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1.   Introduction, definitions and preliminaries

In his survey-cum-expository review article, Srivastava ^[35] included also a brief overview of the classical $q$-analysis versus the so-called $(p, q)$-analysis with an obviously redundant additional parameter $p$ (see, for details, [35,p. 340]). The present sequel to Srivastava's widely-cited review article ^[35], we apply the concept of $q$-convolution in order to introduce and study the general Taylor-Maclaurin coefficient estimates for functions belonging to a new class of normalized analytic and bi-close-to-convex functions in the open unit disk, which we have defined here.

Let $\mathcal{A}$ denote the class of analytic functions of the form:

where $\Delta$ denotes the open unit disk in the complex $z$-plane given by

Also let $\mathcal{S}\subset \mathcal{A}$ consist of functions which are also univalent in $\Delta$.

If the function $f$ is given by (1.1) and the function $\Upsilon \in \mathcal{A}$ is given by

then the Hadamard product (or convolution) of the functions $f$ and $\Upsilon$ is defined by defined by

For $0\leqq \alpha < 1,$ we let $S^{\ast}\left(\alpha \right)$ denote the class of functions $g\in \mathcal{S}$ which are starlike of order $\alpha$ in $\Delta$ such that

We denote by $\mathcal{C}\left(\alpha \right)$ the class of functions $f\in \mathcal{S}$ which are close-to-convex of order $\alpha$ in $\Delta$ such that (see ^[10,24])

where

We note that

by the Bieberbach conjecture or the De Branges Theorem (see ^[3,10]), $\mathbb{N}$ being the set of natural numbers (or the positive integers).

In the above-cited review article, Srivastava ^[35] made use of various operators of $q$-calculus and fractional $q$-calculus. We begin by recalling the definitions and notations as follows (see also ^[33] and [45,pp. 350–351]).

The $q$-shifted factorial is defined, for $\lambda, q\in \mathbb{C}$ and $n \in \mathbb{N}_{0} = \mathbb{N}\cup \{0\}$, by

By using the $q$-gamma function $\Gamma_{q}(z),$ we get

where (see ^[19,33])

We note also that

and that the $q$-gamma function $\Gamma_{q}(z)$ satisfies the following recurrence relation:

where $\left[\lambda\right]_{q}$ denotes the basic (or $q$-) number defined as follows:

Using the definition in (1.3), we have the following consequences:

(ⅰ) For any non-negative integer $n\in \mathbb{N}_0$, the $q$-shifted factorial is given by

(ⅱ) For any positive number $r$, the generalized $q$-Pochhammer symbol is defined by

In terms of the classical (Euler's) gamma function $\Gamma\left(z\right)$, it is easily seen that

We also observe that

where $\left(\lambda\right)_{n}$ is the familiar Pochhammer symbol defined by

For $0 < q < 1$, the $q$-derivative operator (or, equivalently, the $q$-difference operator) $D_{q}$ is defined by (see ^[22]; see also ^[14,16,21])

where, as in the definition (1.3),

Remark 1. Whereas a $q$-extension of the class of starlike functions was introduced in $1990$ in ^[20] by means of the $q$-derivative operator $D_q$, a firm footing of the usage of the $q$-calculus in the context of Geometric Function Theory was actually provided and the generalized basic (or $q$-) hypergeometric functions were first used in Geometric Function Theory in an earlier book chapter published in 1989 by Srivastava (see, for details, ^[34]; see also the recent works ^{[25,27,32,36,37,39,40,46,51,52,53,55,56,57]}).

For $\lambda > -1$ and $0 < q < 1$, El-Deeb et al. ^[14] defined the linear operator $\mathcal{H}_{\Upsilon}^{\lambda, q}:\mathcal{A}\rightarrow \mathcal{A}$ by

where the function $\mathcal{M}_{q, \lambda}(z)$ is given by

A simple computation shows that

From the defining relation (1.5), we can easily verify that the following relations hold true for all $f\in \mathcal{A}$:

Remark 2. If we take different particular cases for the coefficients $\psi_{n},$ we obtain the following special cases for the operator $\mathcal{H}_{h}^{\lambda, q}$:

(ⅰ) For $\psi_{n} = 1$, we obtain the operator $\mathfrak{J}_{q}^{\lambda}$ defined by Arif et al. ^[2] as follows (see also Srivastava ^[47]):

(ⅱ) For

we obtain the operator $\mathcal{N}_{\upsilon, q}^{\lambda}$ defined by El-Deeb and Bulboacǎ ^[12] and El-Deeb ^[11] as follows (see also ^[16]):

where

(ⅲ) For

we obtain the operator $\mathcal{M}_{m, q}^{\lambda, \alpha}$ defined by El-Deeb and Bulboacǎ (see ^[13,43]) as follows:

(ⅳ) For

we obtain a $q$-analogue of the Poisson operator defined in ^[30] by

(ⅴ) For

we get a $q$-analogue $\Psi_{q, \theta}^{\lambda, m}$ of the Pascal distribution operator as follows (see ^[15]):

If $f$ and $F$ are analytic functions in $\Delta$, we say that the function $f$ is subordinate to the function $F$, written as $f(z)\prec F(z)$, if there exists a Schwarz function $s$, which is analytic in $\Delta$ with $s(0) = 0$ and $\left\vert s(z)\right\vert < 1$ for all $z\in \Delta$, such that

Furthermore, if the function $F$ is univalent in $\Delta$, then we have the following equivalence (see, for example, ^[7,28])

The Koebe one-quarter theorem (see ^[10]) asserts that the image of $\Delta$ under every univalent function $f\in \mathcal{S}$ contains the disk of radius $\dfrac{1}{4}$. Therefore, every function $f\in \mathcal{S}$ has an inverse $f^{-1}$ which satisfies the following inequality:

where

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\Delta$ if both $f$ and $f^{-1}$ are univalent in $\Delta$. Let $\Sigma$ denote the class of normalized analytic and bi-univalent functions in $\Delta$ given by (1.1). The class $\Sigma$ of analytic and bi-univalent functions was introduced by Lewin ^[26], where it was shown that

Brannan and Clunie ^[4] improved Lewin's result to the following form:

and, subsequently, Netanyahu ^[29] proved that

It should be noted that the following functions:

together with their corresponding inverses given by

are elements of the analytic and bi-univalent function class $\Sigma$ (see ^[14,48]). A brief history and interesting examples of the analytic and bi-univalent function class $\Sigma$ can be found in (for example) ^[5,48].

Brannan and Taha ^[6] (see also ^[48]) introduced certain subclasses of the bi-univalent function class $\Sigma$ similar to the familiar subclasses $S^{\ast }\left(\alpha \right)$ and $K\left(\alpha \right)$ of starlike and convex functions of order $\alpha \; \left(0\leqq \alpha < 1\right)$, respectively (see ^[5]). Indeed, following Brannan and Taha ^[6], a function $f\in \mathcal{A}$ is said to be in the class $S_{\Sigma}^{\ast }\left(\alpha \right)$ of bi-starlike functions of order $\alpha \; \left(0 < \alpha \leqq 1\right)$, if each of the following conditions is satisfied:

and

where the function $\mathcal{F}$ is the analytic extension of $f^{-1}$ to $\Delta$, given by

A function $f\in A$ is said to be in the class $K_{\Sigma}^{\ast }\left(\alpha \right)$ of bi-convex functions of order $\alpha \; \left(0 < \alpha \leqq 1\right)$, if each of the following conditions is satisfied:

and

The classes $S_{\Sigma}^{\ast}\left(\alpha \right)$ and $K_{\Sigma}\left(\alpha \right)$ of bi-starlike functions of order $\alpha$ in $\Delta$ and bi-convex functions of order $\alpha \; \left(0 < \alpha \leqq 1\right)$ in $\Delta$, corresponding to the function classes $S^{\ast}\left(\alpha \right)$ and $K\left(\alpha \right)$, were also introduced analogously. For each of the function classes $S_{\Sigma}^{\ast}\left(\alpha \right)$ and $K_{\Sigma}\left(\alpha \right)$, non-sharp estimates on the first two Taylor-Maclaurin coefficients $\left\vert a_{2}\right\vert$ and $\left\vert a_{3}\right\vert$ are known (see ^[6,35,48]). In fact, this pioneering work by Srivastava et al. ^[48] happens to be one of the most important studies of the bi-univalent function class $\Sigma$. It not only revived the study of the bi-univalent function class $\Sigma$ in recent years, but it has also inspired remarkably many investigations in this area including the present paper. Some of these many recent papers dealing with problems involving the analytic and bi-univalent functions such as those considered in this article include ^{[1,9,17,23,48]}, and indeed also many other works (see, for example, ^[38,44,54]).

Sakar and Güney ^[31] introduced and studied the following class:

In the same way, we define the following subclass of bi-close-to-convex functions $\mathcal{H}_{\Sigma}^{q, \lambda }\left(\eta, \beta, \Upsilon \right)$ as follows.

Definition 1. For $0\leqq \eta < 1$ and $0\leqq \beta \leqq 1, \;$ a function $f\in \Sigma$ has the form (1.1) and the function $\Upsilon$ given by (1.2), the function $f$ is said to be in the class $\mathcal{H}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon\right)$ if there exists a function $g \in \mathcal{S}^{\ast}$ such that

and

where the function $\mathcal{F}$ is the analytic extension of $f^{-1}$ to $\Delta$, and is given by (1.13), and $\mathcal{G}$ is the analytic extension of $g^{-1}$ to $\Delta$ as follows:

We note that, if $b_{n} = a_{n}\; \; (n\in \mathbb{N}\setminus\{1\})$, $\mathcal{S}_{\Sigma }^{q, \lambda}\left(\eta, \beta, \Upsilon\right)$ becomes the class of bi-starlike functions satisfying the following inequalities:

and

Remark 3. Each of the following limit cases when $q\rightarrow 1{-}$ is worthy of note.

(ⅰ) Putting $q\rightarrow 1{-}$, we obtain

where $\mathcal{P}_{\Sigma}^{\lambda }\left(\eta, \beta, \Upsilon \right)$ represents the functions $f\in \Sigma$ that satisfy (1.14) and (1.15) with $\mathcal{H}_{\Upsilon}^{\lambda, q}$ replaced by $\mathcal{I}_{\Upsilon}^{\lambda}$ as in (1.6).

(ⅱ) Putting

we obtain the class $\mathcal{B}_{\Sigma}^{q, \lambda} \left(\eta, \beta, \upsilon\right)$ representing the functions $f\in \Sigma$ that satisfy (1.14) and (1.15) with $\mathcal{H}_{\Upsilon}^{\lambda, q}$ replaced by $\mathcal{N}_{\upsilon, q}^{\lambda}$ as in (1.8).

(ⅲ) Putting

we obtain the class $\mathcal{L}_{\Sigma}^{\lambda, q} \left(\eta, \beta, m, \alpha \right)$ consisting of the functions $f\in \Sigma$ that satisfy (1.14) and (1.15) with $\mathcal{H} _{\Upsilon}^{\lambda, q}$ replaced by $\mathcal{M}_{m, q}^{\lambda, \alpha }$ as in (1.10).

(ⅳ) Putting

we obtain the class $\mathcal{M}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \rho \right)$ representing the functions $f\in \Sigma$ which satisfy the inequalities in (1.14) and (1.15) with $\mathcal{H}_{\Upsilon}^{\lambda, q}$ replaced by $\mathcal{I}_{q}^{\lambda, \rho}$ as in (1.11).

(ⅴ) Putting

we get the class $\mathcal{W}_{\Sigma}^{q, \lambda}\left(\eta, \beta, m, \theta \right)$ of the functions $f\in \Sigma$ which satisfy the inequalities in (1.14) and (1.15) with $\mathcal{H}_{\Upsilon}^{\lambda, q}$ replaced by $\Psi_{q, \theta}^{\lambda, m}$ occurring in (1.12).

Using the Faber polynomial expansion of functions $f\in \mathcal{A}$ which have the normalized form (1.1), the coefficients of its inverse map may be expressed as follows (see ^[18]):

where

such that $U_{j}$ with $7\leqq j\leqq n$ is a homogeneous polynomial in the variables $a_{2}, a_{3}, \cdots, a_{n}$. Here such expressions as (for example) $(-n)!$ are to be interpreted symbolically by

In particular, the first three terms of $\mathcal{K}_{n-1}^{-n}$ are given by

and

In general, an expansion of $\mathcal{K}_{m}^{-n} \; (n\in \mathbb{N})$ is given by (see ^{[1,8,41,42,47,49,50]})

where

and, alternatively,

where $a_{1} = 1$ and the sum is taken over all non-negative integers $i_{1}, \cdots, i_{m}$ satisfying the following constraints:

and

Evidently, we have

The following Lemma will be needed to prove our results.

The Carathéodory Lemma. (see ^[10]) If $\phi \in \mathfrak{P}$ and

then

This inequality is sharp for all positive integers $n$. Here $\mathfrak{P}$ is the family of all functions $\phi,$ which analytic and have positive real part in $\Delta,$ with $\phi(0) = 1$.

2.   A set of main results

In this section, we apply the above-described Faber polynomial expansion method, we derive bounds for the general Taylor-Maclaurin coefficients of functions in $\mathcal{H}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon \right)$.

Theorem 1. Let the function $f$ given by (1.1) belong to the class $\mathcal{H}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon \right)$. Suppose also that

If $a_{k} = 0$ for $2\leqq k\leqq n-1,$ then

Proof. If $f\in \mathcal{H}_{\Sigma }^{q, \lambda }\left(\eta, \beta, \Upsilon \right)$, then there exists a function $g(z)$, given by

such that

Moreover, by using the Faber polynomial expansion, we have

Also, for the inverse map $\mathcal{F} = f^{-1},$ there exists a function $\mathcal{G}(w)$, given by

such that

the Faber polynomial expansion of the inverse map $\mathcal{F} = f^{-1}$ is given by

so we have

Now, since

there are the following two positive real part functions:

and

for which

so that

and

Now, under the assumption that $a_{k} = 0$ for $0\leqq k\leqq n-1,$ we obtain $A_{n} = -a_{n}.$ Then, by using (2.1) and comparing the corresponding coefficients in (2.3), we obtain

Similarly, by using (2.2) in the Eq (2.4), we find that

and

Taking the moduli of both members of (2.7) and (2.8) for

and applying the Carathéodory Lemma, we conclude that

which completes the proof of Theorem 1.

If we set

in Theorem 1, we obtain the following special case.

Corollary 1. Let the function $f$ given byt (1.1) belong to the class $\mathcal{B}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \upsilon \right)$. Suppose also tha

If $a_{k} = 0$ for $2\leqq k\leqq n-1,$ then

where $\phi_{n}$ is given by (1.9).

Upon putting

in Theorem 1, we obtain the following result.

Corollary 2. Let the function $f$ given by (1.1) belong to the class $\mathcal{L}_{\Sigma}^{q, \lambda}\left(\eta, \beta, m, \alpha \right)$. Suppose also that

If $a_{k} = 0$ for $2\leqq k\leqq n-1,$ then

If we take

in Theorem 1, we obtain the following special case.

Corollary 3. Let the function $f$ given by (1.1) belong to the class $\mathcal{M}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \rho \right)$. Suppose also that

If $a_{k} = 0$ for $2\leqq k\leqq n-1,$ then

Upon setting

in Theorem 1, we are led to the following result for the above-defined class $\mathcal{W}_{\Sigma}^{q, \lambda}\left(\eta, \beta, m, \theta\right)$.

Corollary 4. Let the function $f$ given by (1.1) belong to the following class$:$

If $a_{k} = 0$ for $2\leqq k\leqq n-1,$ then

In particular, if we let $g(z) = f(z)$, we obtain the class $\mathcal{S}_{\Sigma }^{q, \lambda}\left(\eta, \beta, \Upsilon \right)$, which is a subclass of $\mathcal{H}_{\Sigma }^{q, \lambda}\left(\eta, \beta, \Upsilon\right)$. We then give the next theorem, which involves the coefficients of this subclass of the analytic and bi-starlike functions in $\Delta$.

Theorem 2. Let the function $f$ given by (1.1) belong to the class $\mathcal{S}_{\Sigma}^{q, \lambda}\left(\eta, \beta, \Upsilon \right)$. Suppose also that

Then

and

Proof. Putting $n = 2$ and $n = 3$ in (2.5) and (2.6), we have

and

From (2.11) and (2.13), by using the Carathéodory Lemma, we obtain

Also, from (2.12) and (2.14), we have

Thus, by using the Carathéodory Lemma, we obtain

From (2.15) and (2.16), we obtain the desired estimate on the coefficient $\left\vert a_{2}\right\vert$ as asserted in (2.9).

In order to find the bound on the coefficient $\left\vert a_{3}\right\vert,$ we subtract (2.14) from (2.12), so that

that is,

Now, upon substituting the value of $a_{2}^{2}$ from (2.16) into (2.17) and using the Carathéodory Lemma, we find that

Moreover, upon substituting the value of $a_{2}^{2}$ from (2.11) into (2.12), we have

Applying the Carathéodory Lemma, we obtain

Finally, by combining (2.18) and (2.19), we have the desired estimate on the coefficient $\left\vert a_{3}\right\vert$ as asserted in (2.10). The proof of Theorem 2 is thus completed.

3.   Conclusions

In our present investigation, we have made use of the concept of $q$-convolution with a view to introducing a new class of analytic and bi-close-to-convex functions in the open unit disk. For functions belonging to this analytic and bi-univalent function class, we have derived estimates for the general coefficients in their Taylor-Maclaurin series expansions in the open unit disk. Our methodology is based essentially upon the Faber polynomial expansion method. We have also presented a number of corollaries and consequences of our main results.

In his recently-published review-cum-expository review article, in addition to applying the $q$-analysis to Geometric Function Theory of Complex Analysis, Srivastava ^[35] pointed out the fact that the results for the $q$-analogues can easily (and possibly trivially) be translated into the corresponding results for the $(p, q)$-analogues (with $0 < q < p \leqq 1$) by applying some obvious parametric and argument variations, the additional parameter $p$ being redundant. Of course, this exposition and observation of Srivastava [35,p. 340] would apply also to the results which we have considered in our present investigation for $0 < q < 1$.

Acknowledgements

The authors received no funding for the investigation leading to the completion of this article.

Conflict of interest

The authors declare that there is no conflict of interest in respect of this article.