The Fekete–Szegö Inequality for a Certain Subclass of Analytic Functions of Complex Order Related to the $\mathfrak{q}$-Srivastava–Attiya Operator

Abstract

The use of integral and differential operators in geometric function theory has continued to gain interest among researchers in the field of study in recent times. This is due to the wide range of its applications in science, technology and engineering. In this work, therefore, the authors defined and investigated a new subclass of analytic functions in the open unit disk using the $\mathfrak{q}$-Srivastava–Attiya convolution operator and the Jackson’s $\mathfrak{q}$-derivative, by means of the subordination. The authors used two well-known lemmas to determine a sharp upper-bound for the Fekete–Szeg$\ddot{o}$ functional in two different cases. In particular, the authors introduced a new generalized subclass of complex order univalent functions denoted by ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$ and derived the coefficient estimates $a_{\iota }(\iota =2,3)$ of the Taylor–Maclaurin series in this class, as well as the Fekete–Szeg$\ddot{o}$ inequality $\left|a_3-\aleph a_2^2\right|$ for functions in this class. The work generalizes many known results in the literature.

1. Introduction and Preliminaries

Let $\mathcal{A}$ be the class of analytic functions in ℧, that can be expressed in the form:

$$\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{a}_{\iota }{\zeta }^{\iota },\zeta \in \mho ,$$

(1)

where $\mho :=\left\{\zeta \in \mathbb{C}:\left|\zeta \right|<1\right\},$ [1] is the unit disk, normalized by the conditions $\mathfrak{f}\left(0\right)={\mathfrak{f}}^{\prime }\left(0\right)-1=0$. Let $\mathsf{{\rm Y}}$ denote the class of Schwarz functions defined as

$$\mathsf{{\rm Y}}=\{\varpi \in \mathcal{A}:\varpi (0)=0\mathrm{and}|\varpi \left(\xi \right)|<1\mathrm{for}\mathrm{all}\xi \in \mho \}.$$

(2)

If there exists a Schwarz function $\varpi \left(\xi \right)\in \mathsf{{\rm Y}}$, analytic in ℧, such that for analytic functions $\mathfrak{f}$ and $\mathfrak{g}$ in ℧, $\mathfrak{f}$ is said to be subordinate to $\mathfrak{g}$ (denoted $\mathfrak{f}\left(\xi \right)\prec \mathfrak{g}\left(\xi \right)$) if

$$\mathfrak{f}\left(\xi \right)=\mathfrak{g}\left(\varpi \right(\xi \left)\right)\mathrm{for}\mathrm{all}(\xi \in \mho ).$$

If $\mathfrak{g}$ is univalent in ℧, then this subordination $\mathfrak{f}\left(\xi \right)\prec \mathfrak{g}\left(\xi \right)$ is equivalent to $\mathfrak{f}\left(0\right)=\mathfrak{g}\left(0\right)\mathrm{and}\mathfrak{f}(\mho )\subset \mathfrak{g}(\mho ),$ as discussed in [2,3]. For functions $\mathfrak{f}\left(\zeta \right),\mathfrak{g}\left(\zeta \right)\in \mathcal{A}$, which $\mathfrak{g}\left(\zeta \right)$ is given by

$$\mathfrak{g}\left(\zeta \right)=\zeta +\sum _{\iota =2}^{\infty }{b}_{\iota }{\zeta }^{\iota },\zeta \in \mho .$$

Therefore, the functions $\mathfrak{f}\left(\zeta \right)$ and $\mathfrak{g}\left(\zeta \right)$ have a convolution (or Hadamard product), which is given by

$$(\mathfrak{f}\ast \mathfrak{g})\zeta =\zeta +\sum _{\iota =2}^{\infty }{a}_{\iota }{b}_{\iota }{\zeta }^{\iota }=(\mathfrak{g}\ast \mathfrak{f})\zeta ,\zeta \in \mho .$$

The theory of analytic functions is a fundamental area of geometric function theory, where the geometric and analytic behavior of functions defined on the open unit disk ℧ is investigated. A central topic in this field is the study of coefficient problems, since the coefficients of analytic functions carry essential information about the associated mappings. In this paper, we obtain the coefficient estimates, among which the Fekete–Szeg$\ddot{o}$ inequality has emerged as one of the most influential classical results. Introduced by Fekete and Szeg$\ddot{o}$ in 1933 [4], this inequality has since been extensively studied and generalized for a wide range of subclasses of analytic functions, including star-like and convex functions.

In recent years, increasing attention has been directed toward extending the classical framework of geometric function theory by incorporating tools from $\mathfrak{q}$-calculus. Due to its inherent flexibility, $\mathfrak{q}$-calculus provides an effective setting for constructing generalized operators and defining new subclasses of analytic functions. This modern approach has enabled researchers to revisit and refine classical coefficient problems, including those related to the Fekete–Szeg$\ddot{o}$ functional, within a broader and more unified analytical framework, leading to deeper insights and further generalizations of known results. The use of Fekete–Szeg$\ddot{o}$ functional as a useful analytical tool to map the boundary between different behaviours of functions in the complex plane, usually resulting in improved, snugger and better results than those obtained from simple individual coefficient studies is one of the reasons for its application and a major motivation for this study.

Motivated by these developments, the present paper introduces a new general subclass of analytic functions of complex order defined via the principle of subordination in the setting of $\mathfrak{q}$-calculus. For this newly defined class, Fekete–Szeg$\ddot{o}$-type inequalities are established. Several corollaries and special cases are also derived by choosing appropriate values of the involved parameters, demonstrating that a number of existing results in the literature can be obtained as particular cases of our findings. Consequently, the results presented herein contribute to the ongoing development of coefficient problems in modern geometric.

Researchers are highly interested in quantum (or $\mathfrak{q}$-) calculus because of its applications in mathematics and related fields. It is a valuable tool for exploring various families of analytic functions and applying different operators in both $\mathfrak{q}$-calculus and fractional $\mathfrak{q}$-calculus, as Srivastava [5] utilized various operators in $\mathfrak{q}$-calculus and fractional $\mathfrak{q}$-calculus. First, we present some fundamental definitions of the $\mathfrak{q}$ calculus that will be useful in the following sections. We will also introduce some notation and concepts relevant to this study. Initially, as described in [6,7], the $\mathfrak{q}$-derivative operator for the function $\mathfrak{f}$ is given by the following definition:

$$D_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\right):=\left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{f}\left(\zeta \right)-\mathfrak{f}\left(\mathfrak{q}\zeta \right)}{(1-\mathfrak{q})\zeta }},\hfill & \zeta \ne 0\hfill \\ \\ {\mathfrak{f}}^{\prime }\left(0\right),\hfill & \zeta =0.\hfill \end{array}\right.$$

When $\mathfrak{f}$ is differentiable at $\zeta$, we have ${lim}_{\mathfrak{q}\to 1}{D}_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\right)={\mathfrak{f}}^{\prime }\left(\zeta \right)$. Since for $\iota \in \mathbb{N}$ and $\zeta \in \mho$, we have

$$D_{\mathfrak{q}}\left\{\sum _{\iota =1}^{\infty }{a}_{\iota }{\zeta }^{\iota }\right\}=\sum _{\iota =1}^{\infty }{\left[\iota \right]}_{\mathfrak{q}}{a}_{\iota }{\zeta }^{\iota -1}$$

where

$${\left[\iota \right]}_{\mathfrak{q}}={\displaystyle \frac{1-{\mathfrak{q}}^{\iota }}{1-\mathfrak{q}}}(\iota \in \mathbb{N}\setminus \left\{1\right\}\mathrm{and}0<\mathfrak{q}<1),$$

(3)

and as $\mathfrak{q}\to 1^{-}$, we obtain ${\left[\iota \right]}_{\mathfrak{q}}=\iota$ and ${\left[0\right]}_{\mathfrak{q}}=0$. The $\mathfrak{q}$-extension of the Lerch–Hurwitz function, for $\mathfrak{b}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$ and $\mathfrak{s}\in \mathbb{C},$ where $\left|\zeta \right|<1$ and $Re\left(\mathfrak{s}\right)>1$ when $\left|\zeta \right|=1$, is given by

$$\begin{array}{c}\hfill {\varphi }_{\mathfrak{q}}(\mathfrak{s},\mathfrak{b};\zeta )=\sum _{\iota =0}^{\infty }{\displaystyle \frac{{\zeta }^{\iota }}{{[\iota +\mathfrak{b}]}_{\mathfrak{q}}^{\mathfrak{s}}}}.\end{array}$$

(4)

In Ref. [8], Shah and Noor used the normalized form of Equation (4) to define the $\mathfrak{q}$-Srivastava–Attiya operator.

$$\begin{array}{c}\hfill {\Psi }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\left(\zeta \right)={[1+\mathfrak{b}]}_{\mathfrak{q}}^{\mathfrak{s}}\{{\varphi }_{\mathfrak{q}}(\mathfrak{s},\mathfrak{b};\zeta )-{\displaystyle \frac{1}{{\left[\mathfrak{b}\right]}_{\mathfrak{q}}^{\mathfrak{s}}}}\}\\ \hfill =\zeta +\sum _{\iota =2}^{\infty }{\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[\iota +\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}}{\zeta }^{\iota }.\end{array}$$

(5)

Employing the Hadamard product (convolution), together with Equations (1) and (5), Shah and Noor [8] defined the $\mathfrak{q}$-analogue of the Srivastava–Attiya operator as follows.

Definition 1 

([8]). For $\mathfrak{b}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$ and $\mathfrak{s}\in \mathbb{C},$ where $\left|\zeta \right|<1$ and $Re\left(\mathfrak{s}\right)>1$ if $\left|\zeta \right|=1$, the $\mathfrak{q}$-analogue of the Srivastava–Attiya operator ${\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}:\mathcal{A}\to \mathcal{A}$ is given by

$$\begin{array}{cc}\hfill {\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)& ={\Psi }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\left(\zeta \right)\ast \mathfrak{f}\left(\zeta \right)\hfill \\ \hfill & =\zeta +\sum _{\iota =2}^{\infty }{\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[\iota +\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}}{a}_{\iota }{\zeta }^{\iota }=\zeta +\sum _{\iota =2}^{\infty }{\upsilon }_{\iota }{a}_{\iota }{\zeta }^{\iota },\zeta \in \mho \hfill \end{array}$$

(6)

where

$${\upsilon }_{\iota }={\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[\iota +\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}}.$$

The following identity holds for the operator ${\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}$.

$$\zeta D_q\left({\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}+1}\mathfrak{f}\left(\zeta \right)\right)=\left({\displaystyle \frac{[1+\mathfrak{b}]}{{\mathfrak{q}}^{\mathfrak{b}}}}\right){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)-{\displaystyle \frac{\left[\mathfrak{b}\right]}{{\mathfrak{q}}^{\mathfrak{b}}}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}+1}\mathfrak{f}\left(\zeta \right).$$

Applications of operators (differential and integral operators) in geometric functions cannot be over-emphasized. The authors in ref. [1] for example, considered the application of the integral operator associated with Bessel functions and some other applications also highlighted in the current study. More applications are also found in [9,10,11,12,13]. Motivated by some of the cited papers, the authors applying the $\mathfrak{q}$-Srivastava–Attiya convolution operator and the Jackson’s $\mathfrak{q}$-derivative, with the aid of the subordination the authors defined the new subclass of analytic functions in the open unit disk introduced a new subclass of univalent functions denoted by ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$. The aim of the study is to introduce a new class of univalent functions and establish a sharp upper-bound for the Fekete–Szeg$\ddot{o}$ functional using the q-Srivastava–Attiya convolution operator and the Jackson’s q-derivative by means of the subordination principle.

Remark 1. 

The operator ${\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)$ is an extension of several well-known operators that have been examined in previous research, which are reviewed below.

(i) 

${lim}_{\mathfrak{q}\to 1^{-}}{\varphi }_{\mathfrak{q}}(\mathfrak{s},\mathfrak{b};\zeta )$ (the Hurwitz–Lerch zeta function) and ${lim}_{\mathfrak{q}\to 1^{-}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}$ (the Srivastava–Attiya operator), see [14,15].

(ii) 

${\Im }_{\mathfrak{q},\mathfrak{b}}^1\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{\zeta }^b}}{\int }_0^{\zeta }{t}^{\mathfrak{b}-1}\mathfrak{f}\left(t\right)d_{\mathfrak{q}}t$ (the $\mathfrak{q}$-Bernardi operator [16]).

(iii) 

${\Im }_{\mathfrak{q},0}^1\mathfrak{f}\left(\zeta \right)={\int }_0^{\zeta }{\displaystyle \frac{\mathfrak{f}\left(t\right)}{t}}{d}_{\mathfrak{q}}t$ (the $\mathfrak{q}$-Alexander operator) and ${lim}_{\mathfrak{q}\to 1}{\Im }_{\mathfrak{q},0}^1\mathfrak{f}\left(\zeta \right)$, (the Alexander operator) [17].

(iv) 

${\Im }_{\mathfrak{q},1}^1\mathfrak{f}\left(\zeta \right)={\displaystyle \frac{{\left[2\right]}_{\mathfrak{q}}}{\zeta }}{\int }_0^{\zeta }\mathfrak{f}\left(t\right)d_{\mathfrak{q}}t$ (the $\mathfrak{q}$-Libera operator [16]).

(v) 

${\Im }_{\mathfrak{q},0}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)$ (the s$\breve{a}$l$\breve{a}$gean $\mathfrak{q}$-differential operator for negative integers $\mathfrak{s}$) [18].

Now, we introduce the new subclass ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}(\tau ,\Phi )$ of complex order for analytic functions, using the operator ${\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}$ and the concept of subordination. For this subclass, we obtained coefficient estimates, the Fekete–Szeg$\ddot{o}$ inequality.

Definition 2. 

Let P be the class of analytic functions Φ in ℧ such that $\Phi \left(0\right)=1$ and $Re \left\{\Phi \left(\zeta \right)\right\}>0$. A function $\mathfrak{f}\left(\zeta \right)$ of the form (1) is said to be in the class ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}(\tau ,\Phi )$ if it satisfies the following subordination condition:

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left({\displaystyle \frac{(1-\tau )\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{(1-\tau ){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right)\prec \Phi \left(\zeta \right),(\zeta \in \mho , \mathit{and} \mathfrak{h}\in {\mathbb{C}}^{\ast }).\end{array}$$

(7)

where $0\le \tau \le 1$, $0<\mathfrak{q}<1$, $2\le \iota \le \infty$, $\mathfrak{b}\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$ and $\mathfrak{s}\in \mathbb{C},$ where $\left|\zeta \right|<1$ and $Re\left(\mathfrak{s}\right)>1$ if $\left|\zeta \right|=1$.

In the following example, we show that the class ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}(\tau ,\Phi )$ is non-empty.

Example 1. 

Obviously the class of function so defined is non-empty. For example, the identity function,

$$\mathfrak{f}\left(\zeta \right)=\zeta .$$

Also, consider the analytic function

$$\mathfrak{f}\left(\zeta \right)=\zeta +c{\zeta }^2,c\in \mathbb{C}.$$

Using the definition of the $\mathfrak{q}$-analogue of the Srivastava–Attiya operator, we obtain the following. ${\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)=\zeta +K{\zeta }^2,K=c{\upsilon }_2=c{\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[2+\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}}.$

$$\begin{array}{cc}\hfill 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left({\displaystyle \frac{(1-\tau )\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{(1-\tau ){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right)& =\hfill \\ \hfill 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left({\displaystyle \frac{1+A\zeta }{1+B\zeta }}-1\right)=1+{\displaystyle \frac{A-B}{\mathfrak{h}}}{\displaystyle \frac{\zeta }{1+B\zeta }}\end{array}$$

where $A={\left[2\right]}_{\mathfrak{q}}(1+\tau \mathfrak{q})K$ and $B=(1+\tau \mathfrak{q})K$. For $\left|B\right|<1$, the function ${\displaystyle \frac{\zeta }{1+B\zeta }}$ is analytic in ℧. Moreover, if $\left|{\displaystyle \frac{A-B}{\mathfrak{h}}}\right|\le 1$, then the above expression is subordinate to $\Phi \left(\zeta \right)$. Therefore,

$$\left|c\right|=min\{{\displaystyle \frac{1}{1+\tau \mathfrak{q}}},{\displaystyle \frac{\left|\mathfrak{h}\right|}{\mathfrak{q}(1+\tau \mathfrak{q})}}\}{\left({\displaystyle \frac{{[2+\mathfrak{b}]}_{\mathfrak{q}}}{{[1+\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}}$$

which proves that the defined class is non-empty.

Remark 2. 

We note that for certain exceptional cases of the parameters $\tau ,\mathfrak{q},\mathfrak{h}$, and $\Phi \left(\zeta \right)$:

(i) 

${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}(0,\Phi )={\mathcal{S}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\ast ,\mathfrak{s}}\left(\Phi \right)$

$$\begin{array}{c}\hfill =\left\{\mathfrak{f}\in \mathcal{A}:1+{\displaystyle \frac{1}{\mathfrak{h}}}\left({\displaystyle \frac{\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}{{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right)\prec \Phi \left(\zeta \right)\right\},\end{array}$$

${\mathcal{S}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\ast ,\mathfrak{s}}\left(\Phi \right)$ (star-like function of complex order with $\mathfrak{q}$-Srivastava–Attiya operator).

(ii) 

${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}(1,\Phi )={\mathcal{C}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$

$$\begin{array}{c}\hfill =\left\{\mathfrak{f}\in \mathcal{A}:1+{\displaystyle \frac{1}{\mathfrak{h}}}\left({\displaystyle \frac{D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right)\prec \Phi \left(\zeta \right)\right\},\end{array}$$

${\mathcal{C}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$ (convex function of complex order with the $\mathfrak{q}$-Srivastava–Attiya operator).

(iii) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0(\tau ,\Phi )={\mathcal{L}}_{\mathfrak{h}}(\tau ,\Phi )$$(\mathfrak{h}\in {\mathbb{C}}^{\ast })$

$$=\left\{\mathfrak{f}\in \mathcal{A}:1+{\displaystyle \frac{1}{\mathfrak{h}}}\left[{\displaystyle \frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)+\tau {\zeta }^2{\mathfrak{f}}^{\prime \prime }\left(\zeta \right)}{(1-\tau )\mathfrak{f}\left(\zeta \right)+\tau \zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}}-1\right]\prec \Phi \left(\zeta \right)\right\}$$

(iv) 

${\mathcal{L}}_{\mathfrak{q},(1-\alpha )e^{-i\theta }cos\theta }^0(0,\Phi )={\mathcal{S}}_{\mathfrak{q}}^{\ast ,\theta }(\alpha ;\Phi )\left(\left|\theta \right|\le {\displaystyle \frac{\pi }{2}},0\le \alpha <1\right)$

$$=\left\{\mathfrak{f}\in \mathcal{A}:{\displaystyle \frac{e^{i\theta }\frac{\zeta D_{\mathfrak{q}}\mathfrak{f}\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}-\alpha cos\theta -isin\theta }{(1-\alpha )cos\theta }}\prec \Phi \left(\zeta \right)\right\}$$

(v) 

${\mathcal{L}}_{\mathfrak{q},(1-\alpha )e^{-i\theta }cos\theta }^0(0,\Phi )={\mathcal{C}}_{\mathfrak{q}}^{\theta }(\alpha ;\Phi )\left(\left|\theta \right|\le {\displaystyle \frac{\pi }{2}},0\le \alpha <1\right)$

$$=\left\{\mathfrak{f}\in \mathcal{A}:{\displaystyle \frac{e^{i\theta }\frac{D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}\mathfrak{f}\left(\zeta \right)\right)}{D_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\right)}-\alpha cos\theta -isin\theta }{(1-\alpha )cos\theta }}\prec \Phi \left(\zeta \right)\right\}$$

(vi) 

${\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0(0,\Phi )={\mathcal{S}}_{\mathfrak{q},\mathfrak{h}}^{\ast }\left(\Phi \right)$ and ${\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0(1,\Phi )={\mathcal{C}}_{\mathfrak{q},\mathfrak{h}}\left(\Phi \right)$ [19],

(vii) 

${\mathcal{L}}_{\mathfrak{q},1}^0(0,\Phi )={\mathcal{S}}_{\mathfrak{q}}^{\ast }\left(\Phi \right)$ and ${\mathcal{L}}_{\mathfrak{q},1}^0(1,\Phi )={\mathcal{C}}_{\mathfrak{q}}\left(\Phi \right)$ [20],

(viii) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0(0,\Phi )={\mathcal{S}}_{\mathfrak{h}}\left(\Phi \right)$ and ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0(1,\Phi )={\mathcal{C}}_{\mathfrak{h}}\left(\Phi \right)$ [21],

(ix) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},1}^0(0,\Phi )={\mathcal{S}}^{\ast }\left(\Phi \right)$ and ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},1}^0(1,\Phi )=\mathcal{C}\left(\Phi \right)$ [22],

(x) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0\left(0,{\displaystyle \frac{1+(1-2\alpha )\zeta }{1-\zeta }}\right)={\mathcal{S}}_{\alpha }^{\ast }\left(\mathfrak{h}\right)$ and ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0\left(1,{\displaystyle \frac{1+(1-2\alpha )\zeta }{1-\zeta }}\right)={\mathcal{C}}_{\alpha }\left(\mathfrak{h}\right)(0\le \alpha <1)$ [23],

(xi) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0\left(0,{\displaystyle \frac{1+\zeta }{1-\zeta }}\right)={\mathcal{S}}^{\ast }\left(\mathfrak{h}\right)$ ([24]), and ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}}^0\left(1,{\displaystyle \frac{1+\zeta }{1-\zeta }}\right)=\mathcal{C}\left(\mathfrak{h}\right)\left(\mathfrak{h}\in {\mathbb{C}}^{\ast }\right)$ ([25,26]),

(xii) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},(1-\alpha )}^0\left(0,{\displaystyle \frac{1+\zeta }{1-\zeta }}\right)={\mathcal{S}}^{\ast }\left(\alpha \right)$, and ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},(1-\alpha )}^0\left(1,{\displaystyle \frac{1+\zeta }{1-\zeta }}\right)=\mathcal{C}\left(\alpha \right)\left(0\le \alpha <1\right)$, [27],

(xiii) 

${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}{e}^{-i\theta }cos\left(\theta \right)}^0\left(0,{\displaystyle \frac{1+\zeta }{1-\zeta }}\right)={\mathcal{S}}^{\theta }\left(\mathfrak{h}\right)$, and ${lim}_{\mathfrak{q}\to 1^{-}}{\mathcal{L}}_{\mathfrak{q},\mathfrak{h}{e}^{-i\theta }cos\left(\theta \right)}^0\left(1,{\displaystyle \frac{1+\zeta }{1-\zeta }}\right)$$={\mathcal{C}}^{\theta }\left(\mathfrak{h}\right),\left(\left|\theta \right|\le {\displaystyle \frac{\pi }{2}}\right)$ ([28,29]),

where ${\mathcal{S}}^{\ast }$ and $\mathcal{C}$ are, respectively, the classes of star-like and convex functions.

• The following lemmas are important to prove our main result:

Lemma 1 

([22]). Consider $\mathfrak{r}\left(\zeta \right)=1+{\mathfrak{t}}_1\zeta +{\mathfrak{t}}_2{\zeta }^2+\dots \in P$, with $\mathfrak{Re}\left(\mathfrak{r}\right(\zeta \left)\right)>0(\zeta \in \mho )$. For a given complex number ℵ, the following inequality holds as follows:

$$\left|{\mathfrak{t}}_2-\aleph {\mathfrak{t}}_1^2\right|\le 2.max\{1;|2\aleph -1\left|\right\}.$$

The result is sharp for these functions.

$$\mathfrak{r}\left(\zeta \right)={\displaystyle \frac{1+\zeta }{1-\zeta }} \mathit{and} \mathfrak{r}\left(\zeta \right)={\displaystyle \frac{1+{\zeta }^2}{1-{\zeta }^2}}.$$

Lemma 2 

([22]). If $\mathfrak{r}\left(\zeta \right)=1+{\mathfrak{t}}_1\zeta +{\mathfrak{t}}_2{\zeta }^2+\dots$ is an analytic function with a positive real part in ℧, then

$$\left|{\mathfrak{t}}_2-\vartheta {\mathfrak{t}}_1^2\right|\le \left\{\begin{array}{cc}-4\vartheta +2,\hfill & \mathit{for}\vartheta <0,\hfill \\ 2,\hfill & \mathit{for}0\le \vartheta \le 1,\hfill \\ 4\vartheta -2,\hfill & \mathit{for}\vartheta >1.\hfill \end{array}\right.$$

When $\vartheta <0$ or $\vartheta >1$, equality occurs if and only if $\mathfrak{r}\left(\zeta \right)$ is ${\displaystyle \frac{1+\zeta }{1-\zeta }}$ or one of its rotations. For $0<\vartheta <1$, equality holds if and only if $\mathfrak{r}\left(\zeta \right)$ is ${\displaystyle \frac{1+{\zeta }^2}{1-{\zeta }^2}}$ or one of its rotations. When $\vartheta =0$, equality holds if and only if

$$\mathfrak{r}\left(\zeta \right)=\left({\displaystyle \frac{\eta }{2}}+{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{1+\zeta }{1-\zeta }}+\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\eta }{2}}\right){\displaystyle \frac{1-\zeta }{1+\zeta }},(0\le \eta \le 1)$$

or one of its rotations. If $\vartheta =1$, equality holds if and only if p is the reciprocal of a function for which equality holds when $\vartheta =0$.

Additionally, although the upper bound stated above is strict, it can be further enhanced when $0<\vartheta <1:$

$$\left|{\mathfrak{t}}_2-\vartheta {\mathfrak{t}}_1^2\right|+\vartheta {\left|{\mathfrak{t}}_1\right|}^2\le 2(0<\vartheta \le {\displaystyle \frac{1}{2}})$$

and

$$\left|{\mathfrak{t}}_2-\vartheta {\mathfrak{t}}_1^2\right|+(1-\vartheta ){\left|{\mathfrak{t}}_1\right|}^2\le 2({\displaystyle \frac{1}{2}}\le \vartheta <1).$$

2. Main Results

In our first theorem, for the new class ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$, we determine the coefficient estimates $\left|a_{\iota }\right|(\iota =2,3)$ of the Taylor–Maclaurin series in this class, as well as the Fekete–Szeg$\ddot{o}$ inequalities.

Theorem 1. 

Let $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots ,{\mathfrak{B}}_1\ne 0.$ If $\mathfrak{f}\in {\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$ and ℵ is a complex number, then

$$a_2={\displaystyle \frac{{\mathfrak{B}}_1{\mathfrak{t}}_1\mathfrak{h}}{2\mathfrak{q}{\gamma }_2{\upsilon }_2}},a_3={\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{2\mathfrak{q}(1+\mathfrak{q}){\gamma }_3{\upsilon }_3}}\left\{{\mathfrak{t}}_2-{\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{\mathfrak{q}}}-{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}\right]{\mathfrak{t}}_1^2\right\}$$

and

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le {\displaystyle \frac{\left|\mathfrak{h}{\mathfrak{B}}_1\right|}{\mathfrak{q}(1+\mathfrak{q}){\gamma }_3\left|{\upsilon }_3\right|}}max\left\{1;\left|{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}+{\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\gamma }_3}{{\gamma }_2^2}}{\displaystyle \frac{{\upsilon }_3}{{\upsilon }_2^2}}\aleph \right)\right|\right\}.\end{array}$$

(8)

The result is sharp.

$$\begin{array}{c}\hfill \left({\upsilon }_2={\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[2+\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}},{\upsilon }_3={\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[3+\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}},{\gamma }_2=\left(1+\tau \left(\mathfrak{q}+1\right)\right)\mathit{and}{\gamma }_3=\left(1+\tau \mathfrak{q}\left(\mathfrak{q}+1\right)\right)\right).\end{array}$$

Proof of Theorem 1. 

If $\mathfrak{f}\in {\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$, then for the Schwarz function $\varpi$, analytic in $\mathsf{{\rm Y}}$, that satisfies the condition in Equation (2) for $\zeta \in \mho$, such that

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left[{\displaystyle \frac{(1-\tau )\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{(1-\tau ){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right]=\Phi \left(\varpi \left(\zeta \right)\right).\end{array}$$

(9)

Let the function $\mathfrak{r}\left(\zeta \right)$ be defined as follows:

$$\begin{array}{c}\hfill \mathfrak{r}\left(\zeta \right)={\displaystyle \frac{1+\varpi \left(\zeta \right)}{1-\varpi \left(\zeta \right)}}=1+{\mathfrak{t}}_1\zeta +{\mathfrak{t}}_2{\zeta }^2+\dots .\end{array}$$

It follows that $Re\left(\mathfrak{r}\right(\zeta \left)\right)>0$ and $\mathfrak{r}\left(0\right)=1$. Therefore, simplifying the right hand side of Equation (9), we get

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Phi \left(\varpi \right(\zeta \left)\right)& =\Phi \left({\displaystyle \frac{\mathfrak{r}\left(\zeta \right)-1}{\mathfrak{r}\left(\zeta \right)+1}}\right)\hfill \\ & =\Phi \left({\displaystyle \frac{1}{2}}\left[{\mathfrak{t}}_1\zeta +\left({\mathfrak{t}}_2-{\displaystyle \frac{{\mathfrak{t}}_1^2}{2}}\right){\zeta }^2+\left({\mathfrak{t}}_3-{\mathfrak{t}}_1{\mathfrak{t}}_2+{\displaystyle \frac{{\mathfrak{t}}_1^3}{4}}\right){\zeta }^3+\dots \right]\right)\hfill \\ & =1+{\displaystyle \frac{{\mathfrak{B}}_1{\mathfrak{t}}_1}{2}}\zeta +\left[{\displaystyle \frac{{\mathfrak{B}}_1}{2}}\left({\mathfrak{t}}_2-{\displaystyle \frac{{\mathfrak{t}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathfrak{B}}_2{\mathfrak{t}}_1^2}{4}}\right]{\zeta }^2+\dots .\hfill \end{array}\end{array}$$

(10)

From the left hand side of (9), we get

$$\begin{array}{c}\hfill \begin{array}{cc}& 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left[{\displaystyle \frac{(1-\tau )\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{(1-\tau ){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right]\hfill \\ \hfill =& 1+{\displaystyle \frac{({\left[2\right]}_{\mathfrak{q}}-1){\gamma }_2{\upsilon }_2}{\mathfrak{h}}}{a}_2\zeta +{\displaystyle \frac{1}{\mathfrak{h}}}\left(({\left[3\right]}_{\mathfrak{q}}-1){\gamma }_3{\upsilon }_3{a}_3-({\left[2\right]}_{\mathfrak{q}}-1){\gamma }_2^2{\upsilon }_2^2{a}_2^2\right){\zeta }^2\hfill \\ & +{\displaystyle \frac{1}{\mathfrak{h}}}\left({\left[2\right]}_{\mathfrak{q}}{\gamma }_2^3{\upsilon }_2^3{a}_2^3-({\left[3\right]}_{\mathfrak{q}}+{\left[2\right]}_{\mathfrak{q}}-2){\gamma }_2{\gamma }_3{\upsilon }_2{\upsilon }_3{a}_2{a}_3+({\left[4\right]}_{\mathfrak{q}}-1){\gamma }_4{\upsilon }_4{a}_4\right){\zeta }^3+\dots .\hfill \end{array}\end{array}$$

(11)

By equating the coefficients of $\zeta$ and ${\zeta }^2$ in Equations (10) and (11), we obtain

$${\displaystyle \frac{({\left[2\right]}_{\mathfrak{q}}-1){\gamma }_2{\upsilon }_2}{\mathfrak{h}}}{a}_2={\displaystyle \frac{{\mathfrak{B}}_1{\mathfrak{t}}_1}{2}}$$

and

$${\displaystyle \frac{1}{\mathfrak{h}}}\left(({\left[3\right]}_{\mathfrak{q}}-1){\gamma }_3{\upsilon }_3{a}_3-({\left[2\right]}_{\mathfrak{q}}-1){\gamma }_2^2{\upsilon }_2^2{a}_2^2\right)={\displaystyle \frac{{\mathfrak{B}}_1}{2}}\left({\mathfrak{t}}_2-{\displaystyle \frac{{\mathfrak{t}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathfrak{B}}_2{\mathfrak{t}}_1^2}{4}}$$

or, equivalently,

$$a_2={\displaystyle \frac{{\mathfrak{B}}_1{\mathfrak{t}}_1\mathfrak{h}}{2\mathfrak{q}{\gamma }_2{\upsilon }_2}}$$

and

$$a_3={\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{2\mathfrak{q}(1+\mathfrak{q}){\gamma }_3{\upsilon }_3}}\left\{{\mathfrak{t}}_2-{\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{\mathfrak{q}}}-{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}\right]{\mathfrak{t}}_1^2\right\}.$$

Therefore, we have

$$a_3-\aleph a_2^2={\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{2\mathfrak{q}(\mathfrak{q}+1){\gamma }_3{\upsilon }_3}}\left\{{\mathfrak{t}}_2-\vartheta {\mathfrak{t}}_1^2\right\},$$

(12)

where

$$\vartheta ={\displaystyle \frac{1}{2}}\left[1-{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}-{\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\gamma }_3{\upsilon }_3}{{\gamma }_2^2{\upsilon }_2^2}}\aleph \right)\right].$$

(13)

Taking the modulus of both sides of Equation (12) and applying the inequality of Lemma 1, we obtain our result in Equation (8). When $\varpi \left(\zeta \right)=\zeta$ and $\varpi \left(\zeta \right)={\zeta }^2$ in Equation (9), the result is sharp for the functions.

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left[{\displaystyle \frac{(1-\tau )\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{(1-\tau ){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right]=\Phi \left(\zeta \right)\end{array}$$

and

$$\begin{array}{c}\hfill 1+{\displaystyle \frac{1}{\mathfrak{h}}}\left[{\displaystyle \frac{(1-\tau )\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}\left(\zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)\right)}{(1-\tau ){\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)+\tau \zeta D_{\mathfrak{q}}{\Im }_{\mathfrak{q},\mathfrak{b}}^{\mathfrak{s}}\mathfrak{f}\left(\zeta \right)}}-1\right]=\Phi \left({\zeta }^2\right).\end{array}$$

Thus, Theorem 1 is now completely proved. □

Setting $\tau =0$ in Theorem 1, we obtain the Fekete–Szeg$\ddot{o}$ inequality for the star-like class of complex order associated with the $\mathfrak{q}-$Srivastava–Attiya operator ${\mathcal{S}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\ast ,\mathfrak{s}}\left(\Phi \right)$.

Corollary 1. 

Consider $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots$ with ${\mathfrak{B}}_1\ne 0$. If $\mathfrak{f}\in {\mathcal{S}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\ast ,\mathfrak{s}}\left(\Phi \right)$, where $\mathfrak{f}$ is defined by Equation (1), then

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le {\displaystyle \frac{\left|{\mathfrak{B}}_1\mathfrak{h}\right|}{\mathfrak{q}(1+\mathfrak{q})|{\upsilon }_3|}}max\left\{1;\left|{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}+{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\upsilon }_3}{{\upsilon }_2^2}}\aleph \right)\right|\right\}.\aleph \in \mathbb{C}\end{array}$$

The result is sharp.

Setting $\tau =1$ in Theorem 1, we obtain the Fekete–Szeg$\ddot{o}$ inequality for the convex class of complex order associated with the $\mathfrak{q}$-Srivastava–Attiya operator ${\mathcal{C}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$

Corollary 2. 

Consider $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots$ with ${\mathfrak{B}}_1\ne 0$. If $\mathfrak{f}\in {\mathcal{C}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$, where $\mathfrak{f}$ is defined by Equation (1), then

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le {\displaystyle \frac{\left|\mathfrak{h}{\mathfrak{B}}_1\right|}{\mathfrak{q}(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}}max\left\{1;\left|{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}+{\displaystyle \frac{{\mathfrak{B}}_1}{\mathfrak{q}}}\left(1-\left({\displaystyle \frac{{\mathfrak{q}}^2+\mathfrak{q}+1}{\mathfrak{q}+1}}\right){\displaystyle \frac{{\upsilon }_3}{{\upsilon }_2^2}}\aleph \right)\right|\right\},\aleph \in \mathbb{C}.\end{array}$$

The result is sharp.

For $\mathfrak{b}=0$, $\mathfrak{q}\to 1$ and negative integer values of $\mathfrak{s}$, in Theorem 1, we obtain the Fekete–Szeg$\ddot{o}$ inequility for a certain class of analytic functions of complex order associated with the s$\breve{a}$l$\breve{a}$gean operator.

Corollary 3. 

Consider $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots$ with ${\mathfrak{B}}_1\ne 0$. If $\mathfrak{f}\in {\mathcal{L}}_{1,0,\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$, where $\mathfrak{f}$ is defined by Equation (1), then

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le {\displaystyle \frac{\left|\mathfrak{h}{\mathfrak{B}}_1\right|}{3^{\mathfrak{s}}(2+4\tau )}}max\left\{1;\left|{\displaystyle \frac{{\mathfrak{B}}_2}{{\mathfrak{B}}_1}}+{\mathfrak{B}}_1\mathfrak{h}\left({\displaystyle \frac{4^{\mathfrak{s}}{(1+\tau )}^2-3^{\mathfrak{s}}(2+4\tau )\aleph }{4^{\mathfrak{s}}{(1+\tau )}^2}}\right)\right|\right\},\aleph \in \mathbb{C}.\end{array}$$

The result is sharp.

Theorem 2. 

Let $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots$ with ${\mathfrak{B}}_1>0$ and ${\mathfrak{B}}_2\ge 0$. If $\mathfrak{f}\in {\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$ with $\mathfrak{h}>0$ and $\aleph \in \mathbb{R}$, then

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le \left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1){\gamma }_3\left|{\upsilon }_3\right|}}\left[{\mathfrak{B}}_2+{\displaystyle \frac{{\mathfrak{B}}_1^2\mathfrak{h}}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\gamma }_3{\upsilon }_3}{{\gamma }_2^2{\upsilon }_2^2}}\aleph \right)\right],\hfill & \aleph \le {\rho }_1\hfill \\ {\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1){\gamma }_3\left|{\upsilon }_3\right|}},\hfill & {\rho }_1\le \aleph \le {\rho }_2,\hfill \\ {\displaystyle \frac{\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1){\gamma }_3\left|{\upsilon }_3\right|}}\left[-{\mathfrak{B}}_2-{\displaystyle \frac{{\mathfrak{B}}_1^2\mathfrak{h}}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\gamma }_3{\upsilon }_3}{{\gamma }_2^2{\upsilon }_2^2}}\aleph \right)\right],\hfill & \aleph \ge {\rho }_2,\hfill \end{array}\right.\end{array}$$

(14)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\rho }_1& ={\displaystyle \frac{{\gamma }_2^2{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}\left({\mathfrak{B}}_2-{\mathfrak{B}}_1\right)\right]}{(\mathfrak{q}+1){\gamma }_3{\upsilon }_3\mathfrak{h}{\mathfrak{B}}_1^2}},\mathit{and}\hfill \\ \hfill {\rho }_2& ={\displaystyle \frac{{\gamma }_2^2{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}\left({\mathfrak{B}}_2+{\mathfrak{B}}_1\right)\right]}{(\mathfrak{q}+1){\gamma }_3{\upsilon }_3\mathfrak{h}{\mathfrak{B}}_1^2}}.\hfill \end{array}\end{array}$$

(15)

Additionally, the above result can be enhanced as follows when ${\rho }_1\le \aleph \le {\rho }_2$, and

$${\rho }_3={\displaystyle \frac{{\gamma }_2^2\left|{\upsilon }_2^2\right|\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}{\mathfrak{B}}_2\right]}{(\mathfrak{q}+1){\gamma }_3\left|{\upsilon }_3\right|\mathfrak{h}{\mathfrak{B}}_1^2}}.$$

Additionally, if ${\rho }_1\le \aleph \le {\rho }_3$, then

$$\begin{array}{c}\left|a_3-\aleph a_2^2\right|+{\displaystyle \frac{\mathfrak{q}}{(\mathfrak{q}+1){\mathfrak{B}}_1^2\mathfrak{h}}}{\displaystyle \frac{{\gamma }_2^2\left|{\upsilon }_2^2\right|}{{\gamma }_3\left|{\upsilon }_3\right|}}\left[{\mathfrak{B}}_1-{\mathfrak{B}}_2-{\displaystyle \frac{{\mathfrak{B}}_1^2\mathfrak{h}}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\gamma }_3\left|{\upsilon }_3\right|}{{\gamma }_2^2\left|{\upsilon }_2^2\right|}}\aleph \right)\right]{\left|a_2\right|}^2\\ \le {\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\gamma }_3{\upsilon }_3}}.\end{array}$$

If ${\rho }_3\le \aleph \le {\rho }_2$, then

$$\begin{array}{c}\left|a_3-\aleph a_2^2\right|+{\displaystyle \frac{\mathfrak{q}}{(\mathfrak{q}+1){\mathfrak{B}}_1^2\mathfrak{h}}}{\displaystyle \frac{{\gamma }_2^2\left|{\upsilon }_2^2\right|}{{\gamma }_3\left|{\upsilon }_3\right|}}\left[{\mathfrak{B}}_1+{\mathfrak{B}}_2+{\displaystyle \frac{{\mathfrak{B}}_1^2\mathfrak{h}}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\gamma }_3\left|{\upsilon }_3\right|}{{\gamma }_2^2\left|{\upsilon }_2^2\right|}}\aleph \right)\right]{\left|a_2\right|}^2\\ \le {\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\gamma }_3{\upsilon }_3}}.\end{array}$$

The result is sharp.

$$\begin{array}{c}\hfill \left({\upsilon }_2={\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[2+\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}},{\upsilon }_3={\left({\displaystyle \frac{{[1+\mathfrak{b}]}_{\mathfrak{q}}}{{[3+\mathfrak{b}]}_{\mathfrak{q}}}}\right)}^{\mathfrak{s}},{\gamma }_2=\left(1+\tau \left(\mathfrak{q}+1\right)\right)\mathit{and}{\gamma }_3=\left(1+\tau \mathfrak{q}\left(\mathfrak{q}+1\right)\right)\right).\end{array}$$

Proof of Theorem 2. 

For $\mathfrak{f}\in {\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$, then by applying Lemma 2 to Equations (12) and (13) in Theorem 1, we can obtain our results. □

Setting $\tau =0$ in Theorem 2, we obtain the Fekete–Szeg$\ddot{o}$ inequality in case $\aleph \in \mathbb{R}$ for the star-like class of complex order associated with the $\mathfrak{q}$-Srivastava–Attiya operator ${\mathcal{S}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\ast ,\mathfrak{s}}\left(\Phi \right)$.

Corollary 4. 

Let $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots$, where ${\mathfrak{B}}_1>0$ and ${\mathfrak{B}}_2\ge 0$. If $\mathfrak{f}\in {\mathcal{S}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\ast ,\mathfrak{s}}\left(\Phi \right)$ with $\mathfrak{h}>0$ and $\aleph \in \mathbb{R}$, then

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le \left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1)|{\upsilon }_3|}}\left[{\mathfrak{B}}_2+{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\upsilon }_3}{{\upsilon }_2^2}}\aleph \right)\right],\hfill & \aleph \le {\rho }_1,\hfill \\ {\displaystyle \frac{{\mathfrak{B}}_1\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1)|{\upsilon }_3|}},\hfill & {\rho }_1\le \aleph \le {\rho }_2,\hfill \\ {\displaystyle \frac{\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1)|{\upsilon }_3|}}\left[-{\mathfrak{B}}_2-{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\upsilon }_3}{{\upsilon }_2^2}}\aleph \right)\right],\hfill & \aleph \ge {\rho }_2.\hfill \end{array}\right.\end{array}$$

(16)

Let

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\rho }_1& ={\displaystyle \frac{{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}\left({\mathfrak{B}}_2-{\mathfrak{B}}_1\right)\right]}{(\mathfrak{q}+1){\mathfrak{B}}_1^2\mathfrak{h}{\upsilon }_3}},\hfill \\ \hfill {\rho }_2& ={\displaystyle \frac{{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}\left({\mathfrak{B}}_2+{\mathfrak{B}}_1\right)\right]}{(\mathfrak{q}+1){\mathfrak{B}}_1^2\mathfrak{h}{\upsilon }_3}}.\hfill \end{array}\end{array}$$

(17)

Additionally, the above result can be enhanced as follows when ${\rho }_1\le \aleph \le {\rho }_2$, and

$${\rho }_3={\displaystyle \frac{{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}{\mathfrak{B}}_2\right]}{(\mathfrak{q}+1){\mathfrak{B}}_1^2\mathfrak{h}{\upsilon }_3}}.$$

(18)

Additionally, if ${\rho }_1\le \aleph \le {\rho }_3$, then

$$\begin{array}{c}\left|a_3-\aleph a_2^2\right|+{\displaystyle \frac{\mathfrak{q}}{(\mathfrak{q}+1)\mathfrak{h}{\mathfrak{B}}_1^2}}{\displaystyle \frac{|{\upsilon }_2^2|}{|{\upsilon }_3|}}\left[{\mathfrak{B}}_1-{\mathfrak{B}}_2-{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{|{\upsilon }_3|}{|{\upsilon }_2^2|}}\aleph \right)\right]{\left|a_2\right|}^2\\ \le {\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\upsilon }_3}}.\end{array}$$

If ${\rho }_3\le \aleph \le {\rho }_2$, then

$$\begin{array}{c}\left|a_3-\aleph a_2^2\right|+{\displaystyle \frac{\mathfrak{q}}{(\mathfrak{q}+1)\mathfrak{h}{\mathfrak{B}}_1^2}}{\displaystyle \frac{|{\upsilon }_2^2|}{|{\upsilon }_3|}}\left[{\mathfrak{B}}_1+{\mathfrak{B}}_2+{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{|{\upsilon }_3|}{|{\upsilon }_2^2|}}\aleph \right)\right]{\left|a_2\right|}^2\\ \le {\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\upsilon }_3}}.\end{array}$$

Setting $\tau =1$ in Theorem 2, we obtain the Fekete–Szeg$\ddot{o}$ inequality in case $\aleph \in \mathbb{R}$ for the convex class of complex order associated with the $\mathfrak{q}$-Srivastava–Attiya operator ${\mathcal{C}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$.

Corollary 5. 

Let $\Phi \left(\zeta \right)=1+{\mathfrak{B}}_1\zeta +{\mathfrak{B}}_2{\zeta }^2+\dots ,$ where ${\mathfrak{B}}_1>0$ and ${\mathfrak{B}}_2\ge 0$. If $\mathfrak{f}\in {\mathcal{C}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\Phi \right)$ with $\mathfrak{h}>0$ and $\aleph \in \mathbb{R}$, then

$$\begin{array}{c}\hfill \left|a_3-\aleph a_2^2\right|\le \left\{\begin{array}{cc}{\displaystyle \frac{\mathfrak{h}}{\mathfrak{q}(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}}\left[{\mathfrak{B}}_2+{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\left[3\right]}_{\mathfrak{q}}{\upsilon }_3}{{\left[2\right]}_{\mathfrak{q}}^2{\upsilon }_2^2}}\aleph \right)\right],\hfill & \aleph \le {\rho }_1,\hfill \\ {\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}},\hfill & {\rho }_1\le \aleph \le {\rho }_2,\hfill \\ {\displaystyle \frac{1}{\mathfrak{q}(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}}\left[-{\mathfrak{B}}_2-{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\left[3\right]}_{\mathfrak{q}}|{\upsilon }_3}{{\left[2\right]}_{\mathfrak{q}}^2{\upsilon }_2^2}}\aleph \right)\right],\hfill & \aleph \ge {\rho }_2.\hfill \end{array}\right.\end{array}$$

(19)

Let

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\rho }_1& ={\displaystyle \frac{{\left[2\right]}_{\mathfrak{q}}^2{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}\left({\mathfrak{B}}_2-{\mathfrak{B}}_1\right)\right]}{(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}{\upsilon }_3\mathfrak{h}{\mathfrak{B}}_1^2}},\mathit{and}\hfill \\ \hfill {\rho }_2& ={\displaystyle \frac{{\left[2\right]}_{\mathfrak{q}}^2{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}\left({\mathfrak{B}}_2+{\mathfrak{B}}_1\right)\right]}{(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}{\upsilon }_3\mathfrak{h}{\mathfrak{B}}_1^2}}.\hfill \end{array}\end{array}$$

(20)

There is a sharp inequality (19). Additionally, the above result can be enhanced as follows when ${\rho }_1\le \aleph \le {\rho }_2$, and

$${\rho }_3={\displaystyle \frac{{\left[2\right]}_{\mathfrak{q}}^2{\upsilon }_2^2\left[\mathfrak{h}{\mathfrak{B}}_1^2+\mathfrak{q}{\mathfrak{B}}_2\right]}{(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}{\upsilon }_3\mathfrak{h}{\mathfrak{B}}_1^2}}.$$

(21)

Additionally, if ${\rho }_1\le \aleph \le {\rho }_3$, then

$$\begin{array}{c}\left|a_3-\aleph a_2^2\right|+{\displaystyle \frac{\mathfrak{q}}{(\mathfrak{q}+1){\mathfrak{B}}_1^2}}{\displaystyle \frac{{\left[2\right]}_{\mathfrak{q}}^2\left|{\upsilon }_2^2\right|}{{\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}}\left[{\mathfrak{B}}_1-{\mathfrak{B}}_2-{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}{{\left[2\right]}_{\mathfrak{q}}^2\left|{\upsilon }_2^2\right|}}\aleph \right)\right]{\left|a_2\right|}^2\\ \le {\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}{\upsilon }_3}}.\end{array}$$

If ${\rho }_3\le \aleph \le {\rho }_2$, then

$$\begin{array}{c}\left|a_3-\aleph a_2^2\right|+{\displaystyle \frac{\mathfrak{q}}{(\mathfrak{q}+1)\mathfrak{h}{\mathfrak{B}}_1^2}}{\displaystyle \frac{{\left[2\right]}_{\mathfrak{q}}^2\left|{\upsilon }_2^2\right|}{{\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}}\left[{\mathfrak{B}}_1+{\mathfrak{B}}_2+{\displaystyle \frac{\mathfrak{h}{\mathfrak{B}}_1^2}{\mathfrak{q}}}\left(1-(\mathfrak{q}+1){\displaystyle \frac{{\left[3\right]}_{\mathfrak{q}}\left|{\upsilon }_3\right|}{{\left[2\right]}_{\mathfrak{q}}^2\left|{\upsilon }_2^2\right|}}\aleph \right)\right]{\left|a_2\right|}^2\\ \le {\displaystyle \frac{{\mathfrak{B}}_1}{\mathfrak{q}(\mathfrak{q}+1){\left[3\right]}_{\mathfrak{q}}{\upsilon }_3}}.\end{array}$$

These results are sharp.

3. Conclusions

In this study, we introduce a new general subclass of complex order denoted by ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,\Phi \right)$, associated with the $\mathfrak{q}$-Srivastava–Attiya operator defined via subordination. This subclass includes both the class of complex order $\mathfrak{q}$-star-like Srivastava–Attiya operator and the $\mathfrak{q}$-convex Srivastava–Attiya operator. Moreover, for each main result, several interesting corollaries and special cases are presented by choosing appropriate values of the defining parameters. Recent studies by various authors have investigated different results related to the Fekete–Szeg$\ddot{o}$ problem, and this paper further contributes by presenting new results in this area.

The investigation of univalent functions in geometric function theory is an active area of research that involves the development of new and very useful subclasses of univalent functions and in some cases, subclasses of bi-univalent, that is, functions of complex variables in which both the function and its inverse are univalent. These findings have applications in science and engineering [30,31,32,33,34].