Coefficients Inequalities for the Bi-Univalent Functions Related to q-Babalola Convolution Operator

Abstract

This article defines a new operator called the q-Babalola convolution operator by using quantum calculus and the convolution of normalized analytic functions in the open unit disk. We then study a new class of analytic and bi-univalent functions defined in the open unit disk associated with the q-Babalola convolution operator. The main results of the investigation include some upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szego inequalities for the functions in the new class. Many applications of the finds are highlighted in the corollaries based on the various unique choices of the parameters, improving the existing results in Geometric Function Theory.

1. Introduction

The q-calculus is the standard classical calculus without the concept of limits, where q stands for the quantum. Jackson started using the q-calculus in [1,2]. Later, studies on quantum groups led to the recognition of the geometrical meaning of the q-analysis. Additionally, it implies a connection between integrable systems and q-analysis. The q-analogue of the Baskakov–Durrmeyer operator, based on the q-analogue of the beta function, was defined and explored by Aral and Gupta [3,4,5]. The q-Picard and q-Gauss–Weierstrass singular-integral operators, covered in [6,7,8], are two further significant q-generalizations and q-extensions of complex operators. Wongsaijai and Sukantamala [9] methodically generalized a few subclasses of starlike functions by utilizing the q-calculus in the setting of Geometric Function Theory in a significant way. Srivastava’s book chapter introduced the generalized q-hypergeometric functions (see, for details, [10], p. 347). Additionally, Al-Shbeil et al. investigated and applied q-calculus in [11,12,13].

Let $\mathbb{C}$ be the complex plane. One can normalize an analytic function f defined in the open unit disk $\mathcal{U}=\{z\in \mathbb{C}:|z|<1\}$ in the following form by using $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1:$

$$f\left(z\right)=z+\sum _{k=2}^{\infty }{T}_k{z}^k.$$

(1)

Class $\mathfrak{A}$ and its subclass $\mathcal{S}$ are defined as follows (for more details, see [14,15]):

$$\begin{array}{ccc}\hfill \mathfrak{A}& =& \{f:\mathcal{U}⟶\mathbb{C}:f\mathrm{is}\mathrm{analytic}\mathrm{in}\mathcal{U}\mathrm{with}\mathrm{the}\mathrm{form}(\left)\right\},\hfill \\ \hfill \mathcal{S}& =& \{f\in \mathfrak{A}:f\mathrm{is}\mathrm{univalent}\mathrm{in}\mathcal{U}\}.\hfill \end{array}$$

If f and $f^{-1}$ are univalent in $\mathcal{U}$, then we say that f is bi-univalent function in $\mathcal{U}$. Below are some important examples of bi-univalent functions in $\mathcal{U}:$

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

Additionally, examples of univalent functions in $\mathcal{U}$ are

$$z-\frac{z^2}{2}\mathrm{and}\frac{z}{1-z^2}.$$

The Koebe one-quarter theorem [14] ensures that the image of $\mathcal{U}$ under every univalent function $f\in \mathcal{S}$ contains a disk of radius $\frac{1}{4}$. Thus, every univalent function f has an inverse $\mathcal{G}$ satisfying the following for some open disk of radius $r_0\left(f\right)\ge \frac{1}{4}:$

$$\mathcal{G}\left(f\left(z\right)\right)=z,f\left(\mathcal{G}\left(w\right)\right)=w,\left|w\right|<r_0\left(f\right)\mathrm{and}z,w\in \mathcal{U}.$$

By equating the coefficients of $\mathcal{G}$, we can easily obtain the following analytical extension to $\mathcal{U}$:

$$\mathcal{G}\left(w\right)=w-T_2{w}^2+(2T_2^2-T_3)w^3+\cdots .$$

(2)

Now, define another subclass $\Xi$$\subseteq \mathfrak{A}$ of the bi-univalent functions in $\mathcal{U}:$

$$\Xi =\left\{f\in \mathfrak{A}:f\mathrm{is}\mathrm{a}\text{bi-univalent}\mathrm{function}\mathrm{in}\mathcal{U}\right\},$$

which Lewin [16] investigated in 1967 and showed a bound of the coefficient $T_2$ that $|T_2|<1.51$ for $f\in \Xi .$ Following verse, Brannan and Clunie [17] hypothesized that $|T_2|\le \sqrt{2}$ and Netanyahu [18] demonstrated that $max|T_2|=1.3$ for $f\in \Xi$.

Next, an analytic function $s_1$ is called subordinated to an analytic function $s_2,$ denoted by $s_1\prec s_2,$ if there is an analytic function w with $w\left(0\right)=0$ and $\left|w\left(z\right)\right|<1$, such that

$$s_1\left(z\right)=s_2\left(w\left(z\right)\right).$$

In particular, if $s_2$ is a univalent function in $\mathcal{S},$ then $s_1\prec s_2$ if, and only if,

$$s_1\left(0\right)=s_2\left(0\right)\mathrm{and}{s}_1\left(\mathcal{U}\right)\subset s_2\left(\mathcal{U}\right).$$

The convolution of two analytic functions in $\mathfrak{A},$f by (1) and $g\left(z\right)=z+\sum _{k=2}^{\infty }{C}_k{z}^k,$ is defined by

$$(f*g)\left(z\right)=z+\sum _{k=2}^{\infty }{T}_k{C}_k{z}^k.$$

It is clear that $(f*g)\left(z\right)=(g*f)\left(z\right).$

The following q-derivative operator ${\mathfrak{D}}_q$ was introduced by Jackson [1,2] (for more information, see [19,20]).

$${\mathfrak{D}}_{q}f\left(z\right)=\frac{f\left(qz\right)-f\left(z\right)}{z(q-1)}=z^{-1}\left\{z+\sum _{k=2}^{\infty }{\left[k\right]}_q{T}_k{z}^k\right\}.$$

In particular, if $z=0,$ then ${\mathfrak{D}}_{q}f\left(0\right)=f^{\prime }\left(0\right)$. Also if $f\left(z\right)=z^k$ for $k\in \mathbb{N}$, then

$${\mathfrak{D}}_q{z}^k=\frac{{\left(zq\right)}^k-z^k}{z(q-1)}={\left[k\right]}_q{z}^{k-1},\mathrm{for}k\in \mathbb{N},$$

(3)

where

$${\left[k\right]}_q=\frac{q^k-1}{q-1}=1+q+q^2+\cdots +q^{k-1},\mathrm{for}k\in \mathbb{N}.$$

Hence

$$\underset{q⟶1}{lim}{\left[k\right]}_q=\underset{q⟶1}{lim}\frac{q^k-1}{q-1}=k,\mathrm{for}k\in \mathbb{N}.$$

We define a new operator, the q-Babalola convolution operator, in the theory of geometric functions by using the quantum calculus in the following definition, which will be used throughout this paper.

Definition 1.

Let f be an analytic function in $\mathfrak{A}.$ The q-Babalola convolution operator, denote by ${\Lambda }_{\chi ,q}f\left(z\right),$ is defined by

$${\Lambda }_{\chi ,q}f\left(z\right)=({\mu }_{a,q}*{\mu }_{\chi ,q}^{\left(-1\right)}*f)\left(z\right),$$

where

$${\mu }_{\chi ,q}=\frac{z}{{(1-qz)}^{\chi }(1-z)},\chi =a-t>-1,$$

and ${\mu }_{\chi ,q}^{(-1)}$ is given by

$$({\mu }_{a,q}*{\mu }_{\chi ,q}^{(-1)})\left(z\right)=\frac{z}{1-z}.$$

It follows from Definition 1 that

$$\begin{array}{ccc}\hfill {\Lambda }_{\chi ,q}f\left(z\right)& =z+\frac{qa+1}{-tq+qa+1}{T}_2{z}^2+\frac{q^{2}a(a+1)+2qa+2}{q(t-a)\left(q\right(t-a-1)-2)+2}{T}_3{z}^3-\cdots .\hfill & \hfill \end{array}$$

(4)

Which gives that

$${\Lambda }_{\chi ,q}f\left(z\right)=z+\sum _{k=2}^{\infty }{\left(k\right]}_q^c{T}_k{z}^k,$$

(5)

where

$${\left(k\right]}_q^c=\frac{1+q\left(a\right)+q^2\left(a\right)+\cdots +q^{k-1}\left(a\right)}{1+q\left(\chi \right)+q^2\left(\chi \right)+\cdots +q^{k-1}\left(\chi \right)},$$

$$q^{k-1}\left(a\right)=\frac{(a+k-2)!}{(a-1)!}\frac{q^{k-1}}{(k-1)!}\mathrm{and}{q}^{k-1}\left(\chi \right)=\frac{(\chi +k-2)!}{(\chi -1)!}\frac{q^{k-1}}{(k-1)!}.$$

Applying for (3),

$${\mathfrak{D}}_q{\Lambda }_{\chi ,q}f\left(z\right)=1+\sum _{k=2}^{\infty }{\left[k\right]}_q{\left(k\right]}_q^c{T}_k{z}^{k-1}.$$

(6)

Now one can verify that the newly defined q-Babalola convolution operator ${\Lambda }_{\chi ,q}$ generalizes several known operators by taking some specific coefficients ${\left(k\right]}_q^c.$

• If $t=a,$ then operator ${\Lambda }_{\chi ,q}$ derives the operator ${\mathcal{R}}_a^q$ introduced by Aldweby and Darus [21]:

  $$\begin{array}{ccc}\hfill {\mathcal{R}}_a^{q}f\left(z\right)& =& z+(aq+1)T_2{z}^2+\frac{q^{2}a(a+1)+2qa+2}{2}{T}_3{z}^3+\cdots \hfill \\ & =& z+\sum _{k=2}^{\infty }\frac{{[k+a-1]}_q!}{{\left[a\right]}_q!{[k-1]}_q!}{T}_k{z}^k=z+\sum _{k=2}^{\infty }{⋋}_a^q\left(k\right)T_k{z}^k,\hfill \end{array}$$

  where

  $${⋋}_a^q\left(2\right)=aq+1\mathrm{and}{⋋}_a^q\left(3\right)=\frac{q^{2}a(a+1)+2qa+2}{2}.$$
• If taking $q⟶1,$ then operator ${\Lambda }_{\chi ,q}$ derives the operator $L_t^a$ introduced by Babalola [22]:

  $$\begin{array}{ccc}\hfill {\mathcal{L}}_a^{t}f\left(z\right)& =& z+\frac{a+1}{a-t+1}{T}_2{z}^2+\frac{(a+1)(a+2)}{(a-t+1)(a-t+2)}{T}_3{z}^3+\cdots \hfill \\ & =& z+\sum _{k=2}^{\infty }\left[\frac{[a+k-1]!}{a!}\frac{[a-t]!}{[a+k-t-1]!}\right]T_k{z}^k=z+\sum _{k=2}^{\infty }{⨿}_t^a\left(k\right)T_k{z}^k,\hfill \end{array}$$

  where

  $${⨿}_t^a\left(2\right)=\frac{a+1}{a-t+1}\mathrm{and}{⨿}_t^a\left(3\right)=\frac{(a+1)(a+2)}{(a-t+1)(a-t+2)}.$$
• If $t=a$ and $q=1,$ then operator ${\Lambda }_{\chi ,q}$ derives the operator ${\mathcal{R}}_a$ introduced by Ruscheweyh [23]:

  $$\begin{array}{ccc}\hfill {\mathcal{R}}_{a}f\left(z\right)& =& z+(a+1)T_2{z}^2+\frac{(a+1)(a+2)}{2}{T}_3{z}^3+\cdots \hfill \\ & =& z+\sum _{k=2}^{\infty }\frac{\Gamma (a+k)}{(k-1)!\Gamma (a+1)}{T}_k{z}^k=z+\sum _{k=2}^{\infty }{\Omega }_a\left(k\right)T_k{z}^k,\hfill \end{array}$$

  where

  $${\Omega }_a\left(2\right)=(a+1)\mathrm{and}{\Omega }_a\left(3\right)=\frac{(a+1)(a+2)}{2}.$$

The motivation for the current paper came from pioneering work by Srivastava et al. [24], which has brought back interest in the study of analytic and bi-univalent functions in recent years. The success inspired an astonishingly enormous number of sequels by other authors ([25,26,27,28,29,30,31,32]). Recall the following Carathéodory’s lemma at the end of the section.

Lemma 1

([14]). If an analytic function φ has a positive real part $Re\left(\phi \left(z\right)\right)>0$ in the open unit disk $\mathcal{U}$ with $\phi \left(0\right)=1$ and is in the following form

$$\phi \left(z\right)=1+p_{1}z+p_2{z}^2+\cdots ,$$

then

$$\left|p_j\right|\le 2,\mathrm{for}j\in \mathbb{N}.$$

(7)

This inequality is sharp for each $j\in \mathbb{N}$.

2. Upper Bounds for the Coefficients

Srivastava et al. [33] initiated coefficient estimates for a general subclass of Ma-Minda type analytic and bi-univalent functions. To estimate the coefficient, Aldweby and Darus [30] considered a brand-new subclass of analytic and bi-univalent functions connected to the q-Ruscheweyh differential operator. Srivastava’s work was improved and generalized by Ali et al. [34], then by Deniz [35], later by Orhan et al. [36] and Murugusundaramoorthy and Bulut [37].

In this section, we construct new subclasses of Bazilevic functions of complex order from the class $\Xi$ of the analytic and bi-univalent functions in $\mathcal{U}$, associated with the q-Babalola convolution operator, and then find the upper bounds for the coefficients $T_2$ and $T_3$.

Definition 2.

The subclass of Ξ of analytic and bi-univalent functions is called a subclass of Bazilevic functions, denoted by $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta ),$ if each element f satisfying the following subordinations, where $\beta \in \mathbb{C}\setminus \left\{0\right\}$, $\chi =a-t>-1$, and $\psi \ge 0:$

$$1+\frac{1}{\beta }\left(\frac{z^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}{{\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}^{1-\psi }}-1\right)\prec \eta \left(z\right),$$

$$1+\frac{1}{\beta }\left(\frac{w^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}{{\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}^{1-\psi }}-1\right)\prec \eta \left(w\right).$$

In particular, if the analytic function $\eta \left(z\right)=\frac{1+(1-2\delta )z}{1-z}$ for $0\le \delta <1,$ then $f\in O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ if, and only if, it satisfies the following conditions:

$$Re\left[1+\frac{1}{\beta }\left(\frac{z^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}{{\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}^{1-\psi }}-1\right)\right]>\delta ,$$

$$Re\left[1+\frac{1}{\beta }\left(\frac{w^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}{{\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}^{1-\psi }}-1\right)\right]>\delta .$$

One can verify that the newly defined subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ generalizes several known families, including the examples below.

• Aldweby and Darus [21] discovered the following three particular families.
  If $\beta =1$, $t=a$, and $\psi =0$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )={\Xi }_q(a,\eta ).$
  If $\beta =1$, $t=a$, $\psi =0$ and $\eta \left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\tau }$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )={\Xi }_q(a,\tau ).$
  If $\beta =1$, $t=a$, $\psi =0$ and $\eta \left(z\right)=\frac{1+(1-2\delta )z}{1-z}$, $thenthesubclassO{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )={\Xi }_q(a,\delta ).$
• The following three particular families are discovered by Murugusundaramoorthy and Bulut [37]:
  If $t=a$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )={\mathcal{S}}_{\Xi }^q(\beta ,a,\psi ;\eta ).$
  If $t=a$ and $\eta \left(z\right)=\frac{1+(1-2\delta )z}{1-z}$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )={\mathcal{S}}_{\Xi }^q(\beta ,a,\psi ;\delta ).$
  If $t=a$ and $\eta \left(z\right)=\frac{1+Az}{1+Bz}$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )={\mathcal{S}}_{\Xi }^q(\beta ,a,\psi ;A,B)$.
• The following two particular families are discovered by Srivastava et al. [24]:
  If $\beta =1$, $t=a=0$, $\psi =1$, $q⟶1$ and $\eta \left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\tau }$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )=\Xi \left(\tau \right)$.
  If $\beta =1$, $t=a=0$, $\psi =1$, $q⟶1$ and $\eta \left(z\right)=\frac{1+(1-2\delta )z}{1-z}$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )=\Xi \left(\delta \right)$.
• The following particular family is discovered by Ali et al. [34]:
  If $\beta =1$, $t=a=0$, $\psi =1,$ and $q⟶1$, then the subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )=\Xi \left(\eta \right)$.

Theorem 1.

Let f be an analytic and bi-univalent function in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ with the form (1). Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le min\left\{\frac{\left|\beta \right|N_1}{{\left(2\right]}_q^c(q+\psi )},\sqrt{\frac{{2\left|\beta \right|}^2{N}_1^3}{M}}\right\},$$

$$|T_3|\le min\left\{\frac{2\left|\beta \right|N_2}{B_0}+\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )},\frac{{\left|\beta \right|}^2{N}_1^2}{{\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2}+\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )}\right\}.$$

where

$$\begin{array}{ccc}\hfill M& =& |\beta N_1^2(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+2\beta N_1^2{\left(3\right]}_q^c(\psi +q+q^2)-2(N_2-N_1){\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2|,\hfill \\ \hfill B_0& =& (\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+2{\left(3\right]}_q^c(q+q^2+\psi ).\hfill \end{array}$$

Proof. 

Since $f\in O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta ),$ there exist two analytic functions $\nu ,\epsilon :\mathcal{U}⟶\mathcal{U}$ with $\nu \left(0\right)=\epsilon \left(0\right)=0$ satisfying the following conditions.

$$1+\frac{1}{\beta }\left(\frac{z^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}{{\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}^{1-\psi }}-1\right)=\eta \left(\nu \left(z\right)\right),$$

(8)

$$1+\frac{1}{\beta }\left(\frac{w^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}{{\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}^{1-\psi }}-1\right)=\eta \left(\nu \left(w\right)\right).$$

(9)

Define the functions $e\left(z\right)$ and $c\left(z\right)$ by

$$e\left(z\right)=\frac{1+\nu \left(z\right)}{1-\nu \left(z\right)}=1+e_{1}z+e_2{z}^2+\cdots ,$$

$$c\left(z\right)=\frac{1+\epsilon \left(z\right)}{1-\epsilon \left(z\right)}=1+c_{1}z+c_2{z}^2+\cdots .$$

Equivalently,

$$\nu \left(z\right)=\frac{e\left(z\right)-1}{e\left(z\right)+1}=\frac{1}{2}\left\{e_{1}z+\left(e_2-\frac{e_1^2}{2}\right)z^2+\cdots \right\},$$

(10)

$$\epsilon \left(z\right)=\frac{c\left(z\right)-1}{c\left(z\right)+1}=\frac{1}{2}\left\{c_{1}z+\left(c_2-\frac{c_1^2}{2}\right)z^2+\cdots \right\}.$$

(11)

Thus, $e\left(z\right)$ and $c\left(z\right)$ are analytic in $\mathbb{C}$ with $e\left(0\right)=1=c\left(0\right)$. Since $\nu ,\epsilon :\mathcal{U}⟶\mathcal{U}$, the functions $e\left(z\right)$ and $c\left(z\right)$ have a positive real part in $\mathbb{C}$, and, hence, by (7), $|e_k|\le 2$ and $|c_k|\le 2$ for each k.

Applying (10) and (11) in (8) and (9), respectively,

$$1+\frac{1}{\beta }\left(\frac{z^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}{{\left({\Lambda }_{\chi ,q}f\left(z\right)\right)}^{1-\psi }}-1\right)=\eta \left(\frac{1}{2}\left[e_{1}z+\left(e_2-\frac{e_1^2}{2}\right)z^2+\cdots \right]\right),$$

(12)

$$1+\frac{1}{\beta }\left(\frac{w^{1-\psi }{\mathfrak{D}}_q\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}{{\left({\Lambda }_{\chi ,q}\mathcal{G}\left(w\right)\right)}^{1-\psi }}-1\right)=\eta \left(\frac{1}{2}\left[c_{1}z+\left(c_2-\frac{c_1^2}{2}\right)z^2+\cdots \right]\right).$$

(13)

It follows from (5), (6), (12), and (13) that

$$\begin{array}{cc}\hfill 1+\frac{{\left(2\right]}_q^c(q+\psi )}{\beta }{T}_{2}z& +\frac{1}{\beta }\left[{\left(3\right]}_q^c(q+q^2+\psi )T_3+\frac{(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )}{2}{T}_2^2\right]z^2+\cdots \hfill \\ \hfill & =1+\frac{1}{2}{N}_1{e}_{1}z+\left[\frac{1}{2}{N}_1\left(e_2-\frac{e_1^2}{2}\right)+\frac{1}{4}{N}_2{e}_1^2\right]z^2+\cdots ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill 1& -\frac{{\left(2\right]}_q^c(q+\psi )}{\beta }{T}_{2}w+\frac{1}{\beta }[-{\left(3\right]}_q^c(q+q^2+\psi )T_3+(2{\left(3\right]}_q^c(q+q^2+\psi )\hfill \\ \hfill & +\frac{(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )}{2}\left)T_2^2\right]w^2+\cdots =1+\frac{1}{2}{N}_1{c}_{1}w+[\frac{1}{2}{N}_1\left(c_2-\frac{c_1^2}{2}\right)\hfill \\ \hfill & +\frac{1}{4}{N}_2{c}_1^2]w^2+\cdots .\hfill \end{array}$$

Which yield the following relations:

$${\left(2\right]}_q^c(q+\psi )T_2=\frac{\beta }{2}{N}_1{e}_1,$$

(14)

$${\left(3\right]}_q^c(q+q^2+\psi )T_3+\frac{(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )}{2}{T}_2^2=\frac{\beta }{2}{N}_1\left(e_2-\frac{e_1^2}{2}\right)+\frac{\beta }{4}{N}_2{e}_1^2,$$

(15)

$$-{\left(2\right]}_q^c(q+\psi )T_2=\frac{\beta }{2}{N}_1{c}_1,$$

(16)

$$\begin{array}{cc}\hfill -{\left(3\right]}_q^c(q+q^2+\psi )T_3+(2{\left(3\right]}_q^c(q+q^2+\psi )& +\frac{(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )}{2})T_2^2\hfill \\ \hfill & =\frac{\beta }{2}{N}_1\left(c_2-\frac{c_1^2}{2}\right)+\frac{\beta }{4}{N}_2{c}_1^2.\hfill \end{array}$$

(17)

From (14) and (16),

$$e_1=-c_1,$$

(18)

$$8{\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2{T}_2^2={\beta }^2{N}_1^2(e_1^2+c_1^2).$$

(19)

Additionally, from (15) and (17),

$$\begin{array}{cc}\hfill (\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )T_2^2+2{\left(3\right]}_q^c(\psi +q+q^2)T_2^2& =\frac{\beta N_1}{2}(e_2+c_2)\hfill \\ \hfill & +\frac{\beta }{4}(N_2-N_1)(e_1^2+c_1^2).\hfill \end{array}$$

(20)

Using (19), we have

$$T_2^2=\frac{{\beta }^2{N}_1^3(e_2+c_2)}{M_0},$$

(21)

where

$$M_0=2\beta N_1^2(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+4\beta N_1^2{\left(3\right]}_q^c(\psi +q+q^2)-4(N_2-N_1){\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2.$$

In (19) and (21), Lemma 1 gives

$$|T_2|\le \sqrt{\frac{{2\left|\beta \right|}^2{N}_1^3}{M}},$$

(22)

where

$$M=|\beta N_1^2(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+2\beta N_1^2{\left(3\right]}_q^c(\psi +q+q^2)-2(N_2-N_1){\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2|.$$

Thus,

$$|T_2|\le \frac{\left|\beta \right|N_1}{{\left(2\right]}_q^c(q+\psi )}.$$

(23)

So, Equations (22) and (23) give the estimate of $T_2$.

To get the upper bound for $T_3$, subtract (17) from (15) and also from (18), which gives us $e_1^2=c_1^2$.

$$2{\left(3\right]}_q^c(q+q^2+\psi )T_3-2{\left(3\right]}_q^c(\psi +q+q^2)T_2^2=\frac{\beta }{2}{N}_1(e_2-c_2).$$

(24)

Therefore, substituting the value of $T_2^2$ from (19) into (24),

$$T_3=\frac{{\beta }^2{N}_1^2(e_1^2+c_1^2)}{8{\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2}+\frac{\beta N_1(e_2-c_2)}{4{\left(3\right]}_q^c(q+q^2+\psi )}.$$

We have

$$|T_3|\le \frac{{\left|\beta \right|}^2{N}_1^2}{{\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2}+\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )}.$$

Next, substituting the value of $T_2^2$ from (20) into (24),

$$T_3=\frac{\beta N_1(e_2+c_2)}{2B_0}+\frac{\beta (N_2-N_1)(e_1^2+c_1^2)}{4B_0}+\frac{\beta N_1(e_2-c_2)}{4{\left(3\right]}_q^c(q+q^2+\psi )}.$$

Hence

$$|T_3|\le \frac{2\left|\beta \right|N_2}{B_0}+\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )},$$

where

$$B_0=(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+2{\left(3\right]}_q^c(q+q^2+\psi ).$$

□

Remark 1.

The upper bounds for coefficients $T_1$ and $T_2$ in Theorem 1 are sharp. One can verify that the following is an analytic and bi-univalent function in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ with the form (1), which takes the upper bounds.

$$f\left(z\right)=1+\frac{2\beta }{{\left(2\right]}_q^c\left(q+\psi \right)}z+\left(\frac{4\beta }{B_0}+\frac{2\beta }{{\left(2\right]}_q^c\left(q+q^2+\psi \right)}\right)z^2+\cdots .$$

As applications of Theorem 1 about upper bounds for coefficients $T_1$ and $T_2$ for the analytic and bi-univalent functions in the new class $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$, we obtain and improve the known results by Murugusundaramoorthy and Bulut [37] in the following corollaries by setting the particular values of the parameters $\psi ,t,$ and $\eta .$

Corollary 1

([37]). Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $\psi =0$ and $t=a.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le \sqrt{\frac{{2\left|\beta \right|}^2{N}_1^3}{M}},$$

$$|T_3|\le \frac{{\left|\beta \right|}^2{N}_1^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{q}^2}+\frac{\left|\beta \right|N_1}{q{⋋}_a^q\left(3\right)(1+q)},$$

where

$$M=|2\beta N_1^2{\left({⋋}_a^q\left(2\right)\right)}^{2}q+2q\beta N_1^2{⋋}_a^q\left(3\right)(1+q)-2(N_2-N_1){\left({⋋}_a^q\left(2\right)\right)}^2{q}^2|.$$

The known estimates in Corollary 1 are improved by the following corollary.

Corollary 2.

Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $\psi =0$ and $t=a.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le min\left\{\frac{\left|\beta \right|N_1}{q{⋋}_a^q\left(2\right)},\sqrt{\frac{{2\left|\beta \right|}^2{N}_1^3}{M}}\right\},$$

$$|T_3|\le min\left\{\frac{2\left|\beta \right|N_2}{B_0}+\frac{\left|\beta \right|N_1}{q{⋋}_a^q\left(3\right)(1+q)},\frac{{\left|\beta \right|}^2{N}_1^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{q}^2}+\frac{\left|\beta \right|N_1}{q{⋋}_a^q\left(3\right)(1+q)}\right\}.$$

where

$$M=|2\beta N_1^2{\left({⋋}_a^q\left(2\right)\right)}^{2}q+2q\beta N_1^2{⋋}_a^q\left(3\right)(1+q)-2(N_2-N_1){\left({⋋}_a^q\left(2\right)\right)}^2{q}^2|,$$

$$B_0=2q(1+q){⋋}_a^q\left(3\right)-2q{\left({⋋}_a^q\left(2\right)\right)}^2.$$

Corollary 3

([37]). Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $\psi =1$ and $t=a.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le \sqrt{\frac{{2\left|\beta \right|}^2{N}_1^3}{|2\beta N_1^2{⋋}_a^q\left(3\right)(1+q+q^2)-2(N_2-N_1){\left({⋋}_a^q\left(2\right)\right)}^2{(q+1)}^2|}},$$

$$|T_3|\le \frac{{\left|\beta \right|}^2{N}_1^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{(q+1)}^2}+\frac{\left|\beta \right|N_1}{{⋋}_a^q\left(3\right)(q+q^2+1)}.$$

The following also improves the known estimates in Corollary 3.

Corollary 4.

Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $\psi =1$ and $t=a.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le min\left\{\frac{\left|\beta \right|N_1}{{⋋}_a^q\left(2\right)(q+1)},\sqrt{\frac{{2\left|\beta \right|}^2{N}_1^3}{M}}\right\},$$

$$|T_3|\le min\left\{\frac{2\left|\beta \right|N_2}{B_0}+\frac{\left|\beta \right|N_1}{{⋋}_a^q\left(3\right)(q+q^2+1)},\frac{{\left|\beta \right|}^2{N}_1^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{(q+1)}^2}+\frac{\left|\beta \right|N_1}{{⋋}_a^q\left(3\right)(q+q^2+1)}\right\},$$

where

$$M=|2\beta N_1^2{⋋}_a^q\left(3\right)(1+q+q^2)-2(N_2-N_1){\left({⋋}_a^q\left(2\right)\right)}^2{(q+1)}^2|,$$

$$B_0=2{⋋}_a^q\left(3\right)(q+q^2+1).$$

Corollary 5

([37]). Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $t=a$ and $\eta \left(z\right)=\frac{1+Az}{1+Bz}.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le \sqrt{\frac{{2\left|\beta \right|}^2{(A-B)}^2}{M}},$$

$$|T_3|\le \frac{{\left|\beta \right|}^2{(A-B)}^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{(q+\psi )}^2}+\frac{\left|\beta \right|(A-B)}{{⋋}_a^q\left(3\right)(q+q^2+\psi )},$$

where

$$\begin{array}{c}\hfill M=|\beta (A-B)(\psi -1){\left({⋋}_a^q\left(2\right)\right)}^2(2q+\psi )+2\beta (A-B){⋋}_a^q\left(3\right)(\psi +q+q^2)\\ \hfill +2(B+1){\left({⋋}_a^q\left(2\right)\right)}^2{(q+\psi )}^2|.\end{array}$$

The known estimates in Corollary 5 are improved by the following corollary.

Corollary 6.

Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $t=a$ and $\eta \left(z\right)=\frac{1+Az}{1+Bz}.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le min\left\{\frac{\left|\beta \right|(A-B)}{{⋋}_a^q\left(2\right)(q+\psi )},\sqrt{\frac{{2\left|\beta \right|}^2{(A-B)}^2}{M}}\right\},$$

$$|T_3|\le min\left\{\frac{2\left|\beta \right|B(B-A)}{B_0}+\frac{\left|\beta \right|(A-B)}{{⋋}_a^q\left(3\right)(q+q^2+\psi )},\frac{{\left|\beta \right|}^2{(A-B)}^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{(q+\psi )}^2}+\frac{\left|\beta \right|(A-B)}{{⋋}_a^q\left(3\right)(q+q^2+\psi )}\right\},$$

where

$$\begin{array}{c}\hfill M=|\beta (A-B)(\psi -1){\left({⋋}_a^q\left(2\right)\right)}^2(2q+\psi )+2\beta (A-B){⋋}_a^q\left(3\right)(\psi +q+q^2)\\ \hfill +2(B+1){\left({⋋}_a^q\left(2\right)\right)}^2{(q+\psi )}^2|,\end{array}$$

$$B_0=(\psi -1){\left({⋋}_a^q\left(2\right)\right)}^2(2q+\psi )+2{⋋}_a^q\left(3\right)(q+q^2+\psi ).$$

Corollary 7

([37]). Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $t=a$ and $\eta \left(z\right)=\frac{1+(1-2\delta )z}{1-z}.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le \sqrt{\frac{4\left|\beta \right|(1-\delta )}{|(\psi -1){\left({⋋}_a^q\left(2\right)\right)}^2(2q+\psi )+2{⋋}_a^q\left(3\right)(\psi +q+q^2)|}},$$

$$|T_3|\le \frac{{4\left|\beta \right|}^2{(1-\delta )}^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{(q+\psi )}^2}+\frac{2\left|\beta \right|(1-\delta )}{{⋋}_a^q\left(3\right)(q+q^2+\psi )}.$$

The known estimates in Corollary 7 are improved by the following corollary.

Corollary 8.

Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ by taking $t=a$ and $\eta \left(z\right)=\frac{1+(1-2\delta )z}{1-z}.$ Then the upper bounds for two initial Taylor–Maclaurin coefficients are

$$|T_2|\le min\left\{\frac{\left|\beta \right|(1-\delta )}{{⋋}_a^q\left(2\right)(q+\psi )},\sqrt{\frac{4\left|\beta \right|(1-\delta )}{M}}\right\},$$

$$|T_3|\le min\left\{\frac{4\left|\beta \right|(1-\delta )}{B_0}+\frac{2\left|\beta \right|(1-\delta )}{{⋋}_a^q\left(3\right)(q+q^2+\psi )},\frac{{4\left|\beta \right|}^2{(1-\delta )}^2}{{\left({⋋}_a^q\left(2\right)\right)}^2{(q+\psi )}^2}+\frac{2\left|\beta \right|(1-\delta )}{{⋋}_a^q\left(3\right)(q+q^2+\psi )}\right\},$$

where

$$M=|(\psi -1){\left({⋋}_a^q\left(2\right)\right)}^2(2q+\psi )+2{⋋}_a^q\left(3\right)(\psi +q+q^2)|,$$

$$B_0=(\psi -1){\left({⋋}_a^q\left(2\right)\right)}^2(2q+\psi )+2{⋋}_a^q\left(3\right)(q+q^2+\psi ).$$

3. Fekete–Szego Inequalities

The Fekete–Szegö functional $|T_3-\phi T_2^2|$ for analytic and univalent functions of $\mathcal{S}$ arose from the disproof [38] of the Littlewood–Paley conjecture that unity by Fekete and Szegö bounds the coefficients of odd univalent functions. Recently, researchers achieved outstanding achievements in this area (see [25,39,40,41,42,43]). The Fekete–Szego type inequalities for functions of the newly defined subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ are given in the following theorem.

Theorem 2.

Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ for $\phi \in \mathbb{R}.$ Then the upper bound for the Fekete–Szegö functional is

$$\left|T_3-\phi T_2^2\right|\le \left\{\begin{array}{ccc}\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )},\hfill & \hfill & |\phi -1|\le \frac{\left|M\right|}{2\beta N_1^2{\left(3\right]}_q^c(q+q^2+\psi )},\hfill \\ \hfill & \hfill & \\ \frac{{2\left|\beta \right|}^2{N}_1^3(\phi -1)}{\left|M\right|},\hfill & \hfill & |\phi -1|\ge \frac{\left|M\right|}{2\beta N_1^2{\left(3\right]}_q^c(q+q^2+\psi )}.\hfill \end{array}\right.$$

where

$$M=\beta N_1^2(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+2\beta N_1^2{\left(3\right]}_q^c(\psi +q+q^2)-2(N_2-N_1){\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2.$$

The above upper bound for the Fekete–Szegö functional is sharp. The following analytic and bi-univalent function takes the bound.

$$f\left(z\right)=1+\frac{2\beta }{{\left(2\right]}_q^c\left(q+\psi \right)}z+\left(\frac{4\beta }{B_0}+\frac{2\beta }{{\left(2\right]}_q^c\left(q+q^2+\psi \right)}\right)z^2+\cdots .$$

Proof. 

From (21) and (24),

$$\begin{array}{cc}\hfill T_3-\phi T_2^2& =\frac{\beta N_1(e_2-c_2)}{4{\left(3\right]}_q^c(q+q^2+\psi )}+(1-\phi )T_2^2\hfill \\ \hfill & =\frac{\beta N_1(e_2-c_2)}{4{\left(3\right]}_q^c(q+q^2+\psi )}+\frac{{\beta }^2{N}_1^3(e_2+c_2)(1-\phi )}{M_0},\hfill \end{array}$$

where

$$M_0=2\beta N_1^2(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+4\beta N_1^2{\left(3\right]}_q^c(\psi +q+q^2)-4(N_2-N_1){\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2,$$

$$T_2-\phi T_2^2=\frac{\beta }{2}\left[\left(\xi \left(\vartheta \right)+\frac{1}{2{\left(3\right]}_q^c(q+q^2+\psi )}\right)e_2+\left(\xi \left(\vartheta \right)-\frac{1}{2{\left(3\right]}_q^c(q+q^2+\psi )}\right)c_2\right],$$

$$\xi \left(\vartheta \right)=\frac{\beta N_1^2(1-\phi )}{M},$$

$$M=\beta N_1^2(\psi -1){\left({\left(2\right]}_q^c\right)}^2(2q+\psi )+2\beta N_1^2{\left(3\right]}_q^c(\psi +q+q^2)-2(N_2-N_1){\left({\left(2\right]}_q^c\right)}^2{(q+\psi )}^2.$$

Using (7) in Lemma 1,

$$\left|T_3-\phi T_2^2\right|\le \left\{\begin{array}{ccc}\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )},\hfill & \hfill & 0\le \left|\xi \left(\vartheta \right)\right|\le \frac{1}{2{\left(3\right]}_q^c(q+q^2+\psi )},\hfill \\ \hfill & \hfill & \\ N_1\left|\xi \left(\vartheta \right)\right|,\hfill & \hfill & \left|\xi \left(\vartheta \right)\right|\ge \frac{1}{2{\left(3\right]}_q^c(q+q^2+\psi )}.\hfill \end{array}\right.$$

With some calculations, we have

$$\left|T_3-\phi T_2^2\right|\le \left\{\begin{array}{ccc}\frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )},\hfill & \hfill & |\phi -1|\le \frac{\left|M\right|}{2\beta N_1^2{\left(3\right]}_q^c(q+q^2+\psi )},\hfill \\ \hfill & \hfill & \\ \frac{{2\left|\beta \right|}^2{N}_1^3(\phi -1)}{\left|M\right|},\hfill & \hfill & |\phi -1|\ge \frac{\left|M\right|}{2\beta N_1^2{\left(3\right]}_q^c(q+q^2+\psi )}.\hfill \end{array}\right.$$

□

In particular, Theorem 2 gives the following corollary.

Corollary 9.

Let f be an analytic and bi-univalent function with the form (1) in $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ for $\phi =1.$ Then the upper bound for the Fekete–Szegö functional is

$$|T_3-T_2^2|\le \frac{\left|\beta \right|N_1}{{\left(3\right]}_q^c(q+q^2+\psi )}.$$

4. Conclusions

The q-calculus is a broad discipline that has applications in many branches of mathematics and physics, as well as in other fields, such as quantum group theory, analytic number theory, special polynomials, numerical analysis, fractional calculus, and other related theories. The orthogonal polynomials and special functions have played a massive role in mathematics, physics, engineering, and other research disciplines in recent decades.

In this paper, we used q-calculus and q-convolution operators to systematically define new subclasses of functions that were prompted mainly by recent research in Geometric Function Theory. We solved the problems about finding the sharp upper bounds for the Taylor–Maclaurin coefficients $T_2$ and $T_3$ and the inequalities about the Fekete–Szegö functional $|T_3-\phi T_2^2|$ for the newly established subclass $O{B}_{\Xi }(\beta ,\psi ,a,t,q;\eta )$ of analytic and bi-univalent functions in the open unit disk (see Theorem 1 and Theorem 2). The novel q-Babalola convolution operator is used to achieve these new results. Additionally, the paper showed in Corollaries 1–8 how the findings are improved and generalized many existing interesting results and implications by the appropriate specification of the parameters $\beta ,\psi ,a,t,q,\mathrm{and}\eta$, including some recently published results.

Several interesting corollaries from the main theorems can be derived by specifying the operator’s parameters. This is left to the interested readers. The symmetry properties of this newly introduced operator can be studied in future research directions.