Estimates for Coefficients of Bi-Univalent Functions Associated with a Fractional q-Difference Operator

Abstract

In the present paper, we discuss a class of bi-univalent analytic functions by applying a principle of differential subordinations and convolutions. We also formulate a class of bi-univalent functions influenced by a definition of a fractional q-derivative operator in an open symmetric unit disc. Further, we provide an estimate for the function coefficients $|a_2|$ and $|a_3|$ of the new classes. Over and above, we study an interesting Fekete–Szego inequality for each function in the newly defined classes.

1. Preliminaries

In mathematics, functions and symmetric functions are very common in theory and applications. They are applied to various fields including group theory, Lie algebras and the algebraic geometry and may others, to mention but a few. In the concept of geometric functions, the set $\mathcal{A}$ has has been introduced as the set of all analytic class of normalized functions defined on the open symmetric disc $D=\{z\in \mathbb{C},|z|<1\}$ such that they possess the following formula (see [1])

$$\begin{array}{c}\hfill f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n.\end{array}$$

(1)

The univalent class of functions in $\mathcal{A}$ is denoted by S. The best known subclasses of S containing starlike, close to convex and convex functions, respectively, are denoted by ST, C and CV. If f is a function satisfying (1) and g is another function defined by

$$\begin{array}{c}\hfill g\left(z\right)=z+\sum _{n=2}^{\infty }{b}_n{z}^n,\end{array}$$

then the Hadamard product or the convolution of the functions f and g is offered by Ruscheweyh [2] as

$$\begin{array}{c}\hfill (f\ast g)\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{b}_n{z}^n.\end{array}$$

Assume $\mathcal{B}$ represents the class of Schwartz functions $w\left(z\right)$ in the set $\mathcal{A}$ such that

$$\begin{array}{c}\hfill \mathcal{B}=\{w:w\in \mathcal{A},w(0)=0,|w\left(z\right)|<1,(\forall z\in D\left)\right\}.\end{array}$$

Then, for two arbitrary analytic functions f and g in $\mathcal{A}$, we say f is the subordinate function to the function g, expressed as $f\prec g$ or $f\left(z\right)\prec g\left(z\right)$, if there can be found a function $w\in \mathcal{B}$ such that $f\left(z\right)=g\left(w\right(z\left)\right)$ (see [1]). A unified treatment of the familiar subclass of univalent functions has been considered by Ma and Minda [3], following a principle of differential sub-ordinations. Here, we present $\mathcal{P}$ as the class of functions p which are analytic in D provided $p\left(0\right)=1$ and $Rep\left(z\right)>0$ and, for each $z\in D$, the functions $p\left(z\right)$ can be written in the form

$$\begin{array}{c}\hfill p\left(z\right)=1+\sum _{n=1}^{\infty }{p}_n{z}^n.\end{array}$$

(2)

Such a class of functions is the so-called Caratheodory class of functions (see [1]).

Lemma 1

([1]). Let the function $p\in P$ satisfy (2). Then, we have

$$\begin{array}{c}\hfill |p_n|\le 2,(n\in \mathbb{N})\end{array}$$

(3)

and the bounds are sharp.

Due to the one-quarter theorem of Koebe (see [4]), it is shown that every function f in S has an inverse function $f^{-1}$ such that

$$\begin{array}{c}\hfill f^{-1}\left(f\left(z\right)\right)=z\mathrm{and}(z\in D),f\left(f^{-1}\left(w\right)\right)=w\mathrm{provided}\left|w\right|<r_0\left(f\right),r_0\left(f\right)⩾\frac{1}{4},\end{array}$$

where

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

We say a function f in the set $\mathcal{A}$ is bi-univalent in D if $D\subseteq f\left(D\right)$ and both of the functions f and its inverse $f^{-1}$ are univalent functions in D. The class of such bi-univalent functions defined in D is denoted by Σ. For a useful summary and applicable examples in the class of bi-univalent functions we refer to [5]. The class Σ of bi-univalent functions was discussed by Lewin [6] to derive the second coefficient bound. Srivastava et al. [7] have introduced and investigated a new class of analytic and bi-univalent functions. In recent years, some researchers have studied a number of different subclasses of Σ in the context of theory of geometric functions [8,9,10,11].

In recent decades, the fractional q-calculus was applied in the approximation theory which has a new generalization of the classical operator. The concept of q-calculus operator has been broadly been applied in various fields including optimal control, q-difference, hypergeometric series, quantum physics, fractional subdiffusion equations and q-integral equations [12,13,14,15,16]. The concept of p-calculus was first applied in this context by Lupas [17]. Jackson in [18] defined the q-analogues of the ordinary derivative. He also studied the q-integral operator and investigated some applications of this theory.

Nowadays, a number of researchers have studied q-analogues of an analytic class of bi-univalent functions. We may refer readers to Darus [19] who investigated a generalized q-differential operator by utilizing the q-hypergeometric functions. Furthermore, he worked on an operator to derive some useful applications. Zhang et al. [20] discussed a q-Starlike functions associated with generalized conic domain ${\mathsf{\Omega }}_{k,\alpha }$. Kanas et al. [21] defined the operator and later, Arif et al. [22] extended this operator for multivalent functions. Srivastava et al. [23] defined the q-starlike class of functions over conic regions. Furthermore, they presented some of their applications. Srivastava et al. [24] studied q-Noor integral operators and some of their applications. The q-calculus theory in a fractional sense and its real applications in the geometric class of functions of complex analysis and related fields are investigated in [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41].

Let q be a fixed number with $0<q<1$ and $n\in \mathbb{N}$. Then, we recall the following useful definitions and notations (see [18])

$$\begin{array}{c}\hfill {(a;q)}_n=\prod _{i=0}^{n-1}(1-q^{i}a),{(a;q)}_0=1,{(a;q)}_{\infty }=\prod _{i=0}^{\infty }(1-q^{i}a).\end{array}$$

The q-analogue of the Γ-function is defined by

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_q(1-\alpha )=\frac{{(q;q)}_{\infty }}{{(q^{1-\alpha };q)}_{\infty }}{(1-q)}^{\alpha },0<\alpha <1.\end{array}$$

The q-analogue of the difference operator of non-integer order $\alpha$ is defined by

$$\begin{array}{c}\hfill D_q^{\alpha }f\left(z\right)=\frac{1}{{(1-q)}^{\alpha }{z}^{\alpha }}\sum _{n=0}^{\infty }\frac{{(q^{-\alpha };q)}_n}{{(q;q)}_n}{q}^{n}f\left(q^{n}z\right),\end{array}$$

where $\alpha \ne 0$ and $q\in (0,1)$.

Note that $D_q^{\alpha }f\left(z\right)⟶D_{q}f\left(z\right)$ when $\alpha ⟶1$, where $D_f\left(z\right)$ is the q-derivative of f, introduced in [5] as

$$\begin{array}{c}\hfill D_{q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{z(1-q)},z\in D,‘z\ne 0,\left|q\right|<1.\end{array}$$

For $n\in \mathbb{N}$, the following equation is satisfied

$$\begin{array}{c}\hfill D_q^{\alpha }{z}^n=\frac{z^{n-\alpha }}{{\mathrm{\Gamma }}_q(1-\alpha )}\frac{{(q;q)}_n}{{(q^{1-\alpha };q)}_n},0<\alpha <1.\end{array}$$

Therefore for $f\in \mathcal{A}$, we obtain that

$$\frac{{\mathrm{\Gamma }}_q(1-\alpha )}{z^{-\alpha }}\frac{1-q^{1-\alpha }}{1-q}{D}_q^{\alpha }f\left(z\right)=f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right),0<\alpha <1,$$

provided

$$I_{\alpha ,q}^{‡}\left(z\right)=z+\sum _{n=2}^{\infty }\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n(1-q)}{z}^n,0<\alpha <1.$$

Stimulated by the previous result in the paper, we estimate the initial coefficients for a new subclass of bi-univalent functions defined by q-analogue difference operator with non-integer order $\alpha$. Additionally, we establish corresponding Fekete–Szego type fractional inequality for this functions. Furthermore, we estimate bounds for $|a_2|$ and $|a_3|$.

2. Coefficient Estimates for the Function Class ${\mathrm{\Sigma }}_{\mathbf{q}}^{\mathbf{\alpha }}$

In this section, the coefficients $|a_2|$ and $|a_3|$ for a function are estimated in the class ${\mathrm{\Sigma }}_q^{\alpha }$ in some details.

Definition 1.

Let a function f be estimated by (1) and $g\left(w\right)=f^{-1}\left(w\right)$. Then $f\in {\mathrm{\Sigma }}_q^{\alpha },0<\alpha <1$, provided the following subsequent subordination conditions hold:

$$\begin{array}{c}\hfill \frac{1}{z}\left[qf\left(z\right)+(1-q)f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right]\prec \widehat{\psi }\left(z\right),(\alpha >0,z\in D),\end{array}$$

(4)

and

$$\begin{array}{c}\hfill \frac{1}{w}\left[qg\left(w\right)+(1-q)g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right]\prec \widehat{\psi }\left(w\right),(\alpha >0,z\in D),\end{array}$$

(5)

where the functions $\widehat{\psi }\left(z\right):D\to \mathbb{C}$ are analytic of positive real parts. Consequently, we have the appropriate form:

$$\begin{array}{c}\hfill \widehat{\psi }\left(z\right)=1+{\widehat{\psi }}_1\left(z\right)+{\widehat{\psi }}_2\left(z\right)+{\widehat{\psi }}_3\left(z\right)+\cdots ,({\widehat{\psi }}_1>0).\end{array}$$

(6)

Theorem 1.

Let a function f be estimated by (1) and $g=f^{-1}$. If $f\in {\mathrm{\Sigma }}_q^{\alpha },0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_2|\le 2\sqrt{\frac{{\widehat{\psi }}_1^3}{4{\mathsf{\Lambda }}_3{\widehat{\psi }}_1^2+{\mathsf{\Lambda }}_2^2({\widehat{\psi }}_1-{\widehat{\psi }}_2)}},\end{array}$$

(7)

and

$$\begin{array}{c}\hfill |a_3|\le {\widehat{\psi }}_1\left(\frac{4{\widehat{\psi }}_1}{{\mathsf{\Lambda }}_2^2}+\frac{1}{{\mathsf{\Lambda }}_3}\right),\end{array}$$

(8)

where ${\mathsf{\Lambda }}_n=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n},n=2,3$.

Proof. 

Since $g=f^{-1}$, the right-hand side of (4) and (5) have the expansions

$$\begin{array}{c}\hfill \frac{1}{z}\left[qf\left(z\right)+(1-q)f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right]=1+{\mathsf{\Lambda }}_2{a}_{2}z+{\mathsf{\Lambda }}_3{a}_3{z}^2+\dots \end{array}$$

(9)

and

$$\begin{array}{ccc}\hfill \frac{1}{w}\left[qg\left(w\right)+(1-q)g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right]& =& 1-{\mathsf{\Lambda }}_2{a}_{2}w+{\mathsf{\Lambda }}_3(2a_2^2-a_3)w^2\hfill \\ & -& {\mathsf{\Lambda }}_4({52}_2^2-5a_2{a}_3+a_4)w^3+\dots \hfill \end{array}$$

(10)

By using the definition of subordination, we derive

$$\begin{array}{c}\hfill \frac{1}{z}\left[qf\left(z\right)+(1-q)f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right]=\widehat{\psi }\left(\varphi \left(z\right)\right),\end{array}$$

(11)

$$\begin{array}{c}\hfill \frac{1}{w}\left[qg\left(w\right)+(1-q)g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right]=\widehat{\psi }\left(\widehat{\varphi }\left(w\right)\right),\end{array}$$

(12)

where $\varphi \left(z\right),\widehat{\varphi }\left(z\right)\in \mathcal{B}$. We define two functions $u_1,u_2$ as the subsequent form

$$\begin{array}{c}\hfill u_1\left(z\right)=\frac{1+\varphi \left(z\right)}{1-\varphi \left(z\right)},u_2\left(w\right)=\frac{1+\widehat{\varphi }\left(w\right)}{1-\widehat{\varphi }\left(w\right)},(z,w\in D).\end{array}$$

It can be easily shown that $u_1,u_2\in \mathcal{P}$ and write the relations of $\varphi \left(z\right)$ and $\widehat{\varphi }\left(z\right)$ of the restated subsequent form

$$\begin{array}{c}\hfill \varphi \left(z\right)=\frac{u_1\left(z\right)-1}{u_1\left(z\right)+1}=\frac{1}{2}\left({\varphi }_{1}z+\left({\varphi }_2-\frac{{\varphi }_1^2}{2}\right)z^2+\dots \right),\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \widehat{\varphi }\left(w\right)=\frac{u_2\left(w\right)-1}{u_2\left(w\right)+1}=\frac{1}{2}\left({\widehat{\varphi }}_{1}w+\left({\widehat{\varphi }}_2-\frac{{\widehat{\varphi }}_1^2}{2}\right)w^2+\dots \right).\end{array}$$

(14)

Therefore, from Equations (11), (13) and (14), we obtain

$$\begin{array}{c}\hfill \widehat{\psi }\left(\varphi \left(z\right)\right)=\widehat{\psi }\left(\frac{u_1\left(z\right)-1}{u_1\left(z\right)+1}\right)=1+\frac{1}{2}{\widehat{\psi }}_1{\varphi }_{1}z+\left(\frac{1}{2}{\widehat{\psi }}_1\left({\varphi }_2-\frac{{\varphi }_1^2}{4}\right)+\frac{1}{4}{\widehat{\psi }}_2{\varphi }_1^2\right)z^2+\dots \end{array}$$

(15)

and

$$\begin{array}{c}\hfill \widehat{\psi }\left(\widehat{\varphi }\left(w\right)\right)=\widehat{\psi }\left(\frac{u_2\left(w\right)-1}{u_2\left(w\right)+1}\right)=1+\frac{1}{2}{\widehat{\psi }}_1{\widehat{\varphi }}_{1}w+\left(\frac{1}{2}{\widehat{\psi }}_1\left({\widehat{\varphi }}_2-\frac{{\widehat{\varphi }}_1^2}{4}\right)+\frac{1}{4}{\widehat{\psi }}_2{\widehat{\varphi }}_1^2\right)w^2+\dots \end{array}$$

(16)

Equating the corresponding coefficients (9) and (15) reveals

$$\begin{array}{c}\hfill a_2=\frac{1}{{\mathsf{\Lambda }}_2}{\widehat{\psi }}_1{\varphi }_1,\end{array}$$

(17)

$$\begin{array}{c}\hfill a_3{\mathsf{\Lambda }}_3=\frac{1}{2}{\widehat{\psi }}_1({\varphi }_2-\frac{{\varphi }_1^2}{2})+\frac{1}{4}{\widehat{\psi }}_2{\varphi }_1^2.\end{array}$$

(18)

Once again, equating the corresponding coefficients (10) and (16) implies

$$\begin{array}{c}\hfill a_2=-\frac{1}{{\mathsf{\Lambda }}_2}{\widehat{\psi }}_1{\widehat{\varphi }}_1,\end{array}$$

(19)

$$\begin{array}{c}\hfill (2a_2^2-a_3){\mathsf{\Lambda }}_3=\frac{1}{2}{\widehat{\psi }}_1({\widehat{\varphi }}_2-\frac{{\widehat{\varphi }}_1^2}{2})+\frac{1}{4}{\widehat{\psi }}_2{\widehat{\varphi }}_1^2.\end{array}$$

(20)

Comparing the coefficients (17) and (19) yields

$$\begin{array}{c}\hfill {\varphi }_1=-{\widehat{\varphi }}_1\end{array}$$

(21)

and

$$\begin{array}{c}\hfill a_2^2=\frac{{\widehat{\psi }}_1^2}{2{\mathsf{\Lambda }}_2^2}({\varphi }_1^2+{\widehat{\varphi }}_1^2).\end{array}$$

(22)

Moreover, by using (20)–(22), the above reveals

$$\begin{array}{c}\hfill a_2^2=\frac{{\widehat{\psi }}_1^3({\varphi }_2+{\widehat{\varphi }}_2)}{4{\mathsf{\Lambda }}_3{\widehat{\psi }}_1^2+{\mathsf{\Lambda }}_2^2({\widehat{\psi }}_1-{\widehat{\psi }}_2)},\end{array}$$

which, by applying Lemma 1, we derive the desired estimate on $|a_2|$ as presented in (7).

To complete the proof of this theorem, we find the estimate on $|a_3|$. Subtracting Equation (20) from Equation (18) and using (21), we get

$$\begin{array}{c}\hfill a_3=a_2^2+\frac{{\widehat{\psi }}_1({\varphi }_2-{\widehat{\varphi }}_2)}{4{\mathsf{\Lambda }}_3}.\end{array}$$

(23)

Hence, following Equations (22) and (23) gives

$$\begin{array}{c}\hfill a_3=\frac{{\widehat{\psi }}_1^2}{2{\mathsf{\Lambda }}_2^2}({\varphi }_1^2+{\widehat{\varphi }}_1^2)+\frac{{\widehat{\psi }}_1({\varphi }_2-{\widehat{\varphi }}_2)}{4{\mathsf{\Lambda }}_3}.\end{array}$$

Finally, by applying Lemma 1 on the coefficients of ${\varphi }_2$ and ${\widehat{\varphi }}_2$ in the last equation, we derive the desired estimate on $|a_3|$ as presented in (8).  □

In the next theorem, we calculate the Fekete–Szego result for the class ${\mathrm{\Sigma }}_q^{\alpha }$.

Theorem 2.

Let a function f be given by (1), $g=f^{-1}$ and $\lambda \in \mathbb{C}$. If $f\in {\mathrm{\Sigma }}_q^{\alpha },0<\alpha <1$ then

$$\begin{array}{c}\hfill |a_3-\lambda a_2^2|\le \left\{\begin{array}{cc}\frac{p_1}{4{\mathsf{\Lambda }}_3},\hfill & 0\le \psi \left(\lambda \right)<\frac{1}{4{\mathsf{\Lambda }}_3},\hfill \\ 4{\widehat{\psi }}_1\psi \left(\lambda \right),\hfill & \psi \left(\lambda \right)\ge \frac{1}{4{\mathsf{\Lambda }}_3},\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \psi \left(\lambda \right)={\displaystyle \frac{{\widehat{\psi }}_1^2(1-\lambda )}{4{\mathsf{\Lambda }}_3{\widehat{\psi }}_1^2+{\mathsf{\Lambda }}_2^2({\widehat{\psi }}_1-{\widehat{\psi }}_2)}},\end{array}$$

(24)

where ${\mathsf{\Lambda }}_n=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n},n=2,3$.

Proof. 

By using Equations (22) and (23), we get

$$\begin{array}{c}\hfill a_3-\lambda a_2^2=\frac{{\widehat{\psi }}_1(r_2-s_2)}{4{\mathsf{\Lambda }}_3}+(1-\lambda )a_2^2.\end{array}$$

The preceding equation is indeed equivalent to

$$\begin{array}{ccc}\hfill a_3-\lambda a_2^2& =& \frac{{\widehat{\psi }}_1(r_2-s_2)}{4{\mathsf{\Lambda }}_3}+\frac{(1-\lambda ){\widehat{\psi }}_1^3(r_2+s_2)}{4{\mathsf{\Lambda }}_3{\widehat{\psi }}_1^2+{\mathsf{\Lambda }}_2^2({\widehat{\psi }}_1-{\widehat{\psi }}_2)}.\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill a_3-\eta a_2^2={\widehat{\psi }}_1\left[\left(\psi \left(\lambda \right)+\frac{1}{4{\mathsf{\Lambda }}_3}\right)r_2+\left(\psi \left(\lambda \right)-\frac{1}{4{\mathsf{\Lambda }}_3}\right)s_2\right],\end{array}$$

where $\psi \left(\lambda \right)$ is defined in (24). Since ${\widehat{\psi }}_1>0$ and the coefficient ${\widehat{\psi }}_2$ is a real number, then we have

$$\begin{array}{c}\hfill |a_3-\eta a_2^2|={\widehat{\psi }}_1\left|\left(\psi \left(\lambda \right)+\frac{1}{4{\mathsf{\Lambda }}_3}\right)r_2+\left(\psi \left(\lambda \right)-\frac{1}{4{\mathsf{\Lambda }}_3}\right)s_2\right|.\end{array}$$

This completes the proof of Theorem 4. □

Remark 1.

Let $\lambda =1$ then, if $f\in {\mathrm{\Sigma }}_q^{\alpha },0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_3-a_2^2|\le \frac{{\widehat{\psi }}_1}{{\mathsf{\Lambda }}_3},\end{array}$$

where ${\mathsf{\Lambda }}_3=q+\frac{{(q;q)}_3(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_3}$.

This remark is a straightforward consequence of Theorem 4.

Applications of Coefficient Estimates for ${\mathrm{\Sigma }}_q^{\alpha }$

Definition 2.

Let a function f be estimated by (1), $g=f^{-1}$ and $0\le \zeta <1$. Then $f\in {\mathrm{\Sigma }}_q^{\alpha }\left(\zeta \right)$, $0<\alpha <1$ if the following subsequent subordination conditions hold:

$$\begin{array}{c}\hfill \frac{1}{z}\left[qf\left(z\right)+(1-q)f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right]\prec {\widehat{v}}_1\left(z\right),(z\in D),\end{array}$$

and

$$\begin{array}{c}\hfill \frac{1}{w}\left[qg\left(w\right)+(1-q)g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right]\prec {\widehat{v}}_1\left(w\right),(z\in D),\end{array}$$

where ${\widehat{v}}_1$ is function estimated as

$$\begin{array}{c}\hfill {\widehat{v}}_1\left(z\right)=\frac{1+(1-(1+q\left)\zeta \right)z}{1-qz}=1+(1+q)(1-\zeta )z+q(1+q)(1-\zeta )z^2+\cdots \end{array}$$

(25)

Hence, it is evident that

$$\begin{array}{c}\hfill {\widehat{\psi }}_1=(1+q)(1-\zeta ),{\widehat{\psi }}_2=q(1+q)(1-\zeta ).\end{array}$$

Corollary 1.

Let a function f be estimated by (1), $g=f^{-1}$ and $0\le \zeta <1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha }\left(\zeta \right)$, $0<\alpha <1$, then

$$\begin{array}{c}\hfill a_2|\le \sqrt{\frac{{(1+q)}^3{(1-\zeta )}^3}{4{\mathsf{\Lambda }}_3{(1+q)}^2{(1-\alpha )}^2+{\mathsf{\Lambda }}_2^2(1-q^2)(1-\zeta )}},\end{array}$$

(26)

and

$$\begin{array}{c}\hfill |a_3|\le (1+q)(1-\alpha )\left(\frac{4(1+q)(1-\zeta )}{{\mathsf{\Lambda }}_2^2}+\frac{1}{{\mathsf{\Lambda }}_3}\right),\end{array}$$

(27)

where ${\mathsf{\Lambda }}_n=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n},n=2,3$.

Corollary 2.

Let f be a function estimated by (1) and $0\le \zeta <1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha }\left(\zeta \right),0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_3-a_2^2|\le \frac{(1+q)(1-\zeta )}{{\mathsf{\Lambda }}_3},\end{array}$$

(28)

where ${\mathsf{\Lambda }}_3=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_3}$.

Definition 3.

Let a function f be estimated by (1), $g=f^{-1}$ and $0<\gamma \le 1$. The function $f\in {\mathrm{\Sigma }}_q^{\alpha }\left(\gamma \right),0<\alpha <1$ if the following subsequent subordination conditions hold:

$$\begin{array}{c}\hfill \frac{1}{z}\left[qf\left(z\right)+(1-q)f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right]\prec {\widehat{v}}_2\left(z\right),(z\in D),\end{array}$$

and

$$\begin{array}{c}\hfill \frac{1}{w}\left[qg\left(w\right)+(1-q)g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right]\prec {\widehat{v}}_2\left(w\right),(z\in D),\end{array}$$

where the function ${\widehat{v}}_2$ is given by

$$\begin{array}{c}\hfill {\widehat{v}}_2\left(z\right)={\left(\frac{1+z}{1-qz}\right)}^{\gamma }=1+(1+q)\gamma z+\frac{(1+q)\left(\right(1+q)\gamma +q-1)\gamma }{2}{z}^2+\cdots .\end{array}$$

(29)

Hence, it is evident that

$$\begin{array}{c}\hfill {\widehat{\psi }}_1=(1+q)\gamma ,{\widehat{\psi }}_2=\frac{\left[\right(1+q)\gamma +q-1](1+q)\gamma }{2}.\end{array}$$

(30)

Corollary 3.

Let f be a function estimated by (1), $g=f^{-1}$ and $0<\gamma \le 1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha }\left(\gamma \right)$, $0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_2|\le \sqrt{\frac{2{(1+q)}^3{\gamma }^3}{8{\mathsf{\Lambda }}_3{(1+q)}^2{\gamma }^2+{\mathsf{\Lambda }}_2^2\left[\gamma (1+q)(3-(1-q)\gamma -q)\right]}},\end{array}$$

and

$$\begin{array}{c}\hfill |a_3|\le \left(\frac{4(1+q)\gamma }{{\mathsf{\Lambda }}_2^2}+\frac{1}{{\mathsf{\Lambda }}_3}\right),\end{array}$$

where ${\mathsf{\Lambda }}_n=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n},n=2,3$.

Corollary 4.

Let$f\left(z\right)$be a function estimated by (1) and $0<\gamma \le 1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha }\left(\gamma \right),0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_3-a_2^2|\le \frac{(1+q)\gamma }{{\mathsf{\Lambda }}_3},\end{array}$$

(31)

where ${\mathsf{\Lambda }}_3=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_3}$.

3. Coefficient Estimates for the Function Class ${\mathrm{\Sigma }}_{\mathbf{q}}^{\mathbf{\alpha },\mathbf{\beta }}$

In this section, we define the class ${\mathrm{\Sigma }}_q^{\alpha ,\beta }$ and obtain coefficient estimates for $|a_2|$ and $|a_3|$ for the function in this class.

Definition 4.

Let a function f be given by (1), $g=f^{-1}$ and $0\le \beta <1$. The function $f\in \mathrm{\Sigma }$ belongs to the class ${\mathrm{\Sigma }}_q^{\alpha ,\beta },0<\alpha <1$ if the subsequent subordination conditions hold:

$$\begin{array}{c}\hfill Rel\left\{\frac{1}{z}\left(f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right)\right\}>\beta ,(z\in D,0\le \beta <1),\end{array}$$

(32)

and

$$\begin{array}{c}\hfill Rel\left\{\frac{1}{w}\left(g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right)\right\}>\beta ,(w\in D,0\le \beta <1).\end{array}$$

(33)

Theorem 3.

Let f be a function estimated by (1), $g=f^{-1}$ and $0\le \beta <1$. If $f\in \mathrm{\Sigma }$ belongs to the class ${\mathrm{\Sigma }}_q^{\alpha ,\beta },0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_2|\le (1-\beta )\min \left\{\sqrt{\frac{{\widehat{\psi }}_1^2}{2{\mathsf{\Omega }}_2^2}\left(\right|{\varphi }_1{|}^2+|{\widehat{\varphi }}_1{|}^2)},\sqrt{\frac{{\widehat{\psi }}_1^3\left(\right|{\varphi }_2|+|{\widehat{\varphi }}_2\left|\right)}{|4(1-\beta ){\widehat{\psi }}_1^2{\mathsf{\Omega }}_3+({\widehat{\psi }}_1-{\widehat{\psi }}_2){\mathsf{\Omega }}_2^2|}}\right\}\end{array}$$

(34)

and

$$\begin{array}{c}\hfill |a_3|\le \frac{1}{4}(1-\beta ){\widehat{\psi }}_1\left(\right|{\varphi }_2|+|{\widehat{\varphi }}_2\left|\right)\min \left\{\begin{array}{c}\frac{{(1-\beta )}^2{\widehat{\psi }}_1^2}{2{\mathsf{\Omega }}_2^2}\left(\right|{\varphi }_1{|}^2+|{\widehat{\varphi }}_1{|}^2),\hfill \\ \frac{{(1-\beta )}^2{\widehat{\psi }}_1^3\left(\right|{\varphi }_2|+|{\widehat{\varphi }}_2\left)\right|}{|4(1-\beta ){\widehat{\psi }}_1^2{\mathsf{\Omega }}_3+({\widehat{\psi }}_1-{\widehat{\psi }}_2){\mathsf{\Omega }}_2^2|}\hfill \end{array}\right\},\end{array}$$

(35)

where ${\mathsf{\Omega }}_n=\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n(1-q)},n=2,3$.

Proof. 

First, for the argument inequalities in (32) and (33), there exist $\varphi \left(z\right),\widehat{\varphi }\left(z\right)\in \mathcal{B}$ such that

$$\begin{array}{c}\hfill \frac{1}{z}\left(f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right)=\beta +(1-\beta )\widehat{\psi }\left(\varphi \left(z\right)\right),\end{array}$$

(36)

and

$$\begin{array}{c}\hfill \frac{1}{w}\left(g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right)=\beta +(1-\beta )\widehat{\psi }\left(\widehat{\varphi }\left(z\right)\right).\end{array}$$

(37)

We expand the right-hand side of (36) and (37) as

$$\begin{array}{c}\hfill \frac{1}{z}\left(f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right)=1+{\mathsf{\Omega }}_2{a}_{2}z+{\mathsf{\Omega }}_3{a}_3{z}^2+\dots \end{array}$$

(38)

and

$$\begin{array}{ccc}\hfill \frac{1}{w}\left(g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right)& =& 1-{\mathsf{\Omega }}_2{a}_{2}w+{\mathsf{\Omega }}_3(2a_2^2-a_3)w^2\hfill \\ & -& {\mathsf{\Omega }}_4({52}_2^2-5a_2{a}_3+a_4)w^3+\dots .\hfill \end{array}$$

(39)

Now, by equating the coefficients (15) and (38), we establish

$$\begin{array}{c}\hfill a_2=(1-\beta )\frac{{\widehat{\psi }}_1{\varphi }_1}{{\mathsf{\Omega }}_2},\end{array}$$

(40)

$$\begin{array}{c}\hfill a_3{\mathsf{\Omega }}_3=\frac{1}{2}(1-\beta ){\widehat{\psi }}_1\left(r_2-\frac{{\varphi }_1^2}{2}\right)+\frac{1}{4}(1-\beta ){\widehat{\psi }}_2^2{\varphi }_1^2.\end{array}$$

(41)

In the same manner, by equating the coefficients (16) and (39), we obtain

$$\begin{array}{c}\hfill a_2=-(1-\beta )\frac{{\widehat{\psi }}_1{\widehat{\varphi }}_1}{{\mathsf{\Omega }}_2},\end{array}$$

(42)

$$\begin{array}{c}\hfill (2a_2^2-a_3){\mathsf{\Omega }}_3=\frac{1}{2}(1-\beta ){\widehat{\psi }}_1\left({\widehat{\varphi }}_2-\frac{{\widehat{\varphi }}_1^2}{2}\right)+\frac{1}{4}(1-\beta ){\widehat{\psi }}_2^2{\widehat{\varphi }}_1^2.\end{array}$$

(43)

By omparing the coefficients (40) and (42), we get

$$\begin{array}{c}\hfill {\varphi }_1=-{\widehat{\varphi }}_1,\end{array}$$

(44)

$$\begin{array}{c}\hfill a_2^2=\frac{{(1-\beta )}^2{\widehat{\psi }}_1^2}{2{\mathsf{\Omega }}_2^2}({\varphi }_1^2+{\widehat{\varphi }}_1^2).\end{array}$$

(45)

By using (42) and (45), it follows that

$$\begin{array}{c}\hfill a_2^2=\frac{{(1-\beta )}^2{\widehat{\psi }}_1^3({\varphi }_2+{\widehat{\varphi }}_2)}{4(1-\beta ){\widehat{\psi }}_1^2{\mathsf{\Omega }}_3+({\widehat{\psi }}_1-{\widehat{\psi }}_2){\mathsf{\Omega }}_2^2}.\end{array}$$

(46)

So, following (45) and (46), we find

$$\begin{array}{c}\hfill |a_2{|}^2\le \frac{{(1-\beta )}^2{\widehat{\psi }}_1^2}{2{\mathsf{\Omega }}_2^2}(|{\varphi }_1{|}^2+|{\widehat{\varphi }}_1{|}^2),\\ \hfill |a_2{|}^2\le \frac{{(1-\beta )}^2{\widehat{\psi }}_1^3\left(\right|{\varphi }_2|+|{\widehat{\varphi }}_2\left|\right)}{|4(1-\beta ){\widehat{\psi }}_1^2{\mathsf{\Omega }}_3+({\widehat{\psi }}_1-{\widehat{\psi }}_2){\mathsf{\Omega }}_2^2|}.\end{array}$$

Hence, the desired bounds on $|a_2|$ are obtained. The proof will be completed by finding the bound for the coefficient. Therefore, subtracting Equation (43) from Equation (41), we obtain

$$\begin{array}{c}\hfill a_2^2-a_3=\frac{1}{4{\mathsf{\Omega }}_3}(1-\beta ){\widehat{\psi }}_1({\widehat{\varphi }}_2-{\varphi }_2).\end{array}$$

(47)

Put the value of $a_2^2$, given in (45) into (47) to get

$$\begin{array}{c}\hfill a_3=\frac{{(1-\beta )}^2{\widehat{\psi }}_1^2}{2{\mathsf{\Omega }}_2^2}({\varphi }_1^2+{\widehat{\varphi }}_1^2)+\frac{1}{4{\mathsf{\Omega }}_3}(1-\beta ){\widehat{\psi }}_1({\varphi }_2-{\widehat{\varphi }}_2).\end{array}$$

Therefore, we conclude that

$$\begin{array}{c}\hfill |a_3|\le \frac{{(1-\beta )}^2{\widehat{\psi }}_1^2}{2{\mathsf{\Omega }}_2^2}(|{\varphi }_1{|}^2+|{\widehat{\varphi }}_1{|}^2)+\frac{1}{4{\mathsf{\Omega }}_3}(1-\beta ){\widehat{\psi }}_1\left(\right|r_2|+|{\widehat{\varphi }}_2\left|\right).\end{array}$$

Similarly, by putting the value of $a_2^2$ of (46) into (47), we conclude the following bound:

$$\begin{array}{c}\hfill |a_3|\le \frac{{(1-\beta )}^2{\widehat{\psi }}_1^3\left(\right|{\varphi }_2|+|s_2\left)\right|}{|4(1-\beta ){\widehat{\psi }}_1^2{\mathsf{\Omega }}_3+({\widehat{\psi }}_1-{\widehat{\psi }}_2){\mathsf{\Omega }}_2^2|}+\frac{1}{4{\mathsf{\Omega }}_3}(1-\beta ){\widehat{\psi }}_1\left(\right|{\varphi }_2|+|{\widehat{\varphi }}_2\left|\right).\end{array}$$

This finishes the proof of Theorem 3. □

Similarly, from Theorem 4, we get the Fekete–Szego result for the class ${\mathrm{\Sigma }}_q^{\alpha ,\beta }$.

Theorem 4.

Let $f\left(z\right)$ be a function estimated by (1), $g=f^{-1}$, $0\le \beta <1$ and $\lambda \in \mathbb{C}$. If $f\in \mathrm{\Sigma }$ belongs to the class ${\mathrm{\Sigma }}_q^{\alpha ,\beta },0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_3-\lambda a_2^2|\le (1-\beta ){\widehat{\psi }}_1\left\{\begin{array}{cc}\frac{1}{{\mathsf{\Omega }}_3},\hfill & 0\le {\psi }_1\left(\lambda \right)<\frac{1}{4{\mathsf{\Omega }}_3},\hfill \\ 4{\psi }_1\left(\lambda \right),\hfill & {\psi }_1\left(\lambda \right)\ge \frac{1}{4{\mathsf{\Omega }}_3},\hfill \end{array}\right.\end{array}$$

where

$$\begin{array}{c}\hfill {\psi }_1\left(\lambda \right)={\displaystyle \frac{(1-\lambda )(1-\beta ){\widehat{\psi }}_1^2}{(1-\beta ){\widehat{\psi }}_1^2{\mathsf{\Omega }}_3+({\widehat{\psi }}_1-{\widehat{\psi }}_2){\mathsf{\Omega }}_2^2}},\end{array}$$

(48)

and ${\mathsf{\Omega }}_n=\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n(1-q)},n=2,3$.

Applications of Coefficient Estimates for ${\mathrm{\Sigma }}_q^{\alpha ,\beta }$

Substituting $\widehat{\psi }\left(z\right)$ in Theorem 3 by ${\widehat{v}}_1\left(z\right)$, we obtain the following definition.

Definition 5.

Let a function $f\left(z\right)$ be estimated by (1), $g=f^{-1}$, $0\le \beta <1$ and $0\le \zeta <1$. Then $f\in {\mathrm{\Sigma }}_q^{\alpha ,\beta }\left(\zeta \right),0<\alpha <1$ if the subsequent subordination conditions hold:

$$\begin{array}{c}\hfill Rel\left\{\frac{1}{z}\left(f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right)\right\}\prec \beta +(1-\beta ){\widehat{v}}_1\left(z\right),(z\in D,0\le \beta <1),\end{array}$$

and

$$\begin{array}{c}\hfill Rel\left\{\frac{1}{w}\left(g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right)\right\}\prec \beta +(1-\beta ){\widehat{v}}_1\left(w\right),(w\in D,0\le \beta <1).\end{array}$$

where the function ${\widehat{v}}_1$ is given by (25).

Corollary 5.

Let f be a function estimated by (1), $g=f^{-1}$, $0\le \beta <1$ and $0\le \zeta <1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha ,\beta }\left(\zeta \right),0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_2|\le \sqrt{\frac{{(1+q)}^3{(1-\zeta )}^3}{4{\mathsf{\Lambda }}_3{(1+q)}^2{(1-\alpha )}^2+{\mathsf{\Lambda }}_2^2(1-q^2)(1-\zeta )}},\end{array}$$

(49)

and

$$\begin{array}{c}\hfill |a_3|\le (1+q)(1-\alpha )\left(\frac{4(1+q)(1-\zeta )}{{\mathsf{\Lambda }}_2^2}+\frac{1}{{\mathsf{\Lambda }}_3}\right),\end{array}$$

(50)

where ${\mathsf{\Lambda }}_n=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n},n=2,3$.

Corollary 6.

Let f be a function estimated by (1), $0\le \beta <1$ and $0\le \zeta <1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha ,\beta }\left(\zeta \right)$, $0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_3-a_2^2|\le \frac{(1+q)(1-\zeta )}{{\mathsf{\Lambda }}_3},\end{array}$$

(51)

where ${\mathsf{\Lambda }}_3=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_3}$.

Substituting $\widehat{\psi }\left(z\right)$ in Theorem 3 by ${\widehat{v}}_2\left(z\right)$, we obtain the following definition.

Definition 6.

Let f be a function estimated by (1), $g=f^{-1}$, $0\le \beta <1$ and $0<\gamma \le 1$. Then $f\in {\mathrm{\Sigma }}_q^{\alpha ,\beta }\left(\gamma \right),0<\alpha <1$ if the subsequent subordination conditions hold

$$\begin{array}{c}\hfill Rel\left\{\frac{1}{z}\left(f\left(z\right)\ast I_{\alpha ,q}^{‡}\left(z\right)\right)\right\}\prec \beta +(1-\beta ){\widehat{v}}_2\left(z\right),(z\in D,0\le \beta <1),\end{array}$$

and

$$\begin{array}{c}\hfill Rel\left\{\frac{1}{w}\left(g\left(w\right)\ast I_{\alpha ,q}^{‡}\left(w\right)\right)\right\}\prec \beta +(1-\beta ){\widehat{v}}_2\left(w\right),(w\in D,0\le \beta <1),\end{array}$$

where the function ${\widehat{v}}_2$ is given by (29).

Corollary 7.

Let f be a function estimated by (1), $g=f^{-1}$, $0\le \beta <1$ and $0<\gamma \le 1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha ,\beta }\left(\gamma \right),0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_2|\le \sqrt{\frac{2{(1+q)}^3{\gamma }^3}{8{\mathsf{\Lambda }}_3{(1+q)}^2{\gamma }^2+{\mathsf{\Lambda }}_2^2\left[\gamma (1+q)(3-(1-q)\gamma -q)\right]}},\end{array}$$

and

$$\begin{array}{c}\hfill |a_3|\le \left(\frac{4(1+q)\gamma }{{\mathsf{\Lambda }}_2^2}+\frac{1}{{\mathsf{\Lambda }}_3}\right),\end{array}$$

where ${\mathsf{\Lambda }}_n=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_n},n=2,3$.

Corollary 8.

Let f be a function estimated by (1), $g=f^{-1}$, $0\le \beta <1$ and $0<\gamma \le 1$. If $f\in {\mathrm{\Sigma }}_q^{\alpha ,\beta }\left(\gamma \right),0<\alpha <1$, then

$$\begin{array}{c}\hfill |a_3-a_2^2|\le \frac{(1+q)\gamma }{{\mathsf{\Lambda }}_3},\end{array}$$

(52)

where ${\mathsf{\Lambda }}_3=q+\frac{{(q;q)}_n(1-q^{1-\alpha })}{{(q^{1-\alpha };q)}_3}$.

4. Conclusions

In this article, a new analytic subclass of bi-univalent functions was studied on aid of the principle of differential subordination and convolution. Certain set of bi-univalent functions influenced by a q-derivative operator with non-integer order $\alpha$ in the open unit disc were also formulated. Moreover, the Fekete–Szego inequality for each function in the newly defined classes was also obtained.