Applications of the q-Srivastava-Attiya Operator Involving a Certain Family of Bi-Univalent Functions Associated with the Horadam Polynomials

Abstract

In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$ of normalized holomorphic and bi-univalent functions in the open unit disk $\mathbb{U}$, which are associated with the Bazilevič functions and the $\lambda$-pseudo-starlike functions as well as the Horadam polynomials. We estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to the holomorphic and bi-univalent function class, which we introduce here. Furthermore, we establish the Fekete-Szegö inequality for functions in the family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Relevant connections of some of the special cases of the main results with those in several earlier works are also pointed out. Our usage here of the basic or quantum (or q-) extension of the familiar Hurwitz-Lerch zeta function $\Phi (z,s,a)$ is justified by the fact that several members of this family of zeta functions possess properties with local or non-local symmetries. Our study of the applications of such quantum (or q-) extensions in this paper is also motivated by the symmetric nature of quantum calculus itself.

1. Introduction and Preliminaries

We indicate by $\mathcal{A}$ the collection of functions, which are holomorphic in the open unit disk $\mathbb{U}$ given by

$$\mathbb{U}=\left\{z:z\in \mathbb{C}\mathrm{and}\left|z\right|<1\right\}$$

and have the following normalized form:

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

(1)

We denote by $\mathcal{S}$ the sub-collection of the set $\mathcal{A}$ consisting of functions, which are also univalent in $\mathbb{U}$. According to the Koebe one-quarter theorem [1], every function $f\in \mathcal{S}$ has an inverse $f^{-1}$ defined by

$$f^{-1}\left(f\left(z\right)\right)=z(z\in \mathbb{U})$$

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

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

(2)

We say that a function $f\in \mathcal{A}$ is bi-univalent in $\mathbb{U}$ if both f and it inverse $f^{-1}$ are univalent in $\mathbb{U}$. Let $\Sigma$ stand for the family of bi-univalent functions in $\mathbb{U}$ given by (1). Beginning with the pioneering work [2] on the subject by Srivastava et al. [2], a large number of works related to the subject have been (and continue to be) published (see, for example, Refs. [3,4,5,6,7]). From the work of Srivastava et al. [2], we choose to recall the following examples of functions in the family $\Sigma :$

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

We notice that the family $\Sigma$ is not empty. However, the Koebe function is not a member of $\Sigma .$ The problem to find the general coefficient bounds on the Taylor-Maclaurin coefficients

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

for functions $f\in \Sigma$ is still not completely addressed for many of the subfamilies of the bi-univalent function family $\Sigma$.

Finding an upper bound for the functional $\left|a_3-\mu a_2^2\right|$$\left(f\in \mathcal{S}\right)$ constitutes the Fekete-Szegö type inequality (or problem) (see [8]). It originates from their disproof of the Littlewood-Paley conjecture that the coefficients of odd univalent functions are bounded by 1. For some recent developments and examples, see [9,10].

A function $f\in \mathcal{A}$ is called a Bazilevič function in $\mathbb{U}$ if the following inequality holds true (see [11]):

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

On the other hand, a function $f\in \mathcal{A}$ is called a $\lambda$-pseudo-starlike function in $\mathbb{U}$ if the following inequality holds true (see [12]):

$$\Re \left(\frac{z{\left(f^{\prime }\left(z\right)\right)}^{\lambda }}{f\left(z\right)}\right)>0(z\in \mathbb{U};\lambda \geqq 1).$$

Next, we recall the definition of subordination between holomorphic functions. For two functions $f,g\in \mathcal{A}$, we say that the function f is subordinate to g, if there exists a Schwarz function $\omega$, which is holomorphic in $\mathbb{U}$ with the following property:

$$\omega \left(0\right)=0\mathrm{and}\left|\omega \right(z\left)\right|<1(z\in \mathbb{U}),$$

such that

$$f\left(z\right)=g\left(\omega \left(z\right)\right).$$

This subordination is symbolically written as follows:

$$f\prec g\mathrm{or}f\left(z\right)\prec g\left(z\right)(z\in \mathbb{U}).$$

It is well known that if the function g is univalent in $\mathbb{U}$, then the following equivalence holds true (see [13]):

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

Jackson [14,15] introduced the q-derivative operator ${\mathfrak{D}}_q$ of a function f as follows:

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

The following limit relationship is clear:

$$\underset{q\to 1-}{lim}{\mathfrak{D}}_{q}f\left(z\right)=f^{\prime }\left(z\right)\mathrm{and}{\mathfrak{D}}_{q}f\left(0\right)=f^{\prime }\left(0\right).$$

For more conceptual details on the q-derivative operator ${\mathfrak{D}}_q$, see [16,17,18].

For a function $f\in \mathcal{A}$ defined by (1), we deduce the following result:

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

where ${\left[n\right]}_q$, called the q-analogue of $n\in \mathbb{N}$, is given by

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

$\mathbb{N}$ being the set of positive integers.

As $q⟶1-$, we have ${\left[n\right]}_q⟶n$ and ${\left[0\right]}_q=0$.

The widely and extensively studied Srivastava-Attiya operator was defined by Srivastava and Attiya [19] by using the Hurwitz-Lerch zeta function $\Phi (z,s,a)$ which is systematically discussed in the recent survey articles [20,21]. For details about the relationships of the function $\Phi (z,s,a)$ with several important functions of the analytic number theory, the interested reader can refer to Chapter I in [22]).

Shah and Noor [23] (see also [24]) studied the following q-analogue of the Hurwitz-Lerch zeta function $\Phi (z,s,a)$:

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

(3)

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

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

(4)

By using (1) and (4), Shah and Noor [23] defined the q-Srivastava-Attiya operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f:\mathcal{A}⟶\mathcal{A}$ as follows:

Definition 1

(see [23]; see also [24]). The q-Srivastava-Attiya operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f:\mathcal{A}⟶\mathcal{A}$ is defined by

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

where the symbol ∗ stands for the Hadamard product (or convolution).

In recent years, several authors studied many applications of q-calculus associated with various families of holomorphic and univalent (or multivalent) functions (see, for example, [10,25,26,27,28,29,30,31,32,33]).

In his recently-published survey-cum-expository review article, Srivastava [34] explored the mathematical applications of q-calculus, fractional q-calculus and the fractional q-derivative operators in Geometric Function Theory of Complex Analysis. Srivastava [34] also exposed the not-yet-widely-understood fact that the so-called $(p,q)$-variation of classical q-calculus is a rather trivial and inconsequential variation of classical q-calculus, the additional parameter p being redundant or superfluous (see, for details, [34], p. 340).

Here, in this paper, we made use of the basic or quantum (or q-) extension ${\varphi }_q(\mathfrak{s},\mathfrak{t};z)$ which, when $q\to 1-$, yields the familiar Hurwitz-Lerch zeta function $\Phi (z,\mathfrak{s},\mathfrak{t})$. Just as we pointed out above, local or non-local symmetries are known to exist in some properties of several members of the family of the Hurwitz-Lerch zeta functions. Further motivation for our study of the applications of such quantum (or q-) extensions in this paper can be found in the book chapter entitled Symmetric Quantum Calculus, in [35].

Remark 1.

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

• For $q⟶1-$, the function ${\varphi }_q(\mathfrak{s},\mathfrak{t};z)$ reduces to the Hurwitz-Lerch zeta function (see [20,21]) and the operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ coincides with the Srivastava-Attiya operator in [19]. Various applications of the Srivastava-Attiya operator are found in [36,37,38] and in the references cited in each of these earlier works.
• For $\mathfrak{s}=1$, the operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ reduces to the q-Bernardi operator (see [39]).
• For $\mathfrak{s}=\mathfrak{t}=1$, the operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ reduces to the q-Libera operator (see [39]).
• For $q⟶1-$ and $\mathfrak{s}=1$, the operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ reduces to the Bernardi operator (see [40]).
• For $q⟶1-$, $\mathfrak{s}=1$ and $\mathfrak{t}=0$, the operator ${\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}$ reduces to the Alexander operator (see [41]).

Recently, Hörçum and Koçer [42] considered the familiar Horadam polynomials $h_n\left(r\right)$, which are given by Definition 2 below, from Geometric Function Theory of Complex Analysis.

Definition 2

(see [42,43]). The Horadam polynomials $h_n\left(r\right)$ are given by the following recurrence relation:

$$h_n\left(r\right)=\mathfrak{p}r{h}_{n-1}\left(r\right)+\mathfrak{q}{h}_{n-2}\left(r\right)(r\in \mathbb{R};n\in \mathbb{N}=\left\{1,2,3,\dots \right\})$$

(5)

with

$$h_1\left(r\right)=a\mathit{and}{h}_2\left(r\right)=br,$$

for some real constants $a,b,\mathfrak{p}$ and $\mathfrak{q}$. Moreover, the characteristic equation of the recurrence relation $\left(5\right)$ is given by

$$t^2-\mathfrak{p}rt-\mathfrak{q}=0,$$

which has the following two real roots:

$$\alpha =\frac{\mathfrak{p}r+\sqrt{{\mathfrak{p}}^2{r}^2+4\mathfrak{q}}}{2}\mathit{and}\beta =\frac{\mathfrak{p}r-\sqrt{{\mathfrak{p}}^2{r}^2+4\mathfrak{q}}}{2}.$$

Remark 2.

We record here some special cases of the Horadam polynomials $h_n\left(r\right)$ by appropriately choosing the parameters $a,$ $b,$ $\mathfrak{p}$ and $\mathfrak{q}$.

• Taking $a=b=\mathfrak{p}=\mathfrak{q}=1$, we obtain the Fibonacci polynomials $F_n\left(r\right)$.
• Taking $a=2$ and $b=\mathfrak{p}=\mathfrak{q}=1$, we get the Lucas polynomials $L_n\left(r\right)$.
• Taking $a=\mathfrak{q}=1$ and $b=\mathfrak{p}=2$, we have the Pell polynomials $P_n\left(r\right)$.
• Taking $a=b=\mathfrak{p}=2$ and $\mathfrak{q}=1$, we find the Pell-Lucas polynomials $Q_n\left(r\right)$.
• Taking $a=b=1$, $\mathfrak{p}=2$ and $\mathfrak{q}=-1$, we obtain the Chebyshev polynomials $T_n\left(r\right)$ of the first kind.
• Taking $a=1$, $b=\mathfrak{p}=2$ and $\mathfrak{q}=-1$, we have the Chebyshev polynomials $U_n\left(r\right)$ of the second kind.

For widespread usages and applications of various families of orthogonal polynomials and other special functions and specific polynomials, see [43,44,45,46].

The Horadam polynomials $h_n\left(r\right)$ are generated by (see [42]):

$$\Pi (r,z)=\sum _{n=1}^{\infty }{h}_n\left(r\right)z^{n-1}=\frac{a+(b-a\mathfrak{p})rz}{1-\mathfrak{p}rz-\mathfrak{q}{z}^2}.$$

(6)

The Horadam polynomials $h_n\left(r\right)$ were recently applied in a similar context by Srivastava et al. [47]. It was followed by many sequels to [47] (see, for example, [48,49,50,51,52,53,54]).

Remark 3.

The motivation of our present investigation stems, at least in part, from the need for the upper bounds of the Taylor-Maclaurin coefficients of normalized functions belonging to various subclasses of analytic and univalent (or multivalent) functions in the open unit disk $\mathbb{U}$. The proof of the celebrated 68-year-old Bieberbach conjecture, which is attributed to Ludwig Bieberbach $(1886$–$1982)$, by Louis de Branges in the year 1984 has indeed provided impetus to studies on coefficient estimate problems as well as on Fekete-Szegö type coefficient inequalities in recent years.

2. A Set of Main Results

We begin this section by defining the new family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$.

Definition 3.

For $0\leqq \delta \leqq 1$, $\gamma \geqq 0$, $\lambda \geqq 1$ and $r\in \mathbb{R},$ a function $f\in \Sigma$ is said to be in the family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$ if it fulfills the following subordination conditions:

$$\left(1-\delta \right)\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}+\delta \frac{z{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)}\prec \Pi (r,z)+1-a$$

and

$$\left(1-\delta \right)\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{1-\gamma }}+\delta \frac{w{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)}\prec \Pi (r,w)+1-a,$$

where a is real constant and the function $g=f^{-1}$ is given by $\left(2\right)$.

Remark 4.

For brevity and convenience, the notation ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$ for the holomorphic and bi-univalent function class, which we introduced in Definition 3 above, does not include the parameters $a,$ $b,$ $\mathfrak{p}$ and $\mathfrak{q}$ involved in Definition 1 of the Horadam polynomials $h_n\left(r\right)$. In fact, the role of each of these notationally left-out parameters $a,$ $b,$ $\mathfrak{p}$ and $\mathfrak{q},$ which is detailed above in Remark $2,$ is to relate the Horadam polynomials $h_n\left(r\right)$ with many simpler polynomial systems (see also Remark 5 below).

Our first main result is asserted by Theorem 1 below.

Theorem 1.

For $0\leqq \delta \leqq 1,$$\gamma \geqq 0,$$\lambda \geqq 1$ and $r\in \mathbb{R},$ let $f\in \mathcal{A}$ be in the family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Then

$$\left|a_2\right|\leqq \frac{\left|br\right|{[2+\mathfrak{t}]}_q^{\mathfrak{s}}\sqrt{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}}{\Lambda (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q;b.r)}$$

$$\begin{array}{cc}\hfill & (\Lambda (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q;b.r)\hfill \\ \hfill & :=\sqrt{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|})\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq \frac{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]·{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \hfill & +\frac{b^2{r}^2{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{{\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]}^2·{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}},\hfill \end{array}$$

where

$$\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)=\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}},$$

(7)

$$\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)=\left[\frac{1}{2}(1-\delta )(\gamma +2)(\gamma -1)+\delta \left(2\lambda (\lambda -2)+1\right)\right]{[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}$$

(8)

and

$${\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)={\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]}^2{[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}.$$

(9)

Proof. 

Let $f\in {\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Then there are two holomorphic functions $u,v$$:\mathbb{U}⟶\mathbb{U}$ given by

$$u\left(z\right)=u_{1}z+u_2{z}^2+u_3{z}^3+\dots (z\in \mathbb{U})$$

(10)

and

$$v\left(w\right)=v_{1}w+v_2{w}^2+v_3{w}^3+\dots (w\in \mathbb{U}),$$

(11)

with

$$u\left(0\right)=v\left(0\right)=0\mathrm{and}max\left\{\left|u\left(z\right)\right|,\left|v\left(w\right)\right|\right\}<1(z,w\in \mathbb{U}),$$

such that

$$\left(1-\delta \right)\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}+\delta \frac{z{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)}=\Pi (r,u\left(z\right))-a$$

and

$$\left(1-\delta \right)\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{1-\gamma }}+\delta \frac{w{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)}=\Pi (r,v\left(w\right))-a$$

or, equivalently,

$$\begin{array}{cc}\hfill & \left(1-\delta \right)\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}+\delta \frac{z{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)}\hfill \\ \hfill & =1+h_1\left(r\right)+h_2\left(r\right)u\left(z\right)+h_3\left(r\right)u^2\left(z\right)+\dots \hfill \end{array}$$

(12)

and

$$\begin{array}{cc}\hfill & \left(1-\delta \right)\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{1-\gamma }}+\delta \frac{w{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)}\hfill \\ \hfill & =1+h_1\left(r\right)+h_2\left(r\right)v\left(w\right)+h_3\left(r\right)v^2\left(w\right)+\dots .\hfill \end{array}$$

(13)

Combining (10)–(13) yields the following relation:

$$\begin{array}{cc}\hfill & \left(1-\delta \right)\frac{z^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{1-\gamma }}+\delta \frac{z{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}f\left(z\right)}\hfill \\ \hfill & =1+h_2\left(r\right)u_{1}z+\left[h_2\left(r\right)u_2+h_3\left(r\right)u_1^2\right]z^2+\dots \hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill & \left(1-\delta \right)\frac{w^{1-\gamma }{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }}{{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{1-\gamma }}+\delta \frac{w{\left[{\left({\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\mathfrak{J}}_{q,\mathfrak{t}}^{\mathfrak{s}}g\left(w\right)}\hfill \\ \hfill & =1+h_2\left(r\right)v_{1}w+\left[h_2\left(r\right)v_2+h_3\left(r\right)v_1^2\right]w^2+\dots .\hfill \end{array}$$

(15)

It is known that, if

$$max\left\{\left|u\left(z\right)\right|,\left|v\left(w\right)\right|\right\}<1(z,w\in \mathbb{U}),$$

then

$$\left|u_j\right|\leqq 1\mathrm{and}\left|v_j\right|\leqq 1(\forall j\in \mathbb{N}).$$

(16)

Now, by comparing the corresponding coefficients in (14) and (15), we find that

$$\frac{\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{{[2+\mathfrak{t}]}_q^{\mathfrak{s}}}{a}_2=h_2\left(r\right)u_1,$$

(17)

$$\begin{array}{cc}\hfill & \frac{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{a}_3\hfill \\ \hfill & +\frac{\left[\frac{1}{2}(1-\delta )(\gamma +2)(\gamma -1)+\delta \left(2\lambda (\lambda -2)+1\right)\right]{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}{{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{a}_2^2\hfill \\ \hfill & =h_2\left(r\right)u_2+h_3\left(r\right)u_1^2,\hfill \end{array}$$

(18)

$$-\frac{\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{{[2+\mathfrak{t}]}_q^{\mathfrak{s}}}{a}_2=h_2\left(r\right)v_1$$

(19)

and

$$\begin{array}{cc}\hfill & \frac{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\left(2a_2^2-a_3\right)\hfill \\ \hfill & +\frac{\left[\frac{1}{2}(1-\delta )(\gamma +2)(\gamma -1)+\delta \left(2\lambda (\lambda -2)+1\right)\right]{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}{{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{a}_2^2\hfill \\ \hfill & =h_2\left(r\right)v_2+h_3\left(r\right)v_1^2.\hfill \end{array}$$

(20)

It follows from (17) and (19) that

$$u_1=-v_1$$

(21)

and

$$\frac{2{\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]}^2{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}{{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{a}_2^2=h_2^2\left(r\right)(u_1^2+v_1^2).$$

(22)

If we add (18) to (20), we find that

$$\begin{array}{cc}\hfill & (\frac{2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \hfill & +\frac{2\left[\frac{1}{2}(1-\delta )(\gamma +2)(\gamma -1)+\delta \left(2\lambda (\lambda -2)+1\right)\right]{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}{{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}})a_2^2\hfill \\ \hfill & =h_2\left(r\right)(u_2+v_2)+h_3\left(r\right)(u_1^2+v_1^2).\hfill \end{array}$$

(23)

Upon substituting the value of $u_1^2+v_1^2$ from (22) into the right-hand side of (23), we deduce the following result:

$$a_2^2=\frac{h_2^3\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}(u_2+v_2)}{2\left[h_2^2\left(r\right)\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)-h_3\left(r\right){\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]},$$

(24)

where $\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)$, $\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)$ and ${\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)$ are given by (7)–(9), respectively.

By further computations using (5), (16) and (24), we obtain

$$\left|a_2\right|\leqq \frac{\left|br\right|{[2+\mathfrak{t}]}_q^{\mathfrak{s}}\sqrt{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}}{\Lambda (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q;b.r)}$$

$$\begin{array}{cc}\hfill & (\Lambda (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q;b.r)\hfill \\ \hfill & :=\sqrt{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}).\hfill \end{array}$$

Next, if we subtract (20) from (18), we can easily see that

$$\begin{array}{cc}\hfill & \frac{2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}\left(a_3-a_2^2\right)\hfill \\ \hfill & =h_2\left(r\right)(u_2-v_2)+h_3\left(r\right)(u_1^2-v_1^2).\hfill \end{array}$$

(25)

In the light of (21) and (22), we conclude from (25) that

$$\begin{array}{cc}\hfill a_3& =\frac{h_2\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}(u_2-v_2)}{2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \hfill & +\frac{h_2^2\left(r\right){[2+\mathfrak{t}]}_q^{2\mathfrak{s}}(u_1^2+v_1^2)}{2{\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]}^2{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}.\hfill \end{array}$$

Thus, by applying (5), we obtain the following inequality:

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq \frac{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \hfill & +\frac{b^2{r}^2{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}{{\left[(1-\delta )(\gamma +1)+\delta (2\lambda -1)\right]}^2{[1+\mathfrak{t}]}_q^{2\mathfrak{s}}}.\hfill \end{array}$$

This completes the proof of Theorem 1. □

In the next theorem, we present the Fekete-Szegö inequality for ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$.

Theorem 2.

For $0\leqq \delta \leqq 1,$$\gamma \geqq 0,$$\lambda \geqq 1$ and $r,\mu \in \mathbb{R},$ let $f\in \mathcal{A}$ be in the family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Then

$$\left|a_3-\mu a_2^2\right|\leqq \left\{\begin{array}{c}\frac{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \\ \left(\left|\phi -1\right|\leqq \frac{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}{b^2{r}^2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}\right)\hfill \\ \\ \frac{{\left|br\right|}^3{[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\left|\mu -1\right|}{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}\hfill \\ \\ \left(\left|\phi -1\right|\geqq \frac{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}{b^2{r}^2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}\right),\hfill \end{array}\right.$$

where, for convenience,

$$\phi =\phi (\mu ,r):=\frac{h_2^2\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\left(1-\mu \right)}{h_2^2\left(r\right)\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)-h_3\left(r\right){\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)}.$$

Proof. 

It follows from (24) and (25) that

$$\begin{array}{cc}\hfill a_3-\mu a_2^2& =\frac{h_2\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}(u_2-v_2)}{2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}+\left(1-\mu \right)a_2^2\hfill \\ \hfill & =\frac{h_2\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}(u_2-v_2)}{2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \hfill & +\frac{h_2^3\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}(u_2+v_2)\left(1-\mu \right)}{2\left[h_2^2\left(r\right)\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)-h_3\left(r\right){\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]}\hfill \\ \hfill & =\frac{h_2\left(r\right)}{2}[\left(\phi (\mu ,r)+\frac{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right)u_2\hfill \\ \hfill & +\left(\phi (\mu ,r)-\frac{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right)v_2],\hfill \end{array}$$

where, just as stated in Theorem 2,

$$\phi (\mu ,r)=\frac{h_2^2\left(r\right){[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\left(1-\mu \right)}{h_2^2\left(r\right)\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)-h_3\left(r\right){\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)}.$$

Thus, according to (5), we have the following inequality:

$$\left|a_3-\mu a_2^2\right|\leqq \left\{\begin{array}{c}\frac{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \\ \left(0\leqq \left|\phi (\mu ,r)\right|\leqq \frac{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right)\hfill \\ \\ \left|br\right|.\left|\phi (\mu ,r)\right|\hfill \\ \\ \left(\left|\phi (\mu ,r)\right|\geqq \frac{{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\right),\hfill \end{array}\right.$$

which, after simple computation, yields the following inequality:

$$\left|a_3-\mu a_2^2\right|\leqq \left\{\begin{array}{c}\frac{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}\hfill \\ \\ \left(\left|\phi -1\right|\leqq \frac{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)\mathfrak{b}-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}{b^2{r}^2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}\right)\hfill \\ \\ \frac{{\left|br\right|}^3{[3+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}\left|\mu -1\right|}{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}\hfill \\ \\ \left(\left|\phi -1\right|\geqq \frac{\left|\left[\left(\Omega (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)+\Gamma (\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right)b-\mathfrak{p}{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right]b{r}^2-\mathfrak{q}a{\rm Y}(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q)\right|}{b^2{r}^2\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}{[2+\mathfrak{t}]}_q^{2\mathfrak{s}}}\right).\hfill \end{array}\right.$$

We have thus completed the proof of Theorem 2. □

3. Special Cases and Consequences

In this section, we choose to specialize our main results asserted by Theorem 1 and Theorem 2.

By putting $\mu =1$ in Theorem 2, we are led to the following corollary.

Corollary 1.

For $0\leqq \delta \leqq 1,$$\gamma \geqq 0,$$\lambda \geqq 1$ and $r\in \mathbb{R},$ let $f\in \mathcal{A}$ be in the family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Then

$$\left|a_3-a_2^2\right|\leqq \frac{\left|br\right|{[3+\mathfrak{t}]}_q^{\mathfrak{s}}}{\left[(1-\delta )(\gamma +2)+\delta (3\lambda -1)\right]{[1+\mathfrak{t}]}_q^{\mathfrak{s}}}.$$

Remark 5.

The family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$ generalizes several known families of bi-univalent functions. We list them as follows.

• For $\mathfrak{s}=\delta =0$, we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:{\mathcal{N}}_{\Sigma }(\gamma ,r),$$

  where ${\mathcal{N}}_{\Sigma }(\gamma ,r)$ is the bi-univalent function family studied recently by Wanas and Lupas [54].
• For $\mathfrak{s}=\delta =\gamma =0$, we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:S_{\Sigma }^{\ast }\left(r\right),$$

  where $S_{\Sigma }^{\ast }\left(r\right)$ denote the bi-univalent function family studied by Srivastava et al. [47].
• For $\mathfrak{s}=\delta =0$ and $\gamma =1$, we have the following relationship:

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:{\Sigma }^{\prime }\left(r\right),$$

  where ${\Sigma }^{\prime }\left(r\right)$ is the bi-univalent function family introduced by Alamoush [49].
• For $\mathfrak{s}=\delta =0$, $a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$, we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:{\mathcal{B}}_{\Sigma }^{\gamma }\left(t\right),$$

  where ${\mathcal{B}}_{\Sigma }^{\gamma }\left(t\right)$ is the bi-univalent function family introduced by Bulut et al. [55].
• For $\mathfrak{s}=0$, $\delta =a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$, we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:{\mathcal{LB}}_{\Sigma }(\lambda ,t),$$

  where ${\mathcal{LB}}_{\Sigma }(\lambda ,t)$ is the bi-univalent function family investigated by Magesh and Bulut [56].
• For $\mathfrak{s}=\delta =\gamma =0$, $a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$, we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:S_{\Sigma }\left(t\right),$$

  where $S_{\Sigma }\left(t\right)$ is the bi-univalent function family given by Altınkaya and Yalçin [57].
• For $\mathfrak{s}=\delta =0$, $\gamma =a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$, we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:{\mathcal{B}}_{\Sigma }\left(t\right),$$

  where ${\mathcal{B}}_{\Sigma }\left(t\right)$ is the bi-univalent function family given by Bulut et al. [55].
• For $\mathfrak{s}=\delta =0$, $a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$, $r⟶t$ and

  $$\Pi (t,z)={\left(\frac{1}{1-2tz+z^2}\right)}^{\alpha }(0<\alpha \leqq 1),$$

  we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:P_{\Sigma }(\alpha ,\gamma ),$$

  where $P_{\Sigma }(\alpha ,\gamma )$ is the bi-univalent function family considered by Prema and Keerthi [58].
• For $\mathfrak{s}=0$, $\delta =a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$, $r⟶t$ and

  $$\Pi (t,z)={\left(\frac{1}{1-2tz+z^2}\right)}^{\alpha }(0<\alpha \leqq 1),$$

  we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:\mathfrak{L}{B}_{\Sigma }^{\lambda }\left(\alpha \right),$$

  where $\mathfrak{L}{B}_{\Sigma }^{\lambda }\left(\alpha \right)$ is the bi-univalent function family considered by Joshi et al. [59].
• For $\mathfrak{s}=\delta =\gamma =0$, $a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$, $r⟶t$ and

  $$\Pi (t,z)={\left(\frac{1}{1-2tz+z^2}\right)}^{\alpha }(0<\alpha \leqq 1),$$

  we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:S_{\Sigma }^{\ast }\left(\alpha \right),$$

  where $S_{\Sigma }^{\ast }\left(\alpha \right)$ is the bi-univalent function family introduced by Brannan and Taha [60].
• For $\mathfrak{s}=\delta =0$, $\gamma =a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$, $r⟶t$ and

  $$\Pi (t,z)={\left(\frac{1}{1-2tz+z^2}\right)}^{\alpha }(0<\alpha \leqq 1),$$

  we have

  $${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)=:{\mathcal{H}}_{\Sigma }^{\alpha },$$

  where ${\mathcal{H}}_{\Sigma }^{\alpha }$ is the bi-univalent function family considered by Srivastava et al. [2].

Remark 6.

For particular choices of $\mathfrak{s},$ $\delta ,$ $\gamma ,$ $a,$ $b,$ $\mathfrak{p}$ and $\mathfrak{q},$ Theorem 1 and Theorem 2 reduce to a number of known results, which are given below.

• If we put $\mathfrak{s}=\delta =0$ in our Theorems, we have the corresponding results for well-known family ${\mathcal{N}}_{\Sigma }(\gamma ,r)$ of bi-Bazilevič functions which was studied recently by Wanas and Lupas [54].
• If we put $\mathfrak{s}=\delta =\gamma =0$ in our Theorems, we have the corresponding results for the family $S_{\Sigma }^{\ast }\left(r\right)$, which was considered recently by Srivastava et al. [47].
• If we put $\mathfrak{s}=\delta =0$ and $\gamma =1$ in our Theorems, we have the corresponding results for the known family ${\Sigma }^{\prime }\left(r\right)$, which was studied recently by Al-Amoush [49].
• If we put $\mathfrak{s}=\delta =0$, $a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$ in our Theorems, we have the corresponding results for the family of ${\mathcal{B}}_{\Sigma }^{\gamma }\left(t\right)$ of bi-Bazilevič functions, which was discussed recently by Bulut et al. [55].
• If we put $\mathfrak{s}=0$, $\delta =a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$ in our Theorems, we have the corresponding results for the family ${\mathcal{LB}}_{\Sigma }(\lambda ,t)$ of bi-pseudo-starlike functions, which was studied recently by Magesh and Bulut [56].
• If we put $\mathfrak{s}=\delta =\gamma =0$, $a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$ in our Theorems, we obtain the corresponding results for the family $S_{\Sigma }\left(t\right)$ of bi-starlike functions, which was considered recently by Altınkaya and Yalçin [57].
• If we put $\mathfrak{s}=\delta =0$, $\gamma =a=1$, $b=\mathfrak{p}=2$, $\mathfrak{q}=-1$ and $r⟶t$ in our Theorems, we obtain the corresponding results for the family ${\mathcal{B}}_{\Sigma }\left(t\right)$ which was discussed recently by Bulut et al. [55].

4. Conclusions

The fact that we can find many unique and effective usages of a large variety of interesting special functions and specific polynomials in Geometric Function Theory of Complex Analysis provided the primary inspiration and motivation for our analysis in this article. Our main objective was to create a new family ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$ of holomorphic and bi-univalent functions, which is defined by means of the q-Srivastava-Attiya operator and by also using the Horadam polynomial $h_n\left(r\right)$ given by the recurrence relation (5) and by generating function $\Pi (r,z)$ in (6). We derived inequalities for the initial Taylor-Maclaurin coefficients of functions belonging to this newly-introduced holomorphic and bi-univalent function class ${\mathcal{SW}}_{\Sigma }(\delta ,\gamma ,\lambda ,\mathfrak{s},\mathfrak{t},q,r)$. Furthermore, we investigated the celebrated Fekete-Szegö problem for this general holomorphic and bi-univalent function class. We also pointed out several important correlations between our findings and those which were considered in previous studies.

We remark further that, since the additional parameter p is obviously superfluous, Srivastava ([34], p. 340) exposed the so-called $(p,q)$-calculus as a rather trivial and inconsequential variation of the classical q-calculus. So, clearly, while we do encourage and support the q-results of the kind which we have presented in this paper as well as potential q-extensions of other analogous developments in Applicable Mathematical Analysis, we do not encourage and support the so-called $(p,q)$-variations of the suggested q-results by inconsequentially and trivially adding a redundant parameter p.