Faber polynomial coefficients estimates for certain subclasses of $q$-Mittag-Leffler-Type analytic and bi-univalent functions

1.   Introduction, definitions and motivation

Let $\mathcal{A}$ be the class of functions $f$ of the form

which are analytic in the open unit disc

and normalized under the conditions

Furthermore, by $\mathcal{S}$ we shall denote the class of all functions in $\mathcal{A}$ which are univalent in $E$.

Let $f\in \mathcal{A}$ given by (1.1) and $g\in \mathcal{A}$ given by

we define the convolution (or Hadamard product) of $f$ and $g$ as:

Let $f, h\in \mathcal{A}$, $f$ is subordinate to $h$ if there exists a Schwarz function $u$, where

such that

We denote this subordination by

In particular, if the function $h$ is univalent in $E$, the above subordination is equivalent to

We see that (see ^[24]) for the Schwarz function $u(z)$, we have

The Koebe-one quarter Theorem (see ^[24]) shows that the image of $E$ under every univalent function $f\in \mathcal{A}$ contains a disk $\{w:\left \vert w\right \vert < \frac{1}{4}\}$ of radius $\frac{1}{4}$. Every univalent function $f$ has an inverse $f^{-1}$ defined on some disk containing the disk $\{w:\left \vert w\right \vert < \frac{1}{4}\}$ and satisfying:

and

where

A function $f$ is said to be bi-univalent on $E$ if both $f$ and $g = f^{-1}$ are univalent on $E.$ We denote the class of all such functions by $\Sigma$.

Lewin ^[50] studied the class of bi-univalent functions, in fact he obtained the bound

Netanyahu ^[53] showed that $\max \left \vert a_{2}\right \vert = \frac{4 }{3}.$ Brannan and Clunie ^[16] conjectured that $\left \vert a_{2}\right \vert \leq \sqrt{2.}$ In recent years, the pioneering work of Srivastava et al. ^[67] essentially revived the investigation of various subclasses of the analytic and bi-univalent function class $\Sigma$. In fact, in a remarkably large number of sequels to the pioneering work of Srivastava et al. ^[67], several different subclasses of the analytic and bi-univalent function class were introduced and studied analogously by the many authors (see, for example, ^{[11,17,18,19,20,21,29,31,38,39,64,67]}.

In Geometric Function Theory (GFT), the quantum (or $q$-) calculus used as important tools to study different families of analytic function and due to the application in mathematics and some related areas it has inspired a number of well-known mathematicians.

The quantum (or $q$-) calculus is widely applied in various operators which include the $q$-difference ($q$ -derivative) operators and these operators plays an important role in GFT as well as in the theory of hypergeometric series, quantum theory, number theory and statistical mechanics. Jackson ^[35,36] was among the few researchers who defined the $q$-derivative and $q$-integral operator as well as provided some of their applications. Also Ismail et al. ^[34] introduced research work in connection with function theory and $q$-theory. Letter on, by using $q$-beta function Gupta ^[14] introduced $q$ -Baskakov-Durrmeyer operator while $q$-Picard and $q$-Gauss-Weierstrass singular integral operators introduced and studied by Aral in ^[13].

Historically, Srivastava studied univalent function theory by using $q$ -calculus, see for detail ^[62,63]. Moreover, Kanas and Raducanu ^[40] introduced the $q$-analogue of Ruscheweyh differential operator and Arif et al. ^[15] discussed some of its applications for multivalent functions. For detailed study about $q$-analogous of operators we may refer to ^{[1,33,42,43,44,45,46,47,48,49,59,66]}.

Definition 1.1. (see ^[36]) The $q$-number $[t]_{q}$ and $q$-factorial $[n]_{q}!$ for $q\in (0, 1)$ is defined as:

and

where

Definition 1.2. (see ^[36]) The $q$-generalized Pochhammer symbol $[t]_{n, q}$, $t\in \mathbb{C},$ is defined as:

and the $q$-Gamma function be given as:

Definition 1.3. (^[36]) For $f\in \mathcal{A},$ the $q$-derivative operator or $q$ -difference operator be defined as:

Combining (1.1) and (1.3), we have

Note that

We can observe that

Mittag-Leffler introduced Mittag-Leffler function $\mathcal{H}_{\alpha }(z)$ in ^[51,52] as:

and its generalization $\mathcal{H}_{\alpha, \beta }(z)$ introduced by Wiman ^[70] as:

For more study about Mittag-Leffler function see article ^{[12,54,65,68]}.

The $q$-Mittag-Leffler function is defined by (see ^[58])

Note that $q$-Mittag-Leffler function is the specialized case of the $q$ -Fox-Wright function $_{r}\Phi _{s}(z, q)$, (see, for details, ^[60,61]). Since the $q$-Mittag-Leffler function $\mathcal{H}_{\alpha, \beta }(z, q)$ defined by (1.4) does not belong to the normalized analytic function class $\mathcal{A}$.

Now, we define the normalization of this $q$-Mittag-Leffler function $\mathcal{F}_{\alpha, \beta }(z)$ as:

where $z\in E,$ $\Re\alpha > 0$, $\beta \in \mathbb{C} \backslash \left \{ 0, -1, -2, ...\right \}).\ $Corresponding to $\mathcal{F} _{\alpha, \beta }(z, q)$ and for $f\in \mathcal{A}$, we define the following differential operator $\mathcal{D}_{\delta, \mu }^{m, q}(\alpha, \beta): \mathcal{A}\rightarrow \mathcal{A}$ by

If $f(z)$ is given by (1.1), then from (1.5) and (1.6), we have

where

Each of the following special case of the above-mentioned operator $\mathcal{ D}_{\delta, \mu }^{m, q}(\alpha, \beta):\mathcal{A}\rightarrow \mathcal{A}$ is worthy of noted.

(ⅰ) For $\mu = 0,$ $\alpha = 0,$ $\beta = 1$, and $\delta = 1,$ we get Salagean $q$-differential operator introduced by Salagean in ^[27].

(ⅱ) For $q\rightarrow 1-,$ $\mu = 0,$ $\alpha = 0,$ $\beta = 1$, and $\delta = 1,$ we get Salagean differential operator introduced by Salagean in ^[55].

(ⅲ) For $q\rightarrow 1-,$ $\mu = 0,$ $\alpha = 0,$ and $\beta = 1,$ we get Al-Oboudi operator ^[2].

(ⅳ) For $q\rightarrow 1-,$ and $m = 0,$ we have $E_{\alpha, \beta }(z)$ introduced in ^[65].

(ⅴ) For $q\rightarrow 1-,$ $\alpha = 0,$ and $\beta = 1,$ we have Raducanu and Orhan differential operator ^[22] see also ^[23].

The Faber polynomials introduced by Faber ^[25] play an important role in various areas of mathematical sciences, especially in Geometric Function Theory see also ^[28,56,57]. Not much is known about the bounds on general coefficients $\left \vert a_{n}\right \vert,$ for $n\geq 3$ of bi-univalent functions. In the literature only a few work determining the general coefficient $\left \vert a_{n}\right \vert,$ for $n\geq 3$ for the analytic bi-univalent function given by (1.1). For more study see ^{[3,4,30,32,37,69]}.

Here in this paper we define new subclass of bi-univalent functions and determine estimates for the general coefficient bounds $\left \vert a_{n}\right \vert$ for $n\geq 3,$ by using Faber polynomial expansions and newly defined $q$-analogue of differential operator. Throughout in this paper, we assume that

Definition 1.4. A function $f\in \Sigma$ is said to be in the class $\mathcal{B}_{\Sigma }^{m, \lambda, \mu, \delta }(\alpha, \beta, q, A, B)$ if the following subordinations are satisfied:

and

where the function $g$ is given by (1.2).

Remark 1.5. First of all, it ids easy to see that

where $\mathcal{B}_{\Sigma }(m, \lambda, \varphi)$ is the function class introduced and studied by Altinkaya and Yalcin ^[11]. Secindly, we have

where the class $\mathcal{B}_{\Sigma }(\varphi, \lambda)$ was introduced by Frasin and Aouf ^[26].

In this article, we defined certain new subclasses of analytic and bi-univalent functions which involve the differential operator of $q$-Mittag-Leffer functions. Then by applying the method of Faber polynomial expansions, we determined general coefficients bond $|a_{n}|$, for $n\geq 3$. We also highlight some known consequences of our main results.

2.   Main results

By using the Faber polynomial expansion of functions $f$ of the form (1.1), the coefficients of its inverse map $g = f^{-1}$ are given by,

where

and $g = f^{-1}$ given by (1.2), $V_{j}$ with $7\leq j\leq n$ is a homogeneous polynomial in the variables $\left \vert a_{2}\right \vert, \left \vert a_{3}\right \vert, ..., \left \vert a_{n}\right \vert$ (see ^[5])$.$ In particular, the first three terms of $K_{n-1}^{-n}$ are

In general, for any $p\in \mathbb{N}$ and $n\geq 2,$ an expansion of $K_{n-1}^{p}$ (see ^[4]) is$,$

where $E_{n-1}^{p} = E_{n-1}^{p}(a_{2}, a_{3}, ...)$ (see ^[6]) given by

While $a_{1} = 1,$ and the sum is taken over all nonnegative integer $\mu _{1}, ..., \mu _{n}$ satisfying:

and

Evidently, (see ^[3])

or equivalently,

again $a_{1} = 1,$ and the taking the sum over all nonnegative integer $\mu _{1}, ..., \mu _{n}$ satisfying:

It is clear that

the first and last polynomials are

Theorem 2.1. Let $f\in \mathcal{B}_{\Sigma }^{m, \lambda, \mu, \delta }(\alpha, \beta, q, A, B).$ If $a_{i} = 0;\ 2\leq i\leq n-1,$ then

where $\varphi$ is given by (1.7).

Proof. Let $f$ be given by (1.1), we have

and for its inverse map $g = f^{-1},$ we have

where

Since, both the functions $f$ and its inverse map $g = f^{-1}$ are in $\mathcal{B}_{\Sigma }^{m, \lambda, \mu, \delta }(\alpha, \beta, q, A, B),$ by the definition of subordination, for $z, w\in E,$ there exist two Schwarz functions

and

such that

and

where

and

In general ^[3,4] for any $p\in \mathbb{N}$ and $n\geq 2,$ an expansion of $K_{n}^{p}(k_{1, }k_{2, }..., k_{n}, B),$

where $X_{j}$ is a homogeneous polynomial of degree $j$ in the variables $k_{1, }k_{2, }..., k_{n}.$

Comparing the corresponding coefficients of (2.1) and (2.3) yields

and similarly, from (2.2) and (2.4) yields

Note that for $a_{i} = 0;$ $2\leq i\leq n-1,$ we have

and so

Now taking the absolute values of (2.7) and (2.8) and using the fact that

we obtain

If in Theorem 2.1, we take

and let $q\rightarrow 1-,$ we have the following known result.

Corollary 2.2. (^[11]). Let $f\in \mathcal{B}_{\Sigma }(m, \lambda, \varphi).$ If $a_{i} = 0;$ $2\leq i\leq n-1,$ then

Theorem 2.3. Let $f\in \mathcal{B}_{\Sigma }^{m, \lambda, \mu, \delta }(\alpha, \beta, q, A, B).$ Then

and

Proof. Replacing $n$ by $2$ and $3$ in (2.5) and (2.6), respectively, we find that

and

From (2.9) and (2.11) we obtain

Adding (2.10) and (2.12) implies

or equivalently,

From (2.13) and (2.14) we get required assertion.

Now from (2.10), one can easily see that

Next in order to find the bound on the coefficient $\left \vert a_{3}\right \vert,$we subtract (2.12) from (2.10), we thus obtain

Using the fact that $c_{1}^{2} = d_{1}^{2}$ and taking the absolute values of both sides of (2.15), we obtain the desired inequality

Substituting the value of $a_{2}^{2}$ from (2.13) into (2.16), we obtain

Additionally, substituting the value of $a_{2}^{2}$ from (2.14) into (2.16), we obtain

Solving the equation (2.15) for $a_{3}-a_{2}^{2}$, we get the desired inequality as:

Finally we rewrite (2.12) as

and therefore

If in Theorem 2.3, we take

and let $q\rightarrow 1-,$ we have the following known result.

Corollary 2.4. (^[11]). Let $f\in \mathcal{B}_{\Sigma }(p, \lambda, \varphi).$ Then

and

3.   Conclusions

Basic (or $q$-) Calculus is particularly applicable in many deserve areas of mathematics and physics. In our present investigations, we have first introduced the $q$-analogus of generalized differential operator involving $q$-Mittag-Leffler function in open unit disk

and then defined certain new subclasses of analytic and bi-univalent functions. Furthermore, By applying the Faber polynomial expansion method, we have determined general coefficient bounds $|a_{n}|$, for $n\geq 3$. We have also highlight some known consequences of our main results.

Acknowledgment

This work was supported by The Key Scientific Research Project of the Colleges and Universities in Henan Province (NO. 19A110024), Natural Science Foundation of Henan Province (CN) (NO. 212300410204).