2{[\mathfrak{t}+3]}_q^{\mathfrak{s}}}\right|\right\},\end{array}$$

for all $\lambda ,\gamma ,\mathfrak{t},\mathfrak{s},q,x$ such that $0<\lambda \leqq 1$, $\gamma \geqq 0$, $\mathfrak{t}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$, $\mathfrak{s}\in \mathbb{C}$, $0<q<1$, $\eta \in \mathbb{C}$ and $x\in \mathbb{R}$, where $H_{\delta }\left(z,x\right)$ is given by (5).

3. Conclusions

We establish in this work a new family ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ of bi-univalent and holomorphic functions defined by the q-Srivastava–Attiya operator and using the Gegenbauer polynomials ${\mathcal{G}}_n^{\delta }\left(x\right)$, which are given by the recurrence relation (6) and generating the function $H_{\delta }\left(z,x\right)$ in (5). We derived initial Taylor–Maclaurin coefficient inequalities for functions belonging to this newly introduced bi-univalent function family ${\mathcal{J}}_{\mathsf{\Sigma }}(\lambda ,\gamma ,\mathfrak{s},\mathfrak{t},q;h)$ and viewed the famous Fekete–Szegö problem.