Some New Subclasses of Bi-Univalent Functions Related to Quantum Calculus

Abstract

The primary objective of this paper is to introduce and investigate several novel subclasses of bi-univalent functions associated with the q-calculus framework. Using appropriate analytical techniques, we derive coefficient bounds for the initial coefficients of the functions belonging to these newly defined classes. In particular, we provide explicit estimates for the second-order Hankel determinant and address the classical Fekete–Szegö functional problem within the context of these classes under suitable conditions. It is important to note that the findings presented in this work not only contribute to the ongoing development of q-analogs in geometric function theory, but also serve as a unifying generalization of many previously known results, which are obtained as special cases of our main findings.

1. Introduction

Let the Taylor series representation of an analytic function f in the open unit disk V be given by

$$f\left(\nu \right)=\nu +\sum _{t=2}^{\infty }{\mathit{b}}_t{\nu }^t,\nu \in V$$

(1)

and satisfy the normalization conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1.$ The set of all normalized analytic functions with the series representation given by (1) is denoted by A. The class S contains all those normalized analytic functions that are univalent (one-to-one) on the open unit disk V; for details, see [1,2,3]. Two of the most prominent and widely studied subclasses of S are $S^{*}$ (the class of starlike functions) and C (the class of convex functions), each has deep theoretical significance and rich structural properties. For every $f\in S,$ there exists an inverse function $f^{-1},$ which is defined in some neighborhood of the origin. According to the Köebe one-quarter theorem, this neighborhood can be taken as the disk |$\mu$| < $1/4$, where $\mu$ typically denotes the variable in the image domain of f, that is, ($\mu \in f\left(V\right)$). This ensures that $f^{-1}$ is defined and analytic at least for |$\mu$| < $1/4$. In some cases, this inverse function can be extended to the entire unit disk V. For more details, see [4,5]. A calculation reveals that the inverse function $g=f^{-1}$ has the series expansion of the form

$$f^{-1}\left(\mu \right)=g\left(\mu \right)=\mu -{\mathit{b}}_2{\mu }^2+\left(2{\mathit{b}}_2^2-{\mathit{b}}_3\right){\mu }^3-\left(5{\mathit{b}}_2^3-5{\mathit{b}}_2{\mathit{b}}_3+{\mathit{b}}_4\right){\mu }^4+\dots .$$

(2)

A function $f\in A$ is termed bi-univalent in V if both f and its inverse $f^{-1}$ are univalent in V. Let $\mathrm{\Psi }$ denote the class of bi-univalent functions in V (open unit disk) having the series representation provided in (1) (see [6,7] for details). The functions

$${\mathrm{\Phi }}_1\left(\nu \right)={\displaystyle \frac{\nu }{1-\nu }},{\mathrm{\Phi }}_2\left(\nu \right)=-\log \left(1-\nu \right)\mathrm{and}{\mathrm{\Phi }}_3\left(\nu \right)={\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+\nu }{1-\nu }}\right),$$

are examples of bi-univalent functions, with their respective inverses given below:

$${\mathrm{\Psi }}_1\left(\mu \right)={\displaystyle \frac{e^{\mu }-1}{e^{\mu }}},{\mathrm{\Psi }}_2\left(\mu \right)={\displaystyle \frac{\mu }{1+\mu }}\mathrm{and}{\mathrm{\Psi }}_3\left(\mu \right)={\displaystyle \frac{e^{2\mu }-1}{e^{2\mu }+1}}.$$

In coefficient-based problems for bi-univalent functions, Lewin [8] initiated the study by investigating important functionals and estimating inequality bounds. Subsequently, several authors investigated significant and interesting geometrical problems related to subclasses of analytic functions defined within the framework of bi-univalent functions.

In 1933, for $f\in S$ with the series representation given in (1), Fekete and Szegö [9] defined the inequality given by

$$\left|b_3-\kappa b_2^2\right|\le 1+{2e}^{-2\kappa /1-\kappa },\mathrm{if}0\le \kappa \le 1.$$

Here, the expression $\left|b_3-\kappa b_2^2\right|$ is known as the Fekete–Szegö functional.

In 1966, for $f\in A$ having the series representation given in (1), $j\ge 1$ and $t\ge 0,$ the j-th Hankel determinant was defined by Pommerenke [10] (see also [11]) as

$${\mathcal{H}}_{j,t}\left(f\right)=\left|\begin{array}{cccc}{b}_t& b_{t+1}& \cdots & b_{t+j-1}\\ b_{t+1}& b_{t+2}& \cdots & b_{t+j}\\ ⋮& ⋮& ⋮& ⋮\\ b_{t+j-1}& b_{t+j}& \cdots & b_{t+2j-2}\end{array}\right|.$$

(3)

For $j=1$ and $t=2$, we have

$${\mathcal{H}}_{2,1}\left(f\right)=\left|\begin{array}{cc}1& b_2\\ b_2& b_3\end{array}\right|=b_3-b_2^2.$$

The value of ${\mathcal{H}}_{2,1}\left(f\right)$ is a special case of the well-known Fekete–Szegö functional, that is, for $\kappa =1$. For $j=t=2,$ the Hankel determinant of second-order ${\mathcal{H}}_{2,2}\left(k\right)$ is given by

$${\mathcal{H}}_{2,2}\left(f\right)=\left|\begin{array}{cc}{b}_2& b_3\\ b_3& b_4\end{array}\right|=b_2{b}_4-b_3^2.$$

(4)

Hankel determinants are an effective analytic tool for studying coefficient estimates and structural properties of analytic functions arising from the representation of power series. They facilitate the investigation of various geometric aspects of functions, including growth and distortion properties. Beyond their role in geometric function theory, Hankel determinants are central to random matrix theory, where they are connected with special functions by perturbed Gaussian, Laguerre and Jacobi weights playing a key role in the formulation of partition functions. For further relevant applications, see [12,13,14].

Motivated by these theoretical and applied aspects, several researchers have examined the Hankel determinants within the framework of bi-univalent functions theory. Deniz et al. [15] investigated the second-order Hankel determinants for the classes of bi-starlike and bi-convex functions defined via subordination. Subsequently, Orhan et al. [16] and Mustafa et al. [17] derived bounds for the second-order Hankel determinant for various subclasses of bi-univalent functions. More recently, Shaba et al. [18] obtained sharp estimates for the Fekete–Szegö functional and the second-order Hankel determinant for a subclass of bi-univalent functions associated with a differential operator and q-Limaçon domain.

Let $f,ħ$ be functions in A with the series representation provided in (1) and $ħ\left(\nu \right)=\nu +{\sum }_{t=2}^{\infty }{a}_t{\nu }^t$; the convolution or Hadamard product of these functions is represented by $(f\ast ħ)\left(\nu \right)$ and interpreted as

$$(f\ast ħ)\left(\nu \right)=\nu +\sum _{t=2}^{\infty }{\mathit{b}}_t{a}_t{\nu }^t,\nu \in V.$$

(5)

For $\mathrm{\Phi },\chi \in A$, the function $\mathrm{\Phi }$ is said to be subordinate to $\chi$, written as $\left(\mathrm{\Phi }\prec \chi \right)$ if there exists a Schwarz function $F\left(\nu \right)$ satisfying the following conditions:

$$F\left(0\right)=0,\mathrm{and}\left|F\left(\nu \right)\right|\le 1,$$

such that we can write

$$\mathrm{\Phi }\left(\nu \right)=\chi \left(F\left(\nu \right)\right).$$

In 1970, Robertson [19] introduced the theory of quasi-subordination and majorization. A function $f\left(\nu \right)$ is quasi-subordinate to $g\left(\nu \right)$ in V, denoted by

$$\begin{array}{c}\hfill f\left(\nu \right){\prec }_{\mathit{Q}}g\left(\nu \right),\end{array}$$

(6)

if there exist two analytic functions $F\left(\nu \right)$ and $\mathit{m}\left(\nu \right)$ in V, satisfying the conditions $F\left(0\right)=0$, $\left|F\left(\nu \right)\right|<1$ and $\left|\mathit{m}\left(\nu \right)\right|\le 1,$ such that it can be written as

$$f\left(\nu \right)=\mathit{m}\left(\nu \right)g\left(F\left(\nu \right)\right),\nu \in V.$$

(7)

For $\mathit{m}\left(\nu \right)=1$, the quasi-subordination becomes ordinary subordination, as $f\left(\nu \right)=g\left(F\left(\nu \right)\right)$, that is, $f\left(\nu \right)\prec g\left(\nu \right),$ and for $F\left(\nu \right)=\nu ,$ $f\left(\nu \right)$ is majorized by $g\left(\nu \right),$ that is, $f\left(\nu \right)=\mathit{m}\left(\nu \right)g\left(\nu \right),$ and denoted by $f\left(\nu \right)\ll g\left(\nu \right).$ The inclusion of the bounded analytic factor $\mathit{m}\left(\nu \right)$ is what extends the subordination to quasi-subordination and relaxes the strict inclusion relation into a more versatile relation $f\left(\nu \right)=\mathit{m}\left(\nu \right)g\left(F\left(\nu \right)\right)$, applicable to many classical function classes beyond those preserved under ordinary subordination.

In classical calculus, limits form the foundation of analysis, whereas the q-calculus operates independently of limits and is often described as a “calculus without limits”. Jackson [20] introduced the fundamental ideas of q-calculus in 1910 by defining the q-derivative and the q-integral operators. Since then, q-calculus has found wide-ranging applications in mathematics, quantum mechanics, physics, and geometric function theory. Gasper and Rahman [21] provided a comprehensive account of the theory and applications of q-calculus in physics, combinatorics, and number theory. More recently, its connection with geometric function theory has gained considerable attention, where researchers employ q-calculus to define and analyze various subclasses of analytic and univalent functions. In particular, Ismail et al. [22] investigated q-starlike functions, paving the way for further advancement in this direction. This development opened new doors for a more advanced study of q-calculus in geometric function theory. Srivastava [23] provided a detailed exposition of operators arising from basic q-calculus together with their applications in geometric function theory. In particular, the author discussed various q-derivative operators, examined their analytic properties, and demonstrated how these operators are used to study coefficient estimates, subordination results, and geometric characteristics of analytic and univalent functions. Furthermore, Al Dweby and Darus [24] introduced and investigated q-analogs of several convolution-type operators utilizing the Hadamard product. Their work focuses on establishing sufficient conditions, growth and distortion bounds, starlikeness and convexity of analytic functions associated with the proposed operator, thereby extending classical results to q-calculus. Numerous subclasses and results in geometric function theory have since been developed using q-calculus (see, e.g., [25,26,27,28]). Here, we present some fundamental definitions and concepts that are essential for this research work.

The q-derivative (also called the Jackson q-difference operator) of a function f that has the Maclaurin series representation of the form given in (1), is denoted by $D_{q}f\left(\nu \right)$ and defined as:

$$D_{q}f\left(\nu \right)=\left\{\begin{array}{cc}{\displaystyle \frac{f\left(\nu \right)-f\left(q\nu \right)}{\left(1-q\right)\nu }}& \mathrm{for}\nu \ne 0,\\ f^{\prime }\left(0\right)& \mathrm{for}\nu =0.\end{array}\right.$$

(8)

The q-derivative operator reduces to the classical derivative as $q\to 1^{-},$ that is

$$\underset{q\to 1^{-}}{\lim }\left[D_{q}f\left(\nu \right)\right]=f^{\prime }\left(\nu \right).$$

For any analytic function f having the series representation given in (1), the q-derivative operator is defined as:

$$D_q\left(f\left(\nu \right)\right)=1+\stackrel{\infty }{\sum _{n=2}}{b}_n{\left[n\right]}_q{\nu }^{n-1},$$

(9)

where ${\left[n\right]}_q$ is known as the q-number and is defined as

$${\left[n\right]}_q=\left\{\begin{array}{cc}{\displaystyle \frac{1-q^n}{1-q}},\hfill & n\in \mathbb{C},\hfill \\ {\displaystyle \sum _{i=0}^{n-1}}{q}^i=1+q+q^2+\cdots +q^{n-1},\hfill & n\in \mathbb{N},\hfill \\ n,\hfill & q\to 1^{-}.\hfill \end{array}\right.$$

(10)

In recent times, many authors have defined and investigated important q-operators which are used to explore several important subclasses of analytic functions. Some of them are discussed here.

For $f\in A$, Srivastava and Attiya [29] defined the operator ${\mathcal{D}}_{\partial ,\alpha }:A⟶A,$ given as

$${\mathcal{D}}_{\partial ,\alpha }f\left(\nu \right)=\nu +\sum _{t=2}^{\infty }{\left({\displaystyle \frac{1+\alpha }{t+\alpha }}\right)}^{\partial }{\mathit{b}}_t{\nu }^t,\alpha ,\partial \in \mathbb{C}.$$

(11)

For the function $f\in A$, Carlson–Shaffer [30] introduced the operator ${\mathcal{T}}_{y}f\left(\nu \right)$ as given below:

$${\mathcal{T}}_{y}f\left(\nu \right)=\nu +\sum _{t=2}^{\infty }{\displaystyle \frac{{\left(y\right)}_{t-1}}{{\left(c\right)}_{t-1}}}{\mathit{b}}_t{\nu }^t,y,c\in \mathbb{C},$$

(12)

where ${\left(y\right)}_t$ is known as the Pochhammer symbol, which can be defined by

$${\left(y\right)}_t={\displaystyle \frac{\mathrm{\Gamma }(y+t)}{\mathrm{\Gamma }\left(y\right)}}=\left\{\begin{array}{c}1(t=0;y\ne 0),\hfill \\ y(y+1)\dots (y+t-1)(t\in \mathbb{N}).\hfill \end{array}\right.$$

Atshan et al. [31] showed that the convolution of operators ${\mathcal{D}}_{\partial ,\alpha }f\left(\nu \right)$ and ${\mathcal{T}}_{y}f\left(\nu \right)$ is

$${\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)={\mathcal{D}}_{\partial ,\alpha }f\left(\nu \right)\ast {\mathcal{T}}_{y}f\left(\nu \right)=\nu +\sum _{t=2}^{\infty }{\left({\displaystyle \frac{1+\alpha }{t+\alpha }}\right)}^{\partial }{\displaystyle \frac{{\left(y\right)}_{t-1}}{{\left(c\right)}_{t-1}}}{\mathit{b}}_t{\nu }^t,$$

(13)

which can be written as

$${\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)=\nu +\sum _{t=2}^{\infty }{\mathrm{\Omega }}_{t,y}{\mathit{b}}_t{\nu }^t,$$

where

$${\mathrm{\Omega }}_{t,y}={\left({\displaystyle \frac{1+\alpha }{t+\alpha }}\right)}^{\partial }{\displaystyle \frac{{\left(y\right)}_{t-1}}{{\left(c\right)}_{t-1}}}.$$

The presence of the parameters $\partial ,\alpha ,y\mathrm{and}c$ in the operator ${\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)$ imparts sufficient flexibility to induce several well-known differential, integral, and convolution operators as special cases. Few of them are presented in the following remark.

Remark 1.

For some specific values of the parameters, we obtain the following special cases:

1.

For $\partial =0$ and $y=c$, the operator ${\mathbb{B}}_{y,c}^{\partial ,\alpha }$ becomes the identity operator, that is, ${\mathbb{B}}_{y,y}^{0,\alpha }f\left(\nu \right)=f\left(\nu \right)$;

2.

For $\partial =0$, it coincides with Carlson–Shaffer integral operator (see [30]);

3.

For $\alpha =0$, and $\partial =m$, it produces a Salagean-type operator of order m, (see [32]);

4.

For $\alpha =0$, and $\partial =1$, it leads to a Rucheweyh-type convolution operator, (see [33]).

Operators play a fundamental role in modern analysis, with wide-ranging applications across classical and fractional calculus, operator theory, and various branches of mathematics. For details, (see [34,35,36]).

For $f\in A$, Govindaraj and Subramanian [37] (see also [38,39]) defined and investigated the Salagean q-differential operator as given below:

$$\begin{array}{cc}\hfill {\mathbb{J}}_q^{0}f\left(\nu \right)=& f\left(\nu \right)\hfill \\ \hfill {\mathbb{J}}_q^{1}f\left(\nu \right)=& \nu D_{q}f\left(\nu \right),\hfill \\ \hfill ⋮\\ \hfill {\mathbb{J}}_q^{\xi }f\left(\nu \right)=& \nu {\mathbb{J}}_q^{\xi }\left({\mathbb{J}}_q^{\xi -1}f\left(\nu \right)\right)=\nu +\sum _{t=2}^{\infty }{\left[t\right]}_q^{\xi }{\mathit{b}}_t{\nu }^t,\left(\xi \in r_0=\mathbb{N}\cup \left\{0\right\},\nu \in V\right).\hfill \end{array}$$

(14)

The operator ${\mathbb{J}}_q^{\xi }$ is also a convolution-type operator obtained by replacing the classical coefficient multiplier with the q-number, and it serves as an effective q-analog of the Salagean differential operator that has been extensively used in defining subclasses of analytic and bi-univalent functions. Moreover, the operator ${\mathbb{J}}_q^{\xi }f\left(\nu \right)$ reduces to the classical Salagean operator, as $q\to 1^{-}$; see [32].

Let $\zeta$ be an analytic function in the open unit disk V, normalized by $\zeta \left(0\right)=1$ and with series expansion of the form

$$\zeta \left(\nu \right)=1+{\mathrm{\Lambda }}_1\nu +{\mathrm{\Lambda }}_2{\nu }^2+{\mathrm{\Lambda }}_3{\nu }^3+\dots ,{\mathrm{\Lambda }}_1>0,\nu \in V.$$

(15)

Motivated by the above discussion, and in contrast to earlier studies that primarily employed classical differential operators and standard subordination, the present work integrates the generalized convolution operators, Jackson’s q-derivative and quasi-subordination. Within this analytical framework, we introduce several new subclasses of analytic functions associated with bi-univalent functions by employing quasi-subordination, together with the operators defined in (8), (13) and (14) and an independent analytic function $\zeta$, whose series expansion is given in (15).

Definition 1.

Let $f\in A$ be given in (1). Then $f\in {\mathcal{R}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\rho ,\eta ,\zeta ,q\right),$ where $0<q<1$, $0\le \gamma ,\rho \le 1,$ and $\eta \in \mathbb{C}\setminus \left\{0\right\},$ if the following conditions are satisfied:

$${\displaystyle \frac{1}{\eta }}\left[G_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]{\prec }_{\mathit{Q}}\left[\zeta \left(\nu \right)-1\right]$$

(16)

where

$$G_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)={\displaystyle \frac{\nu D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right)+{\nu }^2{D}_q^2\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right)}{\left(1-\gamma \right)\nu +\gamma \nu D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right)}}+\rho \nu D_q^2\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right),$$

and

$${\displaystyle \frac{1}{\eta }}\left[H_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)-1\right]{\prec }_{\mathit{Q}}\left[\zeta \left(\mu \right)-1\right],$$

(17)

where

$$H_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)={\displaystyle \frac{\mu D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right)+{\mu }^2{D}_q^2\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right)}{\left(1-\gamma \right)\mu +\gamma \mu D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right)}}+\rho \mu D_q^2\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right).$$

and g is the inverse of f, defined in (2).

Definition 2.

Let $f\in A$ be given in (1). Then $f\in {\mathcal{K}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\sigma ,\rho ,\zeta ,q\right)$, $0\le \gamma ,$ρ$\ge 1,$ and $\sigma \in \mathbb{C}\setminus \left\{0\right\},$ if the following conditions are satisfied:

$${\displaystyle \frac{1}{\sigma }}\left[M_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]{\prec }_{\mathit{Q}}\left[\zeta \left(\nu \right)-1\right],$$

(18)

where

$$M_{\partial ,c,y}^{\rho ,\gamma ,\alpha }\left(\nu \right)=\left(1-\rho \right){\displaystyle \frac{\nu D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right)}{{\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)}}+\rho D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right)+\gamma \nu D_q^2\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }f\left(\nu \right)\right)$$

and

$${\displaystyle \frac{1}{\sigma }}\left[N_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]{\prec }_{\mathit{Q}}\left[\zeta \left(\mu \right)-1\right],$$

(19)

where

$$N_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)=\left(1-\rho \right){\displaystyle \frac{\mu D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right)}{{\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)}}+\rho D_q\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right)+\gamma \mu D_q^2\left({\mathbb{B}}_{y,c}^{\partial ,\alpha }g\left(\mu \right)\right)$$

and g is the inverse of f, defined in (2).

Definition 3.

Let $0\le \alpha <1$, and $f\in A$ be given in (1). Then for $f\in {\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$, if it satisfies the following condition

$$Re\left\{e^{i\theta }\left[{\displaystyle \frac{{\mathbb{J}}_q^{\xi }f\left(\nu \right)}{{\mathbb{J}}_q^{\delta }f\left(\nu \right)}}\right]\right\}>\alpha ,\left(\nu \in V,\xi >\delta ,\left|\theta \right|<\pi andcos\theta >\alpha \right),$$

and

$$Re\left\{e^{i\theta }\left[{\displaystyle \frac{{\mathbb{J}}_q^{\xi }g\left(\mu \right)}{{\mathbb{J}}_q^{\delta }g\left(\mu \right)}}\right]\right\}>\alpha ,\left(\mu \in V,\xi >\delta ,\left|\theta \right|<\pi andcos\theta >\alpha \right).$$

For the following results, we consider these expressions unless otherwise stated.

$$q_t=1+q+q^2+\cdots +q^{t-2}+q^{t-1},q\in \left(0,1\right)$$

and

$$q_t^{\xi }={\left(1+q+q^2+\cdots +q^{t-2}+q^{t-1}\right)}^{\xi },q\in (0,1)$$

$$q_t^{\delta }={\left(1+q+q^2+\cdots +q^{t-2}+q^{t-1}\right)}^{\delta },q\in \left(0,1\right),$$

where $\xi >\delta$.

For these newly defined classes, the parameters $\gamma$, $\rho$, $\xi$ and $\delta$ are regarded as essential, as they directly influence the structure of operators, whereas the parameters $\eta ,\zeta ,\alpha ,c$ and q serve as auxiliary, as they provide additional flexibility to control the behavior and generalize existing subclasses.

Example 1.

Let $f\in A$ have the series representation given by

$$f\left(\nu \right)=\nu +{\displaystyle \frac{2(1-\alpha )}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}{\nu }^2+{\displaystyle \frac{2(1-\alpha )}{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}{\nu }^3+\dots ,\nu \in V.$$

(20)

Then $f\in {\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$ whenever $cos\theta >\alpha .$ This shows that the class ${\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$ is non-empty.

In this work, our primary focus is to investigate some coefficient-based problems, including coefficient estimates, Fekete–Szegö functionals, and upper bounds on Hankel determinants of functions belonging to these newly defined classes.

The derivation of the main results relies fundamentally on the subsequent Lemmas, which furnish the necessary analytical framework for obtaining the required estimates.

2. A Set of Lemmas

Let the class P contain all the analytic functions that have the series representation given by

$$p\left(\nu \right)=1+r_1\nu +r_2{\nu }^2+\dots ,\nu \in V.$$

(21)

and satisfies the condition $Rep\left(\nu \right)>0$.

Lemma 1

([40]). Let $p\in P$ have the series representation given in (21). Then

$$\begin{array}{cc}\hfill \left|r_t\right|\le & 2,\left(t=1,2,3,\dots \right),\hfill \end{array}$$

the given inequality is sharp.

Lemma 2.

Let $p\in P$ have the series representation given in (21). Then

$$\begin{array}{cc}\hfill 2r_2=& r_1^2+x\left(4-r_1^2\right),\hfill \end{array}$$

(22)

and

$$\begin{array}{cc}\hfill 4r_3=& r_1^3+2\left(4-r_1^2\right)r_{1}x-r_1\left(4-r_1^2\right)x^2+2\left(4-r_1^2\right)\left(1-{\left|x\right|}^2\right)\nu ,\hfill \end{array}$$

(23)

for terms x and ν having inequalities $\left|x\right|\le 1and\left|\nu \right|\le 1.$

The values given in (22) and (23) are due to [40,41] respectively.

3. Main Results

Theorem 1.

Let $f\in {\mathcal{R}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\rho ,\eta ,\zeta ,q\right)$, given in (1). Then

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2\right|\le & {\displaystyle \frac{\eta \left|h_0\right|{\mathrm{\Lambda }}_1\sqrt{{\mathrm{\Lambda }}_1}}{\sqrt{{\mathtt{d}}_1{\mathrm{\Omega }}_{3,y}-{\mathtt{d}}_2{\mathrm{\Omega }}_{2,y}^2}}},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_3\right|\le & {\displaystyle \frac{\eta \left(\right|h_1{|+|h}_0{\left|\right)|\mathrm{\Lambda }}_1}{{\mathtt{d}}_1{\mathrm{\Omega }}_{3,y}}}+{\displaystyle \frac{{\eta }^2{h}_0^2{\mathrm{\Lambda }}_1^2}{{\mathtt{d}}_3{\mathrm{\Omega }}_{2,y}^2}},{\mathrm{\Lambda }}_1>1.\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}\hfill \overline{q}=& 1+q,{\mathtt{d}}_1=\left(1+q\overline{q}\right)\eta h_0{\mathrm{\Lambda }}_1^2\left(1+\overline{q}-\gamma +\overline{q}\rho \right),\hfill \\ \hfill {\mathtt{d}}_2=& {\overline{q}}^2\{\gamma \left(2-\gamma \right)\eta h_0{\mathrm{\Lambda }}_1^2+({\mathrm{\Lambda }}_2-{\mathrm{\Lambda }}_1){\left(2-\gamma +\rho \right)}^2\},{\mathtt{d}}_3={\overline{q}}^2{\left(2-\gamma +\rho \right)}^2,\hfill \\ \hfill {\mathrm{\Omega }}_{2,y}=& {\left({\displaystyle \frac{1+\alpha }{2+\alpha }}\right)}^{\partial }{\displaystyle \frac{{\left(y\right)}_{2-1}}{{\left(c\right)}_{2-1}}},and{\mathrm{\Omega }}_{3,y}={\left({\displaystyle \frac{1+\alpha }{3+\alpha }}\right)}^{\partial }{\displaystyle \frac{{\left(y\right)}_{3-1}}{{\left(c\right)}_{3-1}}}.\hfill \end{array}$$

Proof. 

Let f$\in {\mathcal{R}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\rho ,\eta ,\zeta ,q\right)$, then by the definition of quasi-subordination, we have

$${\displaystyle \frac{1}{\eta }}\left[G_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]=\left[\mathit{m}\left(\nu \right)\left(\zeta \left(F\right(\nu )\right)-1\right],\nu \in V,$$

(26)

and

$${\displaystyle \frac{1}{\eta }}\left[H_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)-1\right]=\left[\mathit{m}\left(\mu \right)\left(\zeta \left(F\right(\mu )\right)-1\right],\mu \in V,$$

(27)

where $F\left(\nu \right)$ and $F\left(\mu \right)$ are known as Schwarz functions and can be written in terms of $r\left(\nu \right),s\left(\mu \right)\in P$ as follows:

$$F\left(\nu \right)={\displaystyle \frac{r\left(\nu \right)-1}{r\left(\nu \right)+1}}={\displaystyle \frac{1}{2}}\left[r_1\nu +\left(r_2-{\displaystyle \frac{r_1^2}{2}}\right){\nu }^2+\dots \right],$$

(28)

and

$$F\left(\mu \right)={\displaystyle \frac{s\left(\mu \right)-1}{s\left(\mu \right)+1}}={\displaystyle \frac{1}{2}}\left[s_1\mu +\left(s_2-{\displaystyle \frac{s_1^2}{2}}\right){\mu }^2+\dots \right].$$

(29)

Let

$$\mathit{m}\left(\nu \right)=h_0+h_1\nu +h_2{\nu }^2+h_3{\nu }^3+\dots .$$

(30)

Taking (15) and using (28)–(30) on the right-hand side of (26) and (27), we have

$$\mathit{m}\left(\nu \right)\left(\zeta \left({\displaystyle \frac{r\left(\nu \right)-1}{r\left(\nu \right)+1}}\right)\right)-1=$$

$${\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1{r}_1\nu +\left\{{\displaystyle \frac{1}{2}}{h}_1{\mathrm{\Lambda }}_1{r}_1+{\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1\left(r_2-{\displaystyle \frac{r_1^2}{2}}\right)+{\displaystyle \frac{1}{4}}{h}_0{\mathrm{\Lambda }}_2{r}_1^2\right\}{\nu }^2+\dots ,$$

(31)

and

$$\mathit{m}\left(\mu \right)\left(\zeta \left({\displaystyle \frac{s\left(\mu \right)-1}{s\left(\mu \right)+1}}\right)\right)-1=$$

$${\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1{s}_1\mu +\left\{{\displaystyle \frac{1}{2}}{h}_1{\mathrm{\Lambda }}_1{s}_1+{\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1\left(s_2-{\displaystyle \frac{s_1^2}{2}}\right)+{\displaystyle \frac{1}{4}}{h}_0{\mathrm{\Lambda }}_2{s}_1^2\right\}{\mu }^2+\dots .$$

(32)

Considering the left-hand sides of (26) and (27), we have

$$\begin{array}{c}{\displaystyle \frac{1}{\eta }}\left[G_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]={\displaystyle \frac{1}{\eta }}[\left(2-\gamma +\rho \right){\left[2\right]}_q{\mathrm{\Omega }}_{2,y}{b}_2\nu \hfill \\ \hfill +\left(\left(1+{\left[2\right]}_q-\gamma +{\left[2\right]}_q\rho \right){\left[3\right]}_q{\mathrm{\Omega }}_{3,y}{b}_3-\gamma (2-\gamma ){\left({\left[2\right]}_q\right)}^2{\mathrm{\Omega }}_{2,y}^2{b}_2^2\right){\nu }^2+\dots ].\end{array}$$

(33)

and

$$\begin{array}{c}{\displaystyle \frac{1}{\eta }}\left[H_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)-1\right]={\displaystyle \frac{1}{\eta }}[-\left(2-\gamma +\rho \right){\left[2\right]}_q{\mathrm{\Omega }}_{2,y}{b}_2\mu \hfill \\ +(\left(1+{\left[2\right]}_q-\gamma +\rho {\left[2\right]}_q\right){\left[3\right]}_q\left(2b_2^2-b_3\right){\mathrm{\Omega }}_{3,y}\hfill \\ -\gamma (2-\gamma ){\left({\left[2\right]}_q\right)}^2{\mathrm{\Omega }}_{2,y}^2{b}_2^2){\mu }^2+\dots ].\hfill \end{array}$$

(34)

Comparing (31) and (33) gives us

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\eta }}\left[\left(2-\gamma +\rho \right){\left[2\right]}_q{\mathrm{\Omega }}_{2,y}{\mathit{b}}_2\right]={\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1{r}_1,\end{array}$$

(35)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\eta }}\left[\left(1+{\left[2\right]}_q-\gamma +{\left[2\right]}_q\rho \right){\left[3\right]}_q{\mathrm{\Omega }}_{3,y}{\mathit{b}}_3-\gamma \left(2-\gamma \right){\left({\left[2\right]}_q\right)}^2{\mathrm{\Omega }}_{2,y}^2{\mathit{b}}_2^2\right]=\hfill \\ \hfill & {\displaystyle \frac{1}{2}}{h}_1{\mathrm{\Lambda }}_1{r}_1+{\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1\left(r_2-{\displaystyle \frac{r_1^2}{2}}\right)+{\displaystyle \frac{1}{4}}{h}_0{\mathrm{\Lambda }}_2{r}_1^2.\hfill \end{array}$$

(36)

Also, by comparing (32) and (34), we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\eta }}\left[-\left(2-\gamma +\rho \right){\left[2\right]}_q{\mathrm{\Omega }}_{2,y}{\mathit{b}}_2\right]={\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1{s}_1,\end{array}$$

(37)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\eta }}\left[{\left(1+{\left[2\right]}_q-\gamma +\rho {\left[2\right]}_q\right)\left[3\right]}_q\left(2{\mathit{b}}_2^2-{\mathit{b}}_3\right){\mathrm{\Omega }}_{3,y}-\gamma \left(2-\gamma \right){\left({\left[2\right]}_q\right)}^2{\mathrm{\Omega }}_{2,y}^2{\mathit{b}}_2^2\right]=\hfill \\ \hfill & {\displaystyle \frac{1}{2}}{h}_1{\mathrm{\Lambda }}_1{s}_1+{\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1\left(s_2-{\displaystyle \frac{s_1^2}{2}}\right)+{\displaystyle \frac{1}{4}}{h}_0{\mathrm{\Lambda }}_2{s}_1^2.\hfill \end{array}$$

(38)

From (35) and (37) we get

$$\begin{array}{c}\hfill {\mathit{b}}_2={\displaystyle \frac{\eta h_0{\mathrm{\Lambda }}_1{r}_1}{2{\left[2\right]}_q\left(2-\gamma +\rho \right){\mathrm{\Omega }}_{2,y}}}={\displaystyle \frac{-\eta h_0{\mathrm{\Lambda }}_1{s}_1}{2{\left[2\right]}_q\left(2-\gamma +\rho \right){\mathrm{\Omega }}_{2,y}}}.\end{array}$$

(39)

Therefore, we have

$$\begin{array}{c}\hfill r_1=-s_1.\end{array}$$

(40)

Taking the square of (39), after simplifying, we have

$$\begin{array}{c}\hfill 8{\left({\left[2\right]}_q\right)}^2{\left(2-\gamma +\rho \right)}^2{\mathrm{\Omega }}_{2,y}^2{\mathit{b}}_2^2={\eta }^2{h}_0^2{\mathrm{\Lambda }}_1^2\left(r_1^2+s_1^2\right).\end{array}$$

(41)

By adding (36) and (38), utilizing (40) and (41), we obtain

$$\begin{array}{cc}\hfill & 8\left[{\left[3\right]}_q{\eta h}_0{\mathrm{\Lambda }}_1^2\left(1+{\left[2\right]}_q-\gamma +{\left[2\right]}_q\rho \right){\mathrm{\Omega }}_{3,y}-{\left({\left[2\right]}_q\right)}^2{\eta h}_0{\mathrm{\Lambda }}_1^2\gamma \left(2-\gamma \right){\mathrm{\Omega }}_{2,y}^2\right]{\mathit{b}}_2^2=\hfill \\ \hfill & {2\eta }^2{h}_0^2{\mathrm{\Lambda }}_1^3\left(r_2+s_2\right)+\left({\mathrm{\Lambda }}_2-{\mathrm{\Lambda }}_1\right)8{\left({\left[2\right]}_q\right)}^2{\left(2-\gamma +\rho \right)}^2{\mathrm{\Omega }}_{2,y}^2{\mathit{b}}_2^2,\hfill \end{array}$$

(42)

which implies

$$\begin{array}{c}\hfill {\mathit{b}}_2^2={\displaystyle \frac{{2\eta }^2{h}_0^2{\mathrm{\Lambda }}_1^3\left(r_2+s_2\right)}{8\left[{\mathtt{d}}_1{\mathrm{\Omega }}_{3,y}-{\mathtt{d}}_2{\mathrm{\Omega }}_{2,y}^2\right]}}.\end{array}$$

(43)

Using Lemma 1 in (43) gives (24).

Now, to examine the estimate of the coefficient $\left|{\mathit{b}}_3\right|$, subtract (36) from (38), and we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{4}{\eta }}\left[-\left(1+{\left[2\right]}_q-\gamma +{\left[2\right]}_q\rho \right){\mathit{b}}_2^2+\left(1+{\left[2\right]}_q-\gamma +{\left[2\right]}_q\rho \right){\mathit{b}}_3\right]{\left[3\right]}_q{\mathrm{\Omega }}_{3,y}=\hfill \\ \hfill & {2h}_1{\mathrm{\Lambda }}_1{r}_1+h_0{\mathrm{\Lambda }}_1\left(r_2-s_2\right),\hfill \end{array}$$

(44)

Applying (41), this leads to

$$\begin{array}{cc}\hfill {\mathit{b}}_3=& {\displaystyle \frac{\eta h_1{\mathrm{\Lambda }}_1{r}_1}{2{\left[3\right]}_q\left({\left[2\right]}_q+1-\gamma +{\left[2\right]}_q\rho \right){\mathrm{\Omega }}_{3,y}}}+{\displaystyle \frac{\eta h_0{\mathrm{\Lambda }}_1(r_2-s_2)}{4{\left[3\right]}_q\left({\left[2\right]}_q+1-\gamma +{\left[2\right]}_q\rho \right){\mathrm{\Omega }}_{3,y}}}+\hfill \\ \hfill & {\displaystyle \frac{{\eta }^2{h}_0^2{\mathrm{\Lambda }}_1^2(r_1^2+s_1^2)}{8{\left[2\right]}_q^2{\left(2-\gamma +\rho \right)}^2{\mathrm{\Omega }}_{2,y}^2}}.\hfill \end{array}$$

(45)

Implementing Lemma 1 in (45) and some simple calculations lead us to (25). □

Remark 2.

By substituting the value $q\to 1^{-}$ in the provided result, this gives a refined and advanced result that matches the findings derived by Atshan et al. [31].

Theorem 2.

Let $f\in A$ be from the subclass ${\mathcal{K}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\sigma ,\rho ,\zeta ,q\right).$ Then

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2\right|\le \min & \left\{{\displaystyle \frac{\sigma {|h}_0{|\mathrm{\Lambda }}_1}{\left(q+\rho +\overline{q}\gamma \right){\mathrm{\Omega }}_{2,y}}},\sqrt{{\displaystyle \frac{\sigma \left|h_0\right|\left({\mathrm{\Lambda }}_1+{|\mathrm{\Lambda }}_2-{\mathrm{\Lambda }}_1|\right)}{\left[\left(\rho +q\overline{q}+{\overline{q}}_o\gamma \right){\mathrm{\Omega }}_{3,y}-q\left(1-\rho \right){\mathrm{\Omega }}_{2,y}^2\right]}}}\right\},\hfill \end{array}$$

(46)

and

$$\begin{array}{c}\hfill \left|{\mathit{b}}_3\right|\le \min \left\{\begin{array}{c}{\displaystyle \frac{\sigma \left(\right|h_1{|\mathrm{\Lambda }}_1+\left|h_0{|\mathrm{\Lambda }}_1\right)}{\left(\rho +q\overline{q}+{\overline{q}}_o\gamma \right){\mathrm{\Omega }}_{3,y}}}+{\displaystyle \frac{{\sigma }^2{h}_0^2{\mathrm{\Lambda }}_1^2}{{\left(q+\rho +\overline{q}\gamma \right)}^2{\mathrm{\Omega }}_{2,y}^2}},\\ {\displaystyle \frac{\sigma \left(\right|h_1{|\mathrm{\Lambda }}_1+\left|h_0{|\mathrm{\Lambda }}_1\right)}{\left(\rho +q\overline{q}+{\overline{q}}_o\gamma \right){\mathrm{\Omega }}_{3,y}}}+{\displaystyle \frac{\sigma |h_0{\left|\right(\mathrm{\Lambda }}_1+|{\mathrm{\Lambda }}_2{-\mathrm{\Lambda }}_1\left|\right)}{\left[\left(\rho +q\overline{q}+{\overline{q}}_o\gamma \right){\mathrm{\Omega }}_{3,y}-q\left(1-\rho \right){\mathrm{\Omega }}_{2,y}^2\right]}}\end{array}\right\},{\mathrm{\Lambda }}_1>1.\end{array}$$

(47)

where

$$\begin{array}{cc}\hfill \overline{q}=& 1+qand{\overline{q}}_o=1+2q+2q^2+q^3.\hfill \end{array}$$

Proof. 

Let f∈${\mathcal{K}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\sigma ,\rho ,\zeta ,q\right)$, then by the definition of quasi-subordination we have

$${\displaystyle \frac{1}{\sigma }}\left[M_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]=\left[\mathit{m}\left(\nu \right)\left(\zeta \left(F\right(\nu )\right)-1\right],$$

(48)

and

$${\displaystyle \frac{1}{\sigma }}\left[N_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)-1\right]=\left[\mathit{m}\left(\mu \right)\left(\zeta \left(F\right(\mu )\right)-1\right],$$

(49)

where the Schwarz functions $F\left(\nu \right)$ and $F\left(\mu \right)$ are defined in (28) and (29) respectively. As the series representations of the right-hand side of Equations (48) and (49) are given in (31) and (33) respectively, now by considering the left-hand side of (48) and (49), we have

$$\begin{array}{c}{\displaystyle \frac{1}{\sigma }}\left[M_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\nu \right)-1\right]={\displaystyle \frac{1}{\sigma }}[\left(\rho +{\left[2\right]}_q\gamma +{\left[2\right]}_q-1\right){\mathrm{\Omega }}_{2,y}{b}_2\nu \hfill \\ +\left(\left(\rho +{\left[3\right]}_q+{\left[2\right]}_q{\left[3\right]}_q\gamma -1\right){\mathrm{\Omega }}_{3,y}{b}_3-\left({\left[2\right]}_q-1\right)(1-\rho ){\mathrm{\Omega }}_{2,y}^2{b}_2^2\right){\nu }^2+\dots ],\hfill \end{array}$$

(50)

and

$$\begin{array}{c}{\displaystyle \frac{1}{\sigma }}\left[N_{\partial ,c,y,q}^{\rho ,\gamma ,\alpha }\left(\mu \right)-1\right]={\displaystyle \frac{1}{\sigma }}[-\left({\left[2\right]}_q-1+\rho +{\left[2\right]}_q\gamma \right){\mathrm{\Omega }}_{2,y}{b}_2\mu \hfill \\ +(\left({\left[3\right]}_q-1+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right)\left(2b_2^2-b_3\right){\mathrm{\Omega }}_{3,y}\hfill \\ -\left({\left[2\right]}_q-1\right)(1-\rho ){\mathrm{\Omega }}_{2,y}^2{b}_2^2){\mu }^2+\dots ].\hfill \end{array}$$

(51)

Equating (31) with (50) results in

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\sigma }}\left[\left(\rho +{\left[2\right]}_q\gamma +{\left[2\right]}_q-1\right){\mathrm{\Omega }}_{2,y}{\mathit{b}}_2\right]={\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1{r}_1,\end{array}$$

(52)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\sigma }}\left[+\left(\rho +{\left[3\right]}_q+{\left[2\right]}_q{\left[3\right]}_q\gamma -1\right){\mathrm{\Omega }}_{3,y}{\mathit{b}}_3+\left(-\left({-1+\left[2\right]}_q\right)(1-\rho )\right){\mathrm{\Omega }}_{2,y}^2{\mathit{b}}_2^2\right]=\hfill \\ \hfill & {\displaystyle \frac{1}{2}}{h}_1{\mathrm{\Lambda }}_1{r}_1+{\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1\left(r_2-{\displaystyle \frac{r_1^2}{2}}\right)+{\displaystyle \frac{1}{4}}{h}_0{\mathrm{\Lambda }}_2{r}_1^2.\hfill \end{array}$$

(53)

Also, equating (32) with (51) gives

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\sigma }}\left[-\left(\rho +{\left[2\right]}_q\gamma +{\left[2\right]}_q-1\right){\mathrm{\Omega }}_{2,y}{\mathit{b}}_2\right]={\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1{s}_1,\end{array}$$

(54)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\sigma }}\left[\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right)\left(2{\mathit{b}}_2^2-{\mathit{b}}_3\right){\mathrm{\Omega }}_{3,y}-\left({\left[2\right]}_q-1\right)\left(1-\rho \right){\mathit{b}}_2^2{\mathrm{\Omega }}_{2,y}^2\right]=\hfill \\ \hfill & {\displaystyle \frac{1}{2}}{h}_1{\mathrm{\Lambda }}_1{s}_1+{\displaystyle \frac{1}{2}}{h}_0{\mathrm{\Lambda }}_1\left(s_2-{\displaystyle \frac{s_1^2}{2}}\right)+{\displaystyle \frac{1}{4}}{h}_0{\mathrm{\Lambda }}_2{s}_1^2.\hfill \end{array}$$

(55)

From (52) and (54) we yield

$$\begin{array}{c}\hfill r_1=-s_1.\end{array}$$

(56)

and

$${8{\left({\left[2\right]}_q-1+\rho +{\left[2\right]}_q\gamma \right)}^2{\mathrm{\Omega }}_{2,y}^2{\mathit{b}}^2}_2={\sigma }^2{h}_0^2{\mathrm{\Lambda }}_1^2\left(r_1^2+s_1^2\right).$$

(57)

Adding (53) and (55) we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{8}{\sigma }}\left[\left(\rho +{\left[3\right]}_q+{\left[2\right]}_q{\left[3\right]}_q\gamma -1\right){\mathrm{\Omega }}_{3,y}{\mathit{b}}_2^2+\left(-\left({\left[2\right]}_q-1\right)(1-\rho )\right){\mathrm{\Omega }}_{2,y}^2{\mathit{b}}_2^2\right]=\hfill \\ \hfill & {2h}_0{\mathrm{\Lambda }}_1\left(r_2+s_2\right)+h_0{(\mathrm{\Lambda }}_2{-\mathrm{\Lambda }}_1{\left)\right(r}_1^2+s_1^2),\hfill \end{array}$$

(58)

which gives

$$\begin{array}{c}\hfill {\mathit{b}}_2^2={\displaystyle \frac{2\sigma h_0{\mathrm{\Lambda }}_1\left(r_2+s_2\right)+\sigma h_0{(\mathrm{\Lambda }}_2{-\mathrm{\Lambda }}_1{\left)\right(r}_1^2+s_1^2)}{8\left[\left(\rho +{\left[3\right]}_q+{\left[2\right]}_q{\left[3\right]}_q\gamma -1\right){\mathrm{\Omega }}_{3,y}-\left({\left[2\right]}_q-1\right)\left(1-\rho \right){\mathrm{\Omega }}_{2,y}^2\right]}}.\end{array}$$

(59)

Implementing Lemma 1 for the coefficients, we have

$$\left|{\mathit{b}}_2\right|\le {\displaystyle \frac{\sigma h_0{\mathrm{\Lambda }}_1}{\left({\left[2\right]}_q-1+\rho +{\left[2\right]}_q\gamma \right){\mathrm{\Omega }}_{2,y}}},$$

and

$$\begin{array}{c}\hfill \left|{\mathit{b}}_2\right|\le \sqrt{{\displaystyle \frac{\sigma \left|h_0\right|\left({\mathrm{\Lambda }}_1+{|\mathrm{\Lambda }}_2-{\mathrm{\Lambda }}_1|\right)}{\left[\left(\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma +{\left[3\right]}_q-1\right){\mathrm{\Omega }}_{3,y}-\left({-1+\left[2\right]}_q\right)\left(1-\rho \right){\mathrm{\Omega }}_{2,y}^2\right]}},}\end{array}$$

This gives the required result.

Now, to find the bound of ${\mathit{b}}_3$, we subtract (53) from (55), and obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{4}{\sigma }}\left[\left({\left[3\right]}_q-1+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right)\left(-{\mathit{b}}_2^2+{\mathit{b}}_3\right){\mathrm{\Omega }}_{3,y}\right]=h_1{\mathrm{\Lambda }}_1{(2r}_1)+h_0{\mathrm{\Lambda }}_1\left(r_2-s_2\right).\end{array}$$

(60)

By using the value of ${\mathit{b}}_2^2$ from (59) in (60) we obtain

$$\begin{array}{c}\hfill {\mathit{b}}_3\le {\displaystyle \frac{2\sigma h_1{\mathrm{\Lambda }}_1{r}_1+{\sigma h}_0{\mathrm{\Lambda }}_1\left(r_2-s_2\right)}{4\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right){\mathrm{\Omega }}_{3,y}}}+{\displaystyle \frac{{\sigma }^2{h}_0^2{\mathrm{\Lambda }}_1^2\left(r_1^2+s_1^2\right)}{8{\left({\left[2\right]}_q-1+\rho +{\left[2\right]}_q\gamma \right)}^2{\mathrm{\Omega }}_{2,y}^2}}.\end{array}$$

(61)

Using the value of ${\mathit{b}}_2^2$ from (59) in (60) gives

$$\begin{array}{cc}\hfill {\mathit{b}}_3\le & {\displaystyle \frac{2\sigma h_1{\mathrm{\Lambda }}_1{r}_1+{\sigma h}_0{\mathrm{\Lambda }}_1\left(r_2-s_2\right)}{4\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right){\mathrm{\Omega }}_{3,y}}}+\hfill \\ \hfill & {\displaystyle \frac{(2\sigma h_0{\mathrm{\Lambda }}_1\left(r_2+s_2\right)+\sigma h_0{(\mathrm{\Lambda }}_2{-\mathrm{\Lambda }}_1{\left)\right(r}_1^2+s_1^2))}{8\left[\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right){\mathrm{\Omega }}_{3,y}-\left({\left[2\right]}_q-1\right)\left(1-\rho \right){\mathrm{\Omega }}_{2,y}^2\right]}}.\hfill \end{array}$$

(62)

Taking modulus and implementing Lemma 1 on (61) and (62) provides

$$|{\mathit{b}}_3|\le {\displaystyle \frac{\sigma \left(\right|h_1{|\mathrm{\Lambda }}_1+\left|h_0{|\mathrm{\Lambda }}_1\right)}{\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right){\mathrm{\Omega }}_{3,y}}}+{\displaystyle \frac{{\sigma }^2{h}_0^2{\mathrm{\Lambda }}_1^2}{{\left({\left[2\right]}_q-1+\rho +{\left[2\right]}_q\gamma \right)}^2{\mathrm{\Omega }}_{2,y}^2}},$$

(63)

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_3\right|\le & {\displaystyle \frac{\sigma \left(\right|h_1{|\mathrm{\Lambda }}_1+\left|h_0{|\mathrm{\Lambda }}_1\right)}{\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right){\mathrm{\Omega }}_{3,y}}}+\hfill \\ \hfill & {\displaystyle \frac{\sigma h_0{(\mathrm{\Lambda }}_1+|{\mathrm{\Lambda }}_2{-\mathrm{\Lambda }}_1\left|\right)}{\left[\left(-1+{\left[3\right]}_q+\rho +{\left[2\right]}_q{\left[3\right]}_q\gamma \right){\mathrm{\Omega }}_{3,y}-\left({\left[2\right]}_q-1\right)\left(1-\rho \right){\mathrm{\Omega }}_{2,y}^2\right]}}.\hfill \end{array}$$

(64)

This provides the bound on $\left|{\mathit{b}}_3\right|,$ given in (47). □

Remark 3.

By setting $q\to 1^{-}$, applying this value in the derived results produces a refined and advanced result that matches the earlier results by Atshan et al. [31].

Theorem 3.

Let $f\in {\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$ for $\xi >\delta +1,0\le \alpha <1,\left|\theta \right|<\pi$ and $cos\theta >\alpha$, with the series representation given in (1). Then

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2\right|& \le {\displaystyle \frac{2X}{c_1}},\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_3\right|& \le {\displaystyle \frac{4X^2}{c_1^2}}+{\displaystyle \frac{2X}{c_2}},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_4\right|& \le {\displaystyle \frac{8\left(c_6+c_3\right)X^3}{c_4{c}_1^4}}+{\displaystyle \frac{10X^2}{c_1{c}_2}}+{\displaystyle \frac{2X}{c_3}}.\hfill \end{array}$$

(67)

and also for any $\kappa \in \mathbb{C},$ we have

$$\begin{array}{c}\hfill \left|{\mathit{b}}_3-\kappa {\mathit{b}}_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{X}{c_2}}0\le \left|T(\kappa ,\xi ,\delta )\right|\le {\displaystyle \frac{1}{2c_2}},\\ 2\left|T(\kappa ,\xi ,\delta )\right|X\left|T(\kappa ,\xi ,\delta )\right|\ge {\displaystyle \frac{1}{2c_2}},\end{array}\right.\end{array}$$

(68)

where

$$\begin{array}{cc}\hfill X=& cos\theta -\alpha ,c_1={(1+q)}^{\xi }-{(1+q)}^{\delta },\hfill \\ \hfill c_2=& {(1+q+q^2)}^{\xi }-{(1+q+q^2)}^{\delta },c_3={(1+q+q^2+q^3)}^{\xi }-{(1+q+q^2+q^3)}^{\delta },\hfill \\ \hfill c_4=& {(1+q)}^{\xi }+{(1+q)}^{\delta },c_5={(1+q)}^{\xi +\delta }-{(1+q)}^{2\delta },\hfill \\ \hfill c_6=& ({(1+q+q^2)}^{\delta }-{(1+q)}^{2\delta })({(1+q)}^{\xi }-{(1+q)}^{\delta }){(1+q)}^{\delta }({(1+q+q^2)}^{\xi }-\hfill \\ \hfill & {(1+q+q^2)}^{\delta }{\displaystyle )}-({(1+q+q^2+q^3)}^{\xi }-{(1+q+q^2+q^3)}^{\delta }),\hfill \\ \hfill T\left(\kappa ,\xi ,\delta \right)=& {\displaystyle \frac{1-\kappa }{2(c_2-c_5)}},\hfill \end{array}$$

Proof. 

Let $f\in {\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$ and g$=f^{-1}$. Define

$${\displaystyle \frac{{\mathbb{J}}_q^{\xi }f\left(\nu \right)}{{\mathbb{J}}_q^{\delta }f\left(\nu \right)}}=1+\sum _{t=1}^{\infty }{h}_t{\nu }^t,{\displaystyle \frac{{\mathbb{J}}_q^{\xi }g\left(\mu \right)}{{\mathbb{J}}_q^{\delta }g\left(\mu \right)}}=1+\sum _{t=1}^{\infty }{h}_t{\mu }^t.$$

Then

$$e^{i\theta }\left[{\displaystyle \frac{{\mathbb{J}}_q^{\xi }f\left(\nu \right)}{{\mathbb{J}}_q^{\delta }f\left(\nu \right)}}\right]-\alpha =e^{i\theta }\left(1+\sum _{t=1}^{\infty }{h}_t{\nu }^t\right)-\alpha ,$$

and similarly for g, we have

$$e^{i\theta }\left[{\displaystyle \frac{{\mathbb{J}}_q^{\xi }g\left(\mu \right)}{{\mathbb{J}}_q^{\delta }g\left(\mu \right)}}\right]-\alpha =e^{i\theta }\left(1+\sum _{t=1}^{\infty }{h}_t{\mu }^t\right)-\alpha .$$

Using $e^{i\theta }=cos\theta +isin\theta$ and some simplifications, we obtain

$$e^{i\theta }\left[{\displaystyle \frac{{\mathbb{J}}_q^{\xi }f\left(\nu \right)}{{\mathbb{J}}_q^{\delta }f\left(\nu \right)}}\right]-\alpha -isin\theta =cos\theta -\alpha {+e}^{i\theta }\left(\sum _{t=1}^{\infty }{h}_t{\nu }^t\right),$$

and

$$e^{i\theta }\left[{\displaystyle \frac{{\mathbb{J}}_q^{\xi }g\left(\mu \right)}{{\mathbb{J}}_q^{\delta }g\left(\mu \right)}}\right]-\alpha -isin\theta =cos\theta -\alpha {+e}^{i\theta }\left(\sum _{t=1}^{\infty }{h}_t{\mu }^t\right),$$

Performing some simple calculations gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{e^{i\theta }\left[\frac{{\mathbb{J}}_q^{\xi }f\left(\nu \right)}{{\mathbb{J}}_q^{\delta }f\left(\nu \right)}\right]-\alpha -isin\theta }{cos\theta -\alpha }}& =1+{\displaystyle \frac{e^{i\theta }\left({\sum }_{t=1}^{\infty }{h}_t{\nu }^t\right)}{cos\theta -\alpha }},\hfill \\ \hfill {\displaystyle \frac{e^{i\theta }\left[\frac{{\mathbb{J}}_q^{\xi }g\left(\mu \right)}{{\mathbb{J}}_q^{\delta }g\left(\mu \right)}\right]-\alpha -isin\theta }{cos\theta -\alpha }}& =1+{\displaystyle \frac{e^{i\theta }\left({\sum }_{t=1}^{\infty }{h}_t{\mu }^t\right)}{cos\theta -\alpha }}.\hfill \end{array}$$

(69)

Since both expressions are analytic in an open unit disk and satisfy the conditions of normalizations for class P, they admit expansion of the form

$$\begin{array}{cc}\hfill {\displaystyle \frac{e^{i\theta }\left[\frac{{\mathbb{J}}_q^{\xi }f\left(\nu \right)}{{\mathbb{J}}_q^{\delta }f\left(\nu \right)}\right]-\alpha -isin\theta }{cos\theta -\alpha }}& =p\left(\nu \right),\hfill \\ \hfill {\displaystyle \frac{e^{i\theta }\left[\frac{{\mathbb{J}}_q^{\xi }g\left(\mu \right)}{{\mathbb{J}}_q^{\delta }g\left(\mu \right)}\right]-\alpha -isin\theta }{cos\theta -\alpha }}& =\lambda \left(\mu \right),\hfill \end{array}$$

(70)

where the functions $p\left(\nu \right)$ and $\lambda \left(\mu \right)$ are from class P and have the series form provided in (21) and

$$\lambda \left(\mu \right)=1+s_1\mu +s_2{\mu }^2+\dots .$$

(71)

Using the series expansions of ${\mathbb{J}}_q^{\xi }f\left(\nu \right),$${\mathbb{J}}_q^{\delta }f\left(\nu \right)$ and their ratio, along with series representations of $p\left(\nu \right)$ and $\lambda \left(\mu \right)$, we equate coefficients of like powers of $\nu$ and $\mu$ on both sides of Equation (70). Comparing coefficients of $\nu$ yields

$$\begin{array}{c}\hfill {\mathit{b}}_2={\displaystyle \frac{r_1\left[cos\theta -\alpha \right]}{e^{i\theta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}},\end{array}$$

(72)

Equating the coefficients of ${\nu }^2$, we obtain

$$\begin{array}{c}\hfill \left[\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right){\mathit{b}}_3-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\mathit{b}}_2^2\right]={\displaystyle \frac{r_2\left(cos\theta -\alpha \right)}{e^{i\theta }}},\end{array}$$

(73)

Similarly, comparing coefficients of ${\nu }^3$, we arrive at

$$\begin{array}{c}\hfill \left[\begin{array}{c}\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right){\mathit{b}}_4+{\left[2\right]}_q^{2\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\mathit{b}}_2^3-\\ {\left[3\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\mathit{b}}_2{\mathit{b}}_3-{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right){\mathit{b}}_2{\mathit{b}}_3\end{array}\right]={\displaystyle \frac{r_3\left(cos\theta -\alpha \right)}{e^{i\theta }}}.\end{array}$$

(74)

Proceeding analogously for $\lambda \left(\mu \right)$, we compare coefficients of $({\mu }^k,k=1,2,3)$ and obtain

$$\begin{array}{c}\hfill -{\mathit{b}}_2={\displaystyle \frac{s_1\left(cos\theta -\alpha \right)}{e^{i\theta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}},\end{array}$$

(75)

$$\begin{array}{c}\hfill \left[2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right){\mathit{b}}_2^2-\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right){\mathit{b}}_3-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\mathit{b}}_2^2\right]={\displaystyle \frac{s_2\left(cos\theta -\alpha \right)}{e^{i\theta }}},\end{array}$$

(76)

$$\begin{array}{cc}\hfill & -\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left(5b_2^3-5b_2{b}_3+b_4\right)\hfill \\ \hfill & -b_2\left(b_3-2b_2^2\right)\left[{\left[3\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)\right]\hfill \\ \hfill & -{\left[2\right]}_q^{2\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)b_2^3={\displaystyle \frac{s_3(\cos \theta -\alpha )}{e^{i\theta }}}.\hfill \end{array}$$

(77)

From (72) and (75), we obtain

$$\begin{array}{c}\hfill r_1=-s_1,\end{array}$$

(78)

and

$$\begin{array}{c}\hfill {\mathit{b}}_2={\displaystyle \frac{r_1\left[cos\theta -\alpha \right]}{e^{i\theta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}.\end{array}$$

(79)

Subtracting (73) and (76) results in

$${\mathit{b}}_3={\displaystyle \frac{r_1^2{(cos\theta -\alpha )}^2}{e^{2i\theta }{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2}}+{\displaystyle \frac{{(r}_2-s_2)\left(cos\theta -\alpha \right)}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{i\theta }}},$$

(80)

Subtracting (74) and (77) gives

$$\begin{array}{c}\hfill {\mathit{b}}_4={\displaystyle \frac{r_1^3{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)\right](cos\theta -\alpha )}^3}{{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)e}^{3i\theta }{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^3}}+\\ \hfill {\displaystyle \frac{5r_1(r_2-s_2){(cos\theta -\alpha )}^2}{4\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{2i\theta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}+{\displaystyle \frac{\left(r_3-s_3\right)\left(cos\theta -\alpha \right)}{2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)e^{i\theta }}}.\end{array}$$

(81)

Taking the modulus and implementing Lemma 1 to (79), (80), and (81), we obtain (65), (66), and (67), respectively.

Now by adding (73) and (76), we have

$$\begin{array}{c}\hfill 2\times \left[\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\right]{\mathit{b}}_2^2={\displaystyle \frac{(r_2+s_2)\left(cos\theta -\alpha \right)}{e^{i\theta }}}.\end{array}$$

(82)

Subtracting (73) from (76), we obtain

$$\begin{array}{c}\hfill {\mathit{b}}_3={\displaystyle \frac{(r_2-s_2)\left(cos\theta -\alpha \right)}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{i\theta }}}+{\mathit{b}}_2^2.\end{array}$$

(83)

From (82) and (83), we have the following results.

$${\mathit{b}}_3-\kappa {\mathit{b}}_2^2={\displaystyle \frac{\left(r_2-s_2\right)\left(cos\theta -\alpha \right)}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{i\theta }}}+(1-\kappa ){\mathit{b}}_2^2,$$

$${\mathit{b}}_3-\kappa {\mathit{b}}_2^2={\displaystyle \frac{\left(r_2-s_2\right)\left(cos\theta -\alpha \right)}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{i\theta }}}+(1-\kappa ){\displaystyle \frac{(r_2+s_2)\left(cos\theta -\alpha \right)}{{2\left[\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\right]e}^{i\theta }}},$$

$$\begin{array}{c}\hfill {\mathit{b}}_3-\kappa {\mathit{b}}_2^2={\displaystyle \frac{(cos\theta -\alpha )}{e^{i\theta }}}\left\{\left({\displaystyle \frac{\left(1-\kappa \right)}{2\left[\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\right]}}+{\displaystyle \frac{1}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}\right)r_2 \right.\\ \hfill \left.+\left({\displaystyle \frac{\left(1-\kappa \right)}{2\left[\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\right]}}-{\displaystyle \frac{1}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}\right)s_2\right\},\end{array}$$

$$\begin{array}{c}\hfill {\mathit{b}}_3-\kappa {\mathit{b}}_2^2={\displaystyle \frac{(cos\theta -\alpha )}{e^{i\theta }}}\left\{\left(T\left(\kappa ,\xi ,\delta \right)+{\displaystyle \frac{1}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}\right)r_2\right. \\ \hfill +\left.\left(T\left(\kappa ,\xi ,\delta \right)-{\displaystyle \frac{1}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}\right)s_2\right\},\end{array}$$

(84)

where

$$T\left(\kappa ,\xi ,\delta \right)={\displaystyle \frac{(1-\kappa )}{2\left[\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-{\left[2\right]}_q^{\delta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\right]}}.$$

(85)

The function $T\left(\kappa ,\xi ,\delta \right)$ defined in (85) is just a normalization constant, and analytically qualifies how the interaction between parameters $\kappa ,\xi \mathrm{and}\delta$ influences the sharpness of coefficient bounds.

By taking the modulus of (84) and applying triangular inequality along with Lemma 1, the required result can be obtained. □

Remark 4.

By letting $q\to 1^{-},$ in derived expressions we obtain a refined form that coincides with the results obtained by Orhan et al. [42], which is presented here as a Corollary 1.

Corollary 1.

Let $f\left(\nu \right)$ given by (1) be from the class $f\in {\mathfrak{S}}_{\mathrm{\Psi }}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$ for $\xi >\delta +1,0\le \alpha <1,\left|\theta \right|<\pi$ and $cos\theta >\alpha .$ Then

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2\right|\le & {\displaystyle \frac{2(cos\theta -\alpha )}{2^{\xi }-2^{\delta }}},\hfill \\ \hfill \left|{\mathit{b}}_3\right|\le & {\displaystyle \frac{4{\left(cos\theta -\alpha \right)}^2}{{\left(2^{\xi }-2^{\delta }\right)}^2}}+{\displaystyle \frac{2\left(cos\theta -\alpha \right)}{3^{\xi }-3^{\delta }}},\hfill \\ \hfill \left|{\mathit{b}}_4\right|\le & {\displaystyle \frac{8{\left(cos\theta -\alpha \right)}^3\left[\left(2^{\xi }-2^{\delta }\right)\left(3^{\delta }-2^{2\delta }\right)+2^{\delta }\left(3^{\xi }-3^{\delta }\right)\right]}{\left(2^{\xi }+2^{\delta }\right){\left(2^{\xi }-2^{\delta }\right)}^4}}+{\displaystyle \frac{10{\left(cos\theta -\alpha \right)}^2}{\left(2^{\xi }-2^{\delta }\right)\left(3^{\xi }-3^{\delta }\right)}}+{\displaystyle \frac{2\left(cos\theta -\alpha \right)}{4^{\xi }-4^{\delta }}},\hfill \end{array}$$

also for any $\kappa \in \mathbb{C}$, we have

$$\begin{array}{c}\hfill \left|{\mathit{b}}_3-\kappa {\mathit{b}}_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{cos\theta -\alpha }{3^{\xi }-3^{\delta }}},0\le \left|T(\kappa ,\xi ,\delta )\right|\le {\displaystyle \frac{1}{2\left(3^{\xi }-3^{\delta }\right)}},\\ 2\left|T(\kappa ,\xi ,\delta )\right|\left[cos\theta -\alpha \right],\left|T(\kappa ,\xi ,\delta )\right|\ge {\displaystyle \frac{1}{2(3^{\xi }-3^{\delta })}},\end{array}\right.\end{array}$$

where

$$T\left(\kappa ,\xi ,\delta \right)={\displaystyle \frac{1-\kappa }{2\left[\left(3^{\xi }-3^{\delta }\right)-\left(2^{\xi +\delta }-2^{2\delta }\right)\right]}}.$$

Theorem 4.

Let $f\left(\nu \right)$ presented by (1) belong to class $f\in {\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right)$ and consider $X=cos\theta -\alpha$ for $\xi >\delta +1,l\ne 0,\left|\theta \right|<\pi$ and $cos\theta >\alpha .$ Then, the second-order Hankel determinant is given by

$${\mathcal{H}}_{2,2}\left(f\right)=\left|{\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{4X^2}{c_3}}\left\{{\displaystyle \frac{4c_6{X}^2}{c_1^4}}+{\displaystyle \frac{1}{c_1}}\right\},X\in \left[0,{\varphi }_{\left(\xi ,\delta \right)}\right],\\ {\displaystyle \frac{X^2}{16c_3}}\left\{{\displaystyle \frac{64c_3}{c_2^2}}-{\displaystyle \frac{{4\left(\mathrm{\Pi }+\frac{2c_{4}X}{c_1{c}_2}\right)}^2}{(\psi +\mathrm{\Sigma })}}\right\},X\in \left[{\varphi }_{\left(\xi ,\delta \right)},1\right],\end{array}\right.$$

where

$$\begin{array}{cc}\hfill {\varphi }_{\left(\xi ,\delta \right)}=& {\displaystyle \frac{c_3{c}_1^2}{8c_2{c}_6}}\left(1+\sqrt{1-{\displaystyle \frac{16c_6(c_3{c}_1-8c_2^2)}{c_1{c}_3^2}}}\right),\mathrm{\Pi }={\displaystyle \frac{6c_2^2-4c_1{c}_3}{c_1{c}_2^2}},\hfill \\ \hfill \mathrm{\Sigma }=& {\displaystyle \frac{X{c}_3}{c_1^2{c}_2}}+{\displaystyle \frac{c_3{c}_1-2c_2^2}{c_1{c}_2^2}},\psi ={\displaystyle \frac{4c_6{X}^2}{c_1^4}},c_1={(1+q)}^{\xi }-{(1+q)}^{\delta },\hfill \\ \hfill c_2=& {(1+q+q^2)}^{\xi }-{(1+q+q^2)}^{\delta },\hfill \\ \hfill c_3=& {(1+q+q^2+q^3)}^{\xi }-{(1+q+q^2+q^3)}^{\delta },\hfill \\ \hfill c_4=& {(1+q)}^{\xi }+{(1+q)}^{\delta },\hfill \\ \hfill c_6=& ({(1+q+q^2)}^{\delta }-{(1+q)}^{2\delta })({(1+q)}^{\xi }-{(1+q)}^{\delta }){(1+q)}^{\delta }({(1+q+q^2)}^{\xi }-\hfill \\ \hfill & {(1+q+q^2)}^{\delta })-({(1+q+q^2+q^3)}^{\xi }-{(1+q+q^2+q^3)}^{\delta }).\hfill \end{array}$$

Proof. 

From (79)–(81) and considering that $X=cos\theta -\alpha$, we obtain

$${\mathit{b}}_2={\displaystyle \frac{r_{1}X}{e^{i\theta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}},{\mathit{b}}_3={\displaystyle \frac{r_1^2{X}^2}{e^{2i\theta }{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2}}+{\displaystyle \frac{(r_2-s_2)X}{2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{i\theta }}},$$

and

$$\begin{array}{cc}\hfill {\mathit{b}}_4=& {\displaystyle \frac{r_1^3{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)\right]X}^3}{{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)e}^{3i\theta }{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^3}}+\hfill \\ \hfill & {\displaystyle \frac{5r_1(r_2-s_2)X^2}{4\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{2i\theta }\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}+{\displaystyle \frac{\left(r_3-s_3\right)X}{2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)e^{i\theta }}}.\hfill \end{array}$$

Substituting these values, we obtain the functional ${\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2$ as

$$\begin{array}{cc}\hfill {\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2=& {\displaystyle \frac{r_1^4{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^4}{{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)e}^{4i\theta }{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}+\hfill \\ \hfill & {\displaystyle \frac{r_1^2(r_2-s_2)X^3}{{4\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{3i\theta }}}+{\displaystyle \frac{r_1\left(r_3-s_3\right)X^2}{2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right){\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)e}^{2i\theta }}}-\hfill \\ \hfill & {\displaystyle \frac{X^2{(r_2-s_2)}^2}{4{{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^{2}e}^{2i\theta }}}.\hfill \end{array}$$

(86)

According to Lemma 2, we have

$$2r_2=r_1^2+x\left(4-r_1^2\right),$$

(87)

and

$$2s_2=s_1^2+y\left(4-s_1^2\right).$$

(88)

Adding (87) and (88) and subtracting (87) from (88) respectively, we have

$$r_2+s_2=r_1^2+{\displaystyle \frac{4-r_1^2}{2}}\left(x+y\right).$$

(89)

and

$$r_2-s_2={\displaystyle \frac{4-r_1^2}{2}}\left(x-y\right),$$

(90)

also

$$\begin{array}{cc}\hfill & 4r_3=r_1^3+2\left(4-r_1^2\right)r_{1}x-r_1\left(4-r_1^2\right)x^2+2\left(4-r_1^2\right)\left(1-{\left|x\right|}^2\right)\nu ,\hfill \\ \hfill & 4s_3=s_1^3+2\left(4-s_1^2\right)s_{1}y-s_1\left(4-s_1^2\right)y^2+2\left(4-s_1^2\right)\left(1-{\left|y\right|}^2\right)\mu .\hfill \end{array}$$

(91)

Similarly,

$$\begin{array}{cc}\hfill r_3-s_3=& {\displaystyle \frac{r_1^3}{2}}+{\displaystyle \frac{\left(4-r_1^2\right)r_1}{2}}\left(x+y\right)-{\displaystyle \frac{r_1\left(4-r_1^2\right)}{4}}{(x}^2+y^2)+\hfill \\ \hfill & {\displaystyle \frac{\left(4-r_1^2\right)}{2}}(\left(1-{\left|x\right|}^2\right)\nu -\left(1-{\left|y\right|}^2\right)\mu ).\hfill \end{array}$$

(92)

We have values $x,y$ and $\nu ,\mu$ satisfying $\left|x\right|\le 1,\left|y\right|\le 1,\left|\nu \right|\le 1,\left|\mu \right|$≤ 1 and $\left|e^{i\theta }\right|=1.$ Using (90), (92), and (89) together with the modulus of (86), we obtain

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2{\mathit{b}}_3-{\mathit{b}}_3^2\right|=& \left|{\displaystyle \frac{r_1^4{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^4}{{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)e}^{4i\theta }{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}\right.\hfill \\ \hfill & +{\displaystyle \frac{r_1^2{\left(4-r_1^2\right)X}^3}{{8\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)e^{3i\theta }}}\left(x-y\right)+\hfill \\ \hfill & {\displaystyle \frac{r_1{X}^2}{2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right){\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)e}^{2i\theta }}}\left({\displaystyle \frac{r_1^3}{2}}+{\displaystyle \frac{\left(4-r_1^2\right)r_1}{2}}\left(x+y\right)-\right.\hfill \\ \hfill & \left.{\displaystyle \frac{r_1\left(4-r_1^2\right)}{4}}{(x}^2+y^2)+{\displaystyle \frac{\left(4-r_1^2\right)}{2}}(\left(1-{\left|x\right|}^2\right)\nu -\left(1-{\left|y\right|}^2\right)\mu )\right)-\hfill \\ \hfill & \left.{\displaystyle \frac{{\left(4-r_1^2\right)}^2{X}^2}{16{{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^{2}e}^{2i\theta }}}{\left(x-y\right)}^2\right|,\hfill \end{array}$$

Now, by the triangular inequality, we get

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2{\mathit{b}}_3-{\mathit{b}}_3^2\right|\le & {\displaystyle \frac{r_1^4{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^4}{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right){\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}+\hfill \\ \hfill & {\displaystyle \frac{r_1^4{X}^2}{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}+{\displaystyle \frac{r_1{\left(4-r_1^2\right)X}^2}{2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}+\hfill \\ \hfill & \left[{\displaystyle \frac{r_1^2{\left(4-r_1^2\right)X}^3}{{8\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{r_1^2{\left(4-r_1^2\right)X}^2}{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}\right]\left(\left|x\right|+\left|y\right|\right)+\hfill \\ \hfill & \left[{\displaystyle \frac{r_1^2{\left(4-r_1^2\right)X}^2}{8\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}-{\displaystyle \frac{r_1{\left(4-r_1^2\right)X}^2}{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}\right]\left({\left|x\right|}^2+{\left|y\right|}^2\right)+\hfill \\ \hfill & {\displaystyle \frac{{\left(4-r_1^2\right)}^2{X}^2}{16{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}{\left(\left|x\right|+\left|y\right|\right)}^2.\hfill \end{array}$$

The function $p\left(e^{i\theta }\nu \right)$, for $\theta \in R$ is in the class P,; therefore, for generality, it can be considered that $r_1=r\in \left[0,2\right]$. Therefore, for $\eta =\left|x\right|\le 1$ and $\beta =\left|y\right|\le 1$, we obtain

$$\left|{\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2\right|\le {\mathcal{Q}}_1+{\mathcal{Q}}_2\left(\eta +\beta \right)+{\mathcal{Q}}_3\left({\eta }^2+{\beta }^2\right)+{\mathcal{Q}}_4{\left(\eta +\beta \right)}^2=F\left(\eta ,\beta \right),$$

where

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_1={\mathcal{Q}}_1\left(r\right)={\displaystyle \frac{X^2}{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left[\left({\displaystyle \frac{1}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}+\right.\right.\hfill \\ \hfill & \left.\left.{\displaystyle \frac{4{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}\right)r^4+{\displaystyle \frac{2r\left(4-r^2\right)}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}\right]\ge 0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{Q}}_2=& {\mathcal{Q}}_2\left(r\right)={\displaystyle \frac{X^2}{24\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}\left[{\displaystyle \frac{r^2\left(4-r^2\right)\left\{3\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)X+6\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\right\}}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\right]\ge 0,\hfill \\ \hfill {\mathcal{Q}}_3=& {\mathcal{Q}}_3\left(r\right)={\displaystyle \frac{X^2}{8\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}\left\{r\left(4-r^2\right)\left(r-2\right)\right\}\le 0,\hfill \\ \hfill {\mathcal{Q}}_4=& {\mathcal{Q}}_4\left(r\right)={\displaystyle \frac{X^2}{16{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}{\left(4-r^2\right)}^2\ge 0.\hfill \end{array}$$

Next, we have to maximize function $F(\eta ,\beta )$ in the closed square

$$\mathbb{S}=\left\{\left(\eta ,\beta \right):0\le \eta \le 1,0\le \beta \le 1\right\},\mathrm{for}r\in \left[0,2\right].$$

Applying the partial derivative of the function $F\left(\eta ,\beta \right)$ leads to

$${\displaystyle \frac{\partial F}{\partial \eta }}={\mathcal{Q}}_2+2{\mathcal{Q}}_3\eta +2{\mathcal{Q}}_4\left(\eta +\beta \right)=0,$$

(93)

and

$${\displaystyle \frac{\partial F}{\partial \beta }}={\mathcal{Q}}_2+2{\mathcal{Q}}_3\beta +2{\mathcal{Q}}_4\left(\eta +\beta \right)=0.$$

(94)

By comparing (93) and (94), we obtain

$$\eta =\beta ,$$

and

$$\eta ={\displaystyle \frac{-{\mathcal{Q}}_2}{2({\mathcal{Q}}_3+2{\mathcal{Q}}_4)}},$$

we need to check the maximum of the function $F(\eta ,\beta )$ on the boundary because the maximum does not exist at the critical points.

Now, for $\eta =0$ and $0\le \eta \le 1$ (similar to $\beta =0$ and $0\le \beta \le 1)$ we obtain

$$F\left(0,\beta \right)={\mathcal{Q}}_1+{\mathcal{Q}}_2\left(\beta \right)+\left({\mathcal{Q}}_3+{\mathcal{Q}}_4\right)\left({\beta }^2\right)=G\left(\beta \right),$$

The interior point of $0\le r\le 2$ for $0\le \beta \le 1$ is gained when ${\mathcal{Q}}_3+{\mathcal{Q}}_4$$\ge 0$. The function ‘F’ possesses a positive slope for the conditions under which the function $G^{\prime }\left(\beta \right)>0$ for $\beta >0$ holds. Therefore, the sharp estimate for functional $\left|{\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2\right|$ reflects $\beta =1$ and $r=0,$ which can be analyzed into $G^{\prime }\left(\beta \right)=2\left({\mathcal{Q}}_3+{\mathcal{Q}}_4\right)\beta +{\mathcal{Q}}_2\ge 0$.

As the maximum of $G\left(\beta \right)$ is located at $\beta =1$ and

$$\max \left\{G\left(\beta \right)\right\}=G\left(1\right)={\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3+{\mathcal{Q}}_4.$$

when ${\mathcal{Q}}_3+{\mathcal{Q}}_4<0,$ it can be recognized that $2\left({\mathcal{Q}}_3+{\mathcal{Q}}_4\right)\beta +{\mathcal{Q}}_2\ge 0$ for $0\le \beta \le 1$ and any fixed r with $0\le r\le 2$. It is clear that

$2\left({\mathcal{Q}}_3+{\mathcal{Q}}_4\right)+{\mathcal{Q}}_2$ < $2\left({\mathcal{Q}}_3+{\mathcal{Q}}_4\right)\beta +{\mathcal{Q}}_2$ < ${\mathcal{Q}}_2$ and thus, $G^{\prime }\left(\beta \right)>0$.

According to $r=2$, we obtain

$$\begin{array}{cc}\hfill F\left(\eta ,\beta \right)=& {\displaystyle \frac{4X^2}{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left\{{\displaystyle \frac{4X^2\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}+\right.\hfill \\ \hfill & \left.{\displaystyle \frac{1}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}\right\}.\hfill \end{array}$$

Now, examining for $\eta =1$ and $0\le \eta \le 1$ (similar to $\beta =1$ and $0\le \beta \le 1$), we obtain

$$\begin{array}{c}\hfill F\left(1,\beta \right)=H\left(\beta \right)={\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3+{\mathcal{Q}}_4+\left({\mathcal{Q}}_2+2{\mathcal{Q}}_4\right)\beta +\left({\mathcal{Q}}_3+{\mathcal{Q}}_4\right){\beta }^2,\end{array}$$

Similarly, for the above cases of ${\mathcal{Q}}_3+{\mathcal{Q}}_4$ where $\beta =1$, we obtain

$$\max \left\{H\left(\beta \right)\right\}=H\left(1\right)={\mathcal{Q}}_1+2{\mathcal{Q}}_2+2{\mathcal{Q}}_3+4{\mathcal{Q}}_4.$$

As $G\left(1\right)\le H\left(1\right),$ we acquired the interior point of $r\in \left[0,2\right]$, where the maximum of F is present at $\eta =1$ and $\beta =1.$ Thus,

$$F\left(\eta ,\beta \right)=F\left(1,1\right)={\mathcal{Q}}_1+2{\mathcal{Q}}_2+2{\mathcal{Q}}_3+4{\mathcal{Q}}_4=I\left(r\right).$$

Substituting the value of ${\mathcal{Q}}_1+{\mathcal{Q}}_2+{\mathcal{Q}}_3+{\mathcal{Q}}_4$ in the function I gives

$$\begin{array}{cc}\hfill I\left(r\right)=& {\displaystyle \frac{X^2}{16\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left\{r^4\left[{\displaystyle \frac{{\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]16X}^4}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}-\right.\right.\hfill \\ \hfill & \left.{\displaystyle \frac{4X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{4\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)-8{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right]+\hfill \\ \hfill & r^2\left[{\displaystyle \frac{16X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{48{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2-32\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right]+\hfill \\ \hfill & \left.{\displaystyle \frac{64\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right\}.\hfill \end{array}$$

To locate the maximum of $I\left(r\right)$ on the interval $r\in [0,2]$, we differentiate it with respect to r, which yields

$$\begin{array}{cc}\hfill I^{\prime }\left(r\right)=& {\displaystyle \frac{X^2}{16\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left\{4r^3\left[{\displaystyle \frac{{16\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}\right.\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{4X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)-8{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right]+\hfill \\ \hfill & \left.2r\left[{\displaystyle \frac{16\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)X}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{48{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2-32\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right]\right\}.\hfill \end{array}$$

Assume that $I\left(r\right)$ attains its maximum at an interior point of $r[0,2]$. Then, necessarily, $I^{\prime }\left(r\right)=0$, which is satisfied only when

$$\begin{array}{cc}\hfill & \left[{\displaystyle \frac{{16\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}-\right.\hfill \\ \hfill & \left.{\displaystyle \frac{4X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)-8{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right]\ge 0,\hfill \end{array}$$

which leads to the restriction

$$X\in \left[0,{\varphi }_{\left(\xi ,\delta \right)}\right].$$

where

$${\varphi }_{(\xi ,\delta )}={\displaystyle \frac{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right){\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2}{8\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)\mathcal{R}}}\left(1+\sqrt{1-{\displaystyle \frac{16\left(\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)-8{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2\right)\mathcal{R}}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}^2}}}\right).$$

where

$$\mathcal{R}={(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)$$

In this case, $I^{\prime }\left(r\right)>0\mathrm{for}\mathrm{all}r\in \left[0,2\right],$ and therefore $I\left(r\right)$ increases in $[0,2]$. Consequently,

$$\max I\left(r\right)=I\left(2\right),$$

which gives

$$\begin{array}{cc}\hfill \left|{\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2\right|\le & {\displaystyle \frac{4X^2}{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left\{{\displaystyle \frac{{4\mathcal{R}X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}+{\displaystyle \frac{1}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}}\right\}.\hfill \end{array}$$

Also, by taking

$$\begin{array}{cc}\hfill & \left[{\displaystyle \frac{{16\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}-\right.\hfill \\ \hfill & \left.{\displaystyle \frac{4X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{4\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)-8{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}\right]\ge 0.\hfill \end{array}$$

We get

$$X\in \left[{\varphi }_{\left(\xi ,\delta \right)},1\right].$$

For a value $r_0<2$, that is $r_0\in [0,2]$, suppose that $I^{\prime }\left(r_0\right)\le 0.$ In this case, the function attains its maximum at the interior point $r=r_0$. Since the interior extremum satisfies $I^{\prime }\left(r\right)=0$, then we obtain

$$r_0=\sqrt{{\displaystyle \frac{-\left[\frac{2\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)X}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}+\frac{6{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2-4\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}\right]}{\left[\begin{array}{c}\frac{{4\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}-\\ \frac{X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}+\frac{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)-2{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}\end{array}\right]}}},$$

Consequently, $\max I\left(r\right)=I\left(r_0\right),$ which gives

$$\begin{array}{c}\hfill \max \left\{I\left(r_0\right)\right\}={\displaystyle \frac{X^2}{16\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left\{{\displaystyle \frac{64\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}-{\displaystyle \frac{{4\left(\frac{2\left({\left[2\right]}_q^{\xi }+{\left[2\right]}_q^{\delta }\right)X}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}+\mathrm{\Pi }\right)}^2}{\left(\psi +\mathrm{\Sigma }\right)}}\right\}.\end{array}$$

where

$$\begin{array}{cc}\hfill \mathrm{\Pi }=& {\displaystyle \frac{6{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2-4\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}},\hfill \\ \hfill \mathrm{\Sigma }=& {\displaystyle \frac{X\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^2\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}}+{\displaystyle \frac{\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)-2{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right){\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \psi =& {\displaystyle \frac{{4\left[{(\left[3\right]}_q^{\delta }-{\left[2\right]}_q^{2\delta })\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)+{\left[2\right]}_q^{\delta }\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)-\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)\right]X}^2}{{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)}^4}}.\hfill \end{array}$$

Therefore, the following estimate holds

$$\begin{array}{c}\hfill \left|{\mathit{b}}_2{\mathit{b}}_4-{\mathit{b}}_3^2\right|\le {\displaystyle \frac{X^2}{16\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}}\left\{{\displaystyle \frac{64\left({\left[4\right]}_q^{\xi }-{\left[4\right]}_q^{\delta }\right)}{{\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}^2}}-{\displaystyle \frac{{4\left(\frac{2\left({\left[2\right]}_q^{\xi }+{\left[2\right]}_q^{\delta }\right)X}{\left({\left[2\right]}_q^{\xi }-{\left[2\right]}_q^{\delta }\right)\left({\left[3\right]}_q^{\xi }-{\left[3\right]}_q^{\delta }\right)}+\mathrm{\Pi }\right)}^2}{\left(\psi +\mathrm{\Sigma }\right)}}\right\}.\end{array}$$

□

Remark 5.

For $q\to 1^{-}$, using this value in the given result provides an advanced result that matches the previous findings of Orhan et al. [42].

4. Conclusions

In this article, we introduced and studied several new subclasses of bi-univalent functions constructed through a generalized operator framework. Utilizing the operator ${\mathbb{B}}_{y,c}^{\partial ,\alpha }$ in conjunction with the concept of quasi-subordination, we derived estimates for the initial Taylor–Maclaurin coefficients of the subclasses ${\mathcal{R}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\rho ,\eta ,\zeta ,q\right)$ and ${\mathcal{K}}_{\mathrm{\Psi },\alpha ,c}^{\partial ,y}\left(\gamma ,\sigma ,\rho ,\zeta ,q\right)$. Furthermore, through the application of the q-generalized Salagean operator, we established upper bounds for the initial coefficients and the second Hankel determinant associated with the subclass ${\mathfrak{S}}_{\mathrm{\Psi },q}^{*}\left(\alpha ,\theta ,\xi ,\delta \right).$ The obtained results generalize, unify, and extend several previously known findings in geometric function theory, thereby contributing to the ongoing development of operator-based approaches in the study of analytic and bi-univalent functions.

The determination of the sharpness of the derived bounds, estimations of higher-order Hankel determinants, the characterization of extremal functions, and the geometric interpretation of the introduced subclasses remain open problems. These directions offer promising avenues for future research.