A Study on q-Starlike Functions Connected with q-Extension of Hyperbolic Secant and Janowski Functions

Abstract

This study introduces a novel subclass of q-starlike functions that is defined by the application of the q-difference operator and q-analogue of hyperbolic secant function. By making certain variations to the parameter “q”, the geometric interpretation of the domain hyperbolic secant function has also been discussed. The primary objective is to investigate and establish key results on the differential subordination of various orders within this newly defined class. Furthermore, convolution properties are explored and coefficient bounds are derived for these functions. A deeper analysis of these coefficients bounds unveils intriguing geometric insights and significant mathematical problems.

1. Introduction

Let $\mathcal{H}\left(\mathcal{D}\right)$ denote the class of all analytic functions $\chi \left(\varsigma \right)$ defined in the open unit disk $\mathcal{D}=\{\varsigma :\varsigma \in \mathbb{C}$ and $\left|\varsigma \right|<1\}$. The set $\mathcal{A}\subseteq$$\mathcal{H}\left(\mathcal{D}\right)$ contains all analytic functions $\chi \left(\varsigma \right)$ with the Taylor series representation:

$$\chi \left(\varsigma \right)=\varsigma +\sum _{t=2}^{\infty }{a}_t{\varsigma }^t,$$

(1)

and fulfills the condition of normalization $\chi \left(0\right)=0$ and ${\chi }^{\prime }\left(0\right)=1$. The set $\mathcal{S}$ contains all the functions that are univalent $\left(\mathrm{one}-\mathrm{one}\right)$ in the open unit disk $\mathcal{D}$. By ${\mathcal{S}}^{*}$ and $\mathcal{C}$, we mean the subclasses of $\mathcal{S}$ that represent the class of starlike and convex functions, respectively.

Let $\chi ,$$\vartheta \in \mathcal{A}$, then $\chi$ is subordinated to $\vartheta$, written as $\left(\chi \prec \vartheta \right)$ if there exists a Schwarz function $\omega \left(\varsigma \right)$, such that $\omega \left(0\right)=0$ and $\left|\omega \left(\varsigma \right)\right|<\left|\varsigma \right|$, and we can write $\chi =\vartheta \left(\omega \left(\varsigma \right)\right)$.

For $\chi$, $\vartheta \in \mathcal{A}$, having the series representation $\chi \left(\varsigma \right)=\varsigma +{\displaystyle \sum _{t=2}^{\infty }}{a}_t{\varsigma }^t,$ and $\vartheta \left(\varsigma \right)=\varsigma +{\displaystyle \sum _{t=2}^{\infty }}{b}_t{\varsigma }^t$ the convolution or $\left(\mathrm{Hadamard}\mathrm{product}\right)$ of $\chi ,$$\vartheta$ is defined as

$$\left(\chi \ast \vartheta \right)\left(\varsigma \right)=\varsigma +\sum _{t=2}^{\infty }{a}_t{b}_t{\varsigma }^t.$$

Let $\mathcal{P}$ denote the class of analytic functions p with the series representation given by:

$$p\left(\varsigma \right)=1+\sum _{t=1}^{\infty }{a}_t{\varsigma }^t,$$

and satisfying the condition that $Rep\left(\varsigma \right)>0$ in the open unit disk $\mathcal{D}$. For $-1\le B<A\le 1$, Janowski [1] introduced the class $\mathcal{P}\left(A,B\right)$, which contains all those functions $p\left(\varsigma \right)$ that satisfy

$$p\left(\varsigma \right)\prec {\displaystyle \frac{1+A\varsigma }{1+B\varsigma }}⟺w\left(\varsigma \right)={\displaystyle \frac{p\left(\varsigma \right)-1}{A-Bp\left(\varsigma \right)}}.$$

Equivalently, we can say $p\in \mathcal{P}\left(A,B\right)$, if and only if

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A-Bp\left(\varsigma \right)}}\right|<1,\varsigma \in \mathcal{D}.$$

Quantum or q-calculus is a branch of mathematics that generalizes classical calculus by introducing a parameter q into the differentiation and integration processes. It is also known as “quantum calculus” named after the mathematician Jackson, who extensively studied its properties in the early 20th century. Q-calculus has its roots in the study of complex functions, special functions, particularly in relation to q-series and q-orthogonal polynomials. Q-calculus has found applications in various fields of mathematics and physics, including number theory, quantum mechanics, and statistical mechanics. In geometric function theory, q-calculus emerges as a key tool. Srivastava [2] and Gasper et al. [3] initially introduced key q-terminologies and formulated a generalized hypergeometric series representation of important functions. Since then, numerous scholars have introduced and explored important subclasses of analytic functions through the application of q-calculus.

Currently, it is important to establish the fundamental definitions and concepts of $q-$calculus to understand the central theme of this article.

Definition 1

([4]). For $q\in \left(0,1\right)$, the q-difference operator of a function $\chi \left(\varsigma \right)$ is defined as:

$$D_q\chi \left(\varsigma \right)={\displaystyle \frac{\chi \left(q\varsigma \right)-\chi \left(\varsigma \right)}{\varsigma (q-1)}}(\varsigma \in \mathcal{D},\varsigma \ne 0).$$

(2)

When ${\displaystyle \underset{q⟶1^{-}}{lim}}{D}_q\chi \left(\varsigma \right)={\chi }^{\prime }\left(\varsigma \right)$ and

$$D_q\left\{\sum _{t=1}^{\infty }{a}_t{\varsigma }^t\right\}=\sum _{t=1}^{\infty }{\left[t\right]}_q{a}_t{\varsigma }^{t-1},$$

where ${\left[t\right]}_q$ is the q-number defined as:

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

In recent years, researchers in this field have shifted their focus towards q-calculus, making use of q-operators and introducing important domains, and have defined and investigated several new subclasses of analytic functions. Building on the groundwork laid by Ma and Minda [5], Seoudy and Aouf [6] expanded on their research by introducing a generalized formulation of q-starlike functions by utilizing the definition of the q-difference operator, as follows:

$${\mathcal{S}}_q^{*}\left(\varphi \right)=\left\{\chi \in \mathcal{A}:{\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}\prec \varphi \left(\varsigma \right),\varsigma \in \mathcal{D}\right\},$$

where $\varphi \left(\varsigma \right)$ is normalized by the conditions $\varphi \left(0\right)=1$, ${\varphi }^{\prime }\left(0\right)>0$ and $Re\varphi \left(\varsigma \right)>0$ in $\mathcal{D}$.

Due to the extensive range of applications and significant importance attributed to the q-calculus, various scholars have set out the distinct domains $\varphi \left(\varsigma \right)$ in terms of q-analogues, which brings fascinating classes and established important geometric findings. By using these creative methods and through analysis, researchers have discovered some basic geometric properties that are important for these new defined classes. Here, we mention a few of them:

• For $\varphi \left(\varsigma \right)=e_q\left(\varsigma \right)$, Srivastava et al. [7] introduced and investigated the class ${\mathcal{S}}_{e_q}^{*}$ for starlike functions.
• For $\varphi \left(\varsigma \right)=1+sin\left(q\varsigma \right)$, Taj et al. [8] defined and studied the class ${\mathcal{S}}_{sq}^{*}$ for starlike functions.
• For $\varphi \left(\varsigma \right)=cos\left(q\varsigma \right)$, Khan [9] defined and investigated ${\mathcal{S}}_{cos}^{*}$ for starlike functions.
• For $\varphi \left(\varsigma \right)=tanh\left(q\varsigma \right)$, Swarup [10] introduced a new subclass of q-starlike functions associated with the q-extension of the hyperbolic tangent function.

For more recent developments on q-subclasses of analytic functions related to q-generalization of certain domains and operators (see [11,12,13,14,15,16,17,18,19]).

Taking inspiration from the studies mentioned above, we introduce the class ${\mathcal{S}}_{\mathrm{sech}q}^{*}$ of q-starlike functions associated with the q-difference operator and q-analogue of the hyperbolic secant function.

Definition 2.

Let $0<q<1$ and $\chi \in \mathcal{A}$ be the same as given in (1). Then, $\chi \in {\mathcal{S}}_{\mathrm{sech}q}^{*},$ if the following subordination condition hold true

$${\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}\prec \mathrm{sech}\left(q\varsigma \right),\varsigma \in \mathcal{D}.$$

(3)

For different values of q, we can see the images of the open unit disc $\mathcal{D}$ under $\mathrm{sech}\left(q\varsigma \right)$, as shown in the figure below.

Figure 1 shows the transformation of the unit disc into convex shapes for certain values of q. These transformations represent an attractive pattern for $\mathrm{sech}\left(q\varsigma \right)$. Furthermore, for each $\chi \in \mathcal{A}$ the image of the unit disc under ${\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}$ lies within the domain of $\mathrm{sech}\left(q\varsigma \right)$.

Let $p\left(\varsigma \right)$ and $h\left(\varsigma \right)$ be two analytic functions in an open unit disc $\mathcal{D}$ and $\mathsf{{\rm Y}}:{\mathbb{C}}^2⟶\mathbb{C}$ be a complex-valued function, then the condition

$$\mathsf{{\rm Y}}\left(p\left(\varsigma \right),\varsigma p^{\prime }\left(\varsigma \right)\right)\prec h\left(\varsigma \right),$$

is known as first-order differential subordination. This concept generalizes differential inequalities in real variables and has been extensively studied in the field of analytic and univalent functions. Miller and Mocanu [20] laid the groundwork by delving into the general theory of differential subordination, inspiring subsequent research to explore various extensions and applications (see [21,22]).

In recent years, differential subordination problems have been investigated within the framework of q-calculus. Notably, Hadia et al. [23] introduced a new subclass of analytic functions by setting the q-analogue of the exponential function, and addressed some differential subordination and superordination problems related to the q-operator. Haq et al. [24] advanced the frontier by proving q-analogues of differential subordination results pertinent to the cardioid and limaçon domains. Furthermore, Raza et al. [25] defined the q-analogue of differential subordination linked to the lemniscate of Bernoulli, while Zainab et al. [26] contributed by establishing the q-analogue of differential subordination results associated with the cardioid domain. Khan et al. [27] also made notable strides by demonstrating significant subordination results for subfamilies of analytic functions within symmetric image domains. Andrei et al. [28] further enriched the field by establishing differential subordination results for q-convex functions, connected with the Ruscheweyh differential operator. Ali et al. [29] explored the differential subordination results entwined with the lemniscate of Bernoulli, while Cho et al. [30] tackled both differential and radius problems associated with the Booth lemniscate.

This research focuses primarily on establishing key geometric properties for functions belonging to the newly defined class ${\mathcal{S}}_{\mathrm{sech}q}^{*},$ by using differential subordination and convolution techniques, specifically for q-analogues of Janowski-type functions and the hyperbolic secant function.

To illustrate our main findings, we need the following q-analogue of Jack’s Lemma.

Lemma 1

([31]). Let w be an analytic function in $\mathcal{D}$ with $w\left(0\right)=0$ and if w attains its maximum value on the circle $\left|\varsigma \right|=r$ at a point ${\varsigma }_0\in \mathcal{D}$, then for $0<q<1$, we have

$${\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)=mw\left({\varsigma }_0\right),m\ge 1.$$

2. Main Results

Theorem 1.

Assume that

$$\left|\gamma \right|\ge {\displaystyle \frac{A_q-B_q}{q\left(\mathrm{sech}\left(q\right)tanh\left(q\right)-\left|B_q\right|sec\left(q\right)tan\left(q\right)\right)}},0<q<1,$$

(4)

and

$$-1\le B_q<{\displaystyle \frac{tanh\left(q\right)\mathrm{sech}\left(q\right)}{tan\left(q\right)sec\left(q\right)}}\le A_q\le 1,$$

where

$$A_q={\displaystyle \frac{\left(A+1\right)+\left(A-1\right)q}{2}},B_q={\displaystyle \frac{\left(B+1\right)+\left(B-1\right)q}{2}}\mathrm{for}-1\le B<A\le 1.$$

(5)

Let h be an analytic function, such that $h\left(0\right)=1$, and it satisfies the condition

$$1+\gamma \varsigma D_{q}h\left(\varsigma \right)\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},$$

that is

$$1+\gamma \varsigma D_{q}h\left(\varsigma \right)={\displaystyle \frac{1+A_{q}w\left(\varsigma \right)}{1+B_{q}w\left(\varsigma \right)}},$$

(6)

where $w\left(\varsigma \right)$ is an analytic function with $w\left(0\right)=0$, then

$$h\left(\varsigma \right)\prec \mathrm{sech}\left(q\varsigma \right).$$

(7)

Proof. 

Let

$$h\left(\varsigma \right)=\mathrm{sech}\left(qw\left(\varsigma \right)\right),$$

(8)

where $w\left(\varsigma \right)$ is known as Schwarz function in $\mathcal{D}$. For the required results, we have to show that $\left|w\left(\varsigma \right)\right|<1$. From (6), consider

$$p\left(\varsigma \right)=1+\gamma \varsigma D_{q}h\left(\varsigma \right),$$

(9)

where p is an analytic function and $p\left(0\right)=1$, which can be written as

$$w\left(\varsigma \right)={\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}.$$

(10)

From (8) and (9), we have

$$p\left(\varsigma \right)=1-q\gamma \varsigma \left[\mathrm{sech}\left(qw\left(\varsigma \right)\right)tanh\left(qw\left(\varsigma \right)\right)D_{q}w\left(\varsigma \right)\right].$$

(11)

Using (11) in (10), we obtain

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}\right|=\left|{\displaystyle \frac{q\gamma \varsigma \left[\mathrm{sech}\left(qw\left(\varsigma \right)\right)tanh\left(qw\left(\varsigma \right)\right)D_{q}w\left(\varsigma \right)\right]}{A_q-B_q+q{B}_q\gamma \varsigma \left[\mathrm{sech}\left(qw\left(\varsigma \right)\right)tanh\left(qw\left(\varsigma \right)\right)D_{q}w\left(\varsigma \right)\right]}}\right|.$$

(12)

On the contrary, assume that there exists a point ${\varsigma }_0\in \mathcal{D}$, such that

$$\underset{\left|\varsigma \right|\le \left|{\varsigma }_0\right|}{max}\left|w\left(\varsigma \right)\right|=\left|w\left({\varsigma }_0\right)\right|=1,$$

then by Lemma 1, there exists $m\ge 1$, such that ${\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)=mw\left({\varsigma }_0\right)$. Suppose $w\left(z_0\right)=e^{i\theta }$ where $\theta \in \left[-\pi ,\pi \right]$, then for ${\varsigma }_0\in \mathcal{D}$, we have

$$\begin{array}{ccc}\hfill \left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|& =& \left|{\displaystyle \frac{q\gamma {\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)\left[\mathrm{sech}\left(qw\left({\varsigma }_0\right)\right)tanh\left(qw\left({\varsigma }_0\right)\right)\right]}{A_q-B_q+q{B}_q\gamma {\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)\left[\mathrm{sech}\left(qw\left({\varsigma }_0\right)\right)tanh\left(qw\left({\varsigma }_0\right)\right)\right]}}\right|\hfill \\ & =& \left|{\displaystyle \frac{q\gamma m{e}^{i\theta }\left[\mathrm{sech}\left(q{e}^{i\theta }\right)tanh\left(q{e}^{i\theta }\right)\right]}{A_q-B_q+q{B}_q\gamma m{e}^{i\theta }\left[\mathrm{sech}\left(q{e}^{i\theta }\right)tanh\left(q{e}^{i\theta }\right)\right]}}\right|\hfill \\ & \ge & {\displaystyle \frac{qm\left|\gamma \right|\left|\mathrm{sech}\left(q{e}^{i\theta }\right)\right|\left|tanh\left(q{e}^{i\theta }\right)\right|}{A_q-B_q+qm\left|\gamma \right|\left|B_q\right|\left|\mathrm{sech}\left(q{e}^{i\theta }\right)\right|\left|tanh\left(q{e}^{i\theta }\right)\right|}}.\hfill \end{array}$$

(13)

Computation gives

$$\begin{array}{ccc}\hfill {\left|\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}^2& =& {\left|{\displaystyle \frac{cos\left(qsin\theta \right)cosh\left(qcos\theta \right)}{{sinh}^2\left(qcos\theta \right)+{cos}^2\left(qsin\theta \right)}}-i{\displaystyle \frac{sin\left(qsin\theta \right)sinh\left(qcos\theta \right)}{{sinh}^2\left(qcos\theta \right)+{cos}^2\left(qsin\theta \right)}}\right|}^2.\hfill \\ & =& {\displaystyle \frac{1}{{cosh}^2\left(qcos\theta \right)+{cos}^2\left(qsin\theta \right)-1}}.\hfill \\ & =& {\chi }_1\left(\theta \right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\left|tanh\left(q{e}^{i\theta }\right)\right|}^2& =& {\left|{\displaystyle \frac{sinh\left(qcos\theta \right)cosh\left(qcos\theta \right)}{{sinh}^2\left(qcos\theta \right)+{cos}^2\left(qsin\theta \right)}}-i{\displaystyle \frac{sin\left(qsin\theta \right)cos\left(qsin\theta \right)}{{sinh}^2\left(qcos\theta \right)+{cos}^2\left(qsin\theta \right)}}\right|}^2.\hfill \\ & =& {\displaystyle \frac{{cosh}^2\left(qcos\theta \right)-{cos}^2\left(qsin\theta \right)}{{cosh}^2\left(qcos\theta \right)+{cos}^2\left(qsin\theta \right)-1}}.\hfill \\ & =& {\chi }_2\left(\theta \right).\hfill \end{array}$$

Since ${\chi }_i\left(\theta \right)={\chi }_i\left(-\theta \right)$ for $i=1,2,$ we consider $\theta \in \left[0,\pi \right]$. It can be easily seen that ${\chi }_i\left(\theta \right)$, $i=1,2,$ are increasing functions in the interval $\left[0,{\displaystyle \frac{\pi }{2}}\right]$ and decreasing functions in $\left[{\displaystyle \frac{\pi }{2}},\pi \right],$ so

$$\begin{array}{ccc}\hfill max\left\{{\chi }_1\left(\theta \right)\right\}& =& {\chi }_1\left({\displaystyle \frac{\pi }{2}}\right)={sec}^2\left(q\right),\hfill \\ \hfill min\left\{{\chi }_1\left(\theta \right)\right\}& =& {\chi }_1\left(0\right)={\chi }_1\left(\pi \right)={\mathrm{sech}}^2\left(q\right),\hfill \\ \hfill max\left\{{\chi }_2\left(\theta \right)\right\}& =& {\chi }_2\left({\displaystyle \frac{\pi }{2}}\right)={tan}^2\left(q\right),\hfill \\ \hfill min\left\{{\chi }_2\left(\theta \right)\right\}& =& {\chi }_2\left(0\right)={\chi }_2\left(\pi \right)={tanh}^2\left(q\right).\hfill \end{array}$$

Therefore

$$\mathrm{sech}\left(q\right)\le \left|\mathrm{sech}\left(q{e}^{i\theta }\right)\right|\le sec\left(q\right),$$

(14)

and

$$tanh\left(q\right)\le \left|tanh\left(q{e}^{i\theta }\right)\right|\le tan\left(q\right),$$

(15)

Using (14) and (15) in (13), we have

$$\left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|\ge {\displaystyle \frac{qm\left|\gamma \right|\mathrm{sech}\left(q\right)tanh\left(q\right)}{A_q-B_q+qm\left|\gamma \right|\left|B_q\right|sec\left(q\right)tan\left(q\right)}}=\psi \left(m\right).$$

Then

$${\psi }^{\prime }\left(m\right)={\displaystyle \frac{\left(A_q-B_q\right)q\left|\gamma \right|sech\left(q\right)tanh\left(q\right)}{{\left(A_q-B_q+qm\left|\gamma \right|\left|B_q\right|sec\left(q\right)tan\left(q\right)\right)}^2}}\ge 0,$$

this shows that $\psi \left(m\right)$ is an increasing function of $m\in \left[1,\infty \right).$ Thus, $\psi \left(m\right)\ge \psi \left(1\right)$ and using (4), we have

$$\left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|\ge {\displaystyle \frac{q\left|\gamma \right|sech\left(q\right)tanh\left(q\right)}{A_q-B_q+q\left|\gamma \right|\left|B_q\right|sec\left(q\right)tan\left(q\right)}}\ge 1.$$

The contradiction arises from the hypothesis of the Theorem that there is a point ${\varsigma }_0\in \mathcal{D},$ such that $\left|w\left({\varsigma }_0\right)\right|=1$. Consequently, it follows that $\left|w\left(\varsigma \right)\right|<1$ for all $\varsigma \in \mathcal{D}$, which gives the required result. □

Corollary 1.

Let $\chi \in \mathcal{A}$ and $p\left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}},$ satisfying the subordination

$$1+\gamma \varsigma D_q\left({\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}\right)\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},\varsigma \in \mathcal{D},$$

and

$$\left|\gamma \right|>{\displaystyle \frac{A_q-B_q}{q\left(\mathrm{sech}\left(q\right)tanh\left(q\right)-\left|B_q\right|sec\left(q\right)tan\left(q\right)\right)}}.$$

Then $\chi \in {\mathcal{S}}_{\mathrm{sech}q}^{*}$.

Theorem 2.

Assume that

$$\left|\gamma \right|>{\displaystyle \frac{A_q-B_q}{q\left(tanh\left(q\right)-\left|B_q\right|tan\left(q\right)\right)}},0<q<1,$$

(16)

and

$$-1\le B_q<{\displaystyle \frac{tanh\left(q\right)}{tan\left(q\right)}}\le A_q\le 1,$$

where $A_q$ and $B_q$ are defined in (5). Let h be an analytic function, such that $h\left(0\right)=1$, and suppose that

$$1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h\left(\varsigma \right)}}={\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},$$

(17)

such that

$$1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h\left(\varsigma \right)}}={\displaystyle \frac{1+A_q\left(w\left(\varsigma \right)\right)}{1+B_q\left(w\left(\varsigma \right)\right)}},$$

(18)

then

$$h\left(\varsigma \right)\prec \mathrm{sech}\left(q\varsigma \right).$$

(19)

Proof. 

Let

$$p\left(\varsigma \right)=1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h\left(\varsigma \right)}},$$

(20)

where $p\left(0\right)=1$. Now, consider

$$h\left(\varsigma \right)=\mathrm{sech}\left(qw\left(\varsigma \right)\right),$$

(21)

then

$$p\left(\varsigma \right)=1-\gamma q\varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right).$$

(22)

To prove our required result, we have to show that $\left|w\left(\varsigma \right)\right|<1.$ From (21) and (22), we have

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}\right|=\left|{\displaystyle \frac{q\gamma \varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right)}{A_q-B_q+q{B}_q\left(\gamma \varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right)\right)}}\right|.$$

Let ${\varsigma }_0\in \mathcal{D}$, such that ${\displaystyle \underset{\left|\varsigma \right|\le \left|{\varsigma }_0\right|}{max}}\left|w\left(\varsigma \right)\right|=\left|w\left({\varsigma }_0\right)\right|=1$, then in view of Lemma 1 there exist $m\ge 1$ such that ${\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)=mw\left({\varsigma }_0\right)$. Let $w\left({\varsigma }_0\right)=e^{i\theta }$, $\theta \in \left[-\pi ,\pi \right]$, then

$$\begin{array}{ccc}\hfill \left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|& =& \left|{\displaystyle \frac{q\gamma {\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}{A_q-B_q+q\gamma B_q{\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{q\gamma mw\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}{A_q-B_q+q\gamma B_{q}mw\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{q\gamma m{e}^{i\theta }tanh\left(q{e}^{i\theta }\right)}{A_q-B_q+q\gamma B_{q}m{e}^{i\theta }tanh\left(q{e}^{i\theta }\right)}}\right|\hfill \\ & \ge & {\displaystyle \frac{qm\left|\gamma \right|\left|tanh\left(q{e}^{i\theta }\right)\right|}{A_q-B_q+qm\left|\gamma \right|\left|B_q\right|\left|tanh\left(q{e}^{i\theta }\right)\right|}}.\hfill \end{array}$$

(23)

Using (15) in (23), we have

$$\left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|\ge {\displaystyle \frac{qm\left|\gamma \right|tanh\left(q\right)}{A_q-B_q+qm\left|\gamma \right|\left|B_q\right|tan\left(q\right)}}=\psi \left(m\right).$$

It can easily be seen that $\psi \left(m\right)$ is an increasing function of $m\in \left[1,\infty \right)$. Then, by using (16), we have

$$\left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|\ge {\displaystyle \frac{q\left|\gamma \right|tanh\left(q\right)}{A_q-B_q+q\left|\gamma \right|\left|B_q\right|tan\left(q\right)}}\ge 1.$$

This contradicts the assumption of the Theorem. Hence, we obtain the required result. □

Corollary 2.

Let $\chi \in \mathcal{A}$ and $p\left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}},$ satisfying

$$1+\gamma \varsigma \left({\displaystyle \frac{\chi \left(\varsigma \right)}{\varsigma D_q\chi \left(\varsigma \right)}}\right)D_q\left({\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}\right)\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},\varsigma \in \mathcal{D},$$

and

$$\left|\gamma \right|>{\displaystyle \frac{A_q-B_q}{q\left(tanh\left(q\right)-\left|B_q\right|tan\left(q\right)\right)}}.$$

Then $\chi \in {\mathcal{S}}_{sechq}^{*}$.

Theorem 3.

Assume that

$$\left|\gamma \right|>{\displaystyle \frac{\left(A_q-B_q\right)sec\left(q\right)}{q\left(tanh\left(q\right)-\left|B_q\right|tan\left(q\right)\right)}},0<q<1,$$

(24)

and

$$-1\le B_q<{\displaystyle \frac{tanh\left(q\right)}{tan\left(q\right)}}\le A_q\le 1,$$

where $A_q$ and $B_q$ are given in (5). Let h be an analytic function, such that $h\left(0\right)=1$, and suppose

$$1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h^2\left(\varsigma \right)}}\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},$$

(25)

such that

$$1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h^2\left(\varsigma \right)}}={\displaystyle \frac{1+A_q\left(w(\varsigma \right))}{1+B_q\left(w(\varsigma \right))}},$$

(26)

then

$$h\left(\varsigma \right)\prec \mathrm{sech}\left(q\varsigma \right).$$

(27)

Proof. 

Let

$$p\left(\varsigma \right)=1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h^2\left(\varsigma \right)}},$$

(28)

where p is analytic function and $p\left(0\right)=1$. Now, consider

$$h\left(\varsigma \right)=\mathrm{sech}\left(qw\left(\varsigma \right)\right),$$

(29)

From (28) and (29), we have

$$p\left(\varsigma \right)=1-{\displaystyle \frac{\gamma q\varsigma D_{q}w\left(\varsigma \right)tanhqw\left(\varsigma \right)}{\mathrm{sech}\left(qw\left(\varsigma \right)\right)}}.$$

(30)

To prove the required result, we have to show that $\left|w\left(\varsigma \right)\right|<1$. For this, let

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}\right|=\left|{\displaystyle \frac{\gamma q\varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right)}{\left(A_q-B_q\right)\mathrm{sech}\left(qw\left(\varsigma \right)\right)+q{B}_q\left(\gamma \varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right)\right)}}\right|.$$

(31)

On the contrary, suppose that for any ${\varsigma }_0\in \mathcal{D}$, such that ${\displaystyle \underset{\left|\varsigma \right|\le \left|{\varsigma }_0\right|}{max}}\left|w\left(\varsigma \right)\right|=\left|w\left({\varsigma }_0\right)\right|=1$, then in view of Lemma 1, there exists $m\ge 1$, such that ${\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)=mw\left({\varsigma }_0\right)$. Let $w\left({\varsigma }_0\right)=e^{i\theta }$, $\theta \in \left[-\pi ,\pi \right]$, then

$$\begin{array}{ccc}\hfill \left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|& =& \left|{\displaystyle \frac{\gamma q{\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}{\left(A_q-B_q\right)\mathrm{sech}\left(qw\left({\varsigma }_0\right)\right)+q\gamma B_q\left(\varsigma D_{q}w\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{\gamma qmw\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}{\left(A_q-B_q\right)\mathrm{sech}\left(qw\left({\varsigma }_0\right)\right)+q{B}_q\left(\gamma mw\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{\gamma qm{e}^{i\theta }tanh\left(q{e}^{i\theta }\right)}{\left(A_q-B_q\right)\mathrm{sech}\left(q{e}^{i\theta }\right)+q{B}_q\gamma m{e}^{i\theta }tanh\left(q{e}^{i\theta }\right)}}\right|\hfill \\ & \ge & {\displaystyle \frac{mq\left|\gamma \right|\left|tanh\left(q{e}^{i\theta }\right)\right|}{\left(A_q-B_q\right)\left|\mathrm{sech}\left(q{e}^{i\theta }\right)\right|+mq\left|\gamma \right|\left|B_q\right|\left|tanh\left(q{e}^{i\theta }\right)\right|}}\hfill \end{array}$$

(32)

Using (14) and (15) in (32), we have

$$\left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|\ge {\displaystyle \frac{mq\left|\gamma \right|tanh\left(q\right)}{\left(A_q-B_q\right)sec\left(q\right)+mq\left|\gamma \right|\left|B_q\right|tan\left(q\right)}}=\psi \left(m\right).$$

It can easily be seen that $\psi \left(m\right)$ is an increasing function of $m\in \left[1,\infty \right)$, also by (24), we conclude that

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}\right|\ge {\displaystyle \frac{q\left|\gamma \right|tanh\left(q\right)}{\left(A_q-B_q\right)sec\left(q\right)+q\left|\gamma \right|\left|B_q\right|tan\left(q\right)}}\ge 1.$$

This contradicts the assumption of the Theorem. Hence, the result is proved. □

Corollary 3.

Let $\chi \in \mathcal{A}$ and $p\left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}},$ satisfying the subordination

$$1+\gamma \varsigma {\left({\displaystyle \frac{\chi \left(\varsigma \right)}{\varsigma D_q\chi \left(\varsigma \right)}}\right)}^2{D}_q\left({\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}\right)\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},\varsigma \in \mathcal{D},$$

and

$$\left|\gamma \right|>{\displaystyle \frac{\left(A_q-B_q\right)sec\left(q\right)}{q\left(tanh\left(q\right)-\left|B_q\right|tan\left(q\right)\right)}}.$$

Then $\chi \in {\mathcal{S}}_{sechq}^{*}$.

Theorem 4.

Assume that

$$\left|\gamma \right|>{\displaystyle \frac{\left(A_q-B_q\right){\left(secq\right)}^{n-1}}{q\left(tanh\left(q\right)-\left|B_q\right|tan\left(q\right)\right)}},0<q<1,$$

(33)

and

$$-1\le B_q<{\displaystyle \frac{tanh\left(q\right)}{tan\left(q\right)}}\le A_q\le 1,$$

where $A_q$ and $B_q$ are given in (5). Let h be an analytic function, such that $h\left(0\right)=1$, and suppose

$$1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h^n\left(\varsigma \right)}}\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},$$

(34)

such that

$$1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h^n\left(\varsigma \right)}}={\displaystyle \frac{1+A_q\left(w(\varsigma \right))}{1+B_q\left(w(\varsigma \right))}},$$

(35)

then

$$h\left(\varsigma \right)\prec \mathrm{sech}\left(q\varsigma \right).$$

(36)

Proof. 

Let

$$p\left(\varsigma \right)=1+\gamma {\displaystyle \frac{\varsigma D_{q}h\left(\varsigma \right)}{h^n\left(\varsigma \right)}},$$

(37)

where p is an analytic function and $p\left(0\right)=1$. Now, consider

$$h\left(\varsigma \right)=\mathrm{sech}\left(qw\left(\varsigma \right)\right),$$

(38)

From (37) and (38), we have

$$p\left(\varsigma \right)=1-\gamma {\displaystyle \frac{q\varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right)}{{\left(\mathrm{sech}\left(qw\left(\varsigma \right)\right)\right)}^{n-1}}}.$$

(39)

To prove our required result, we have to show that $\left|w\left(\varsigma \right)\right|<1$. For this, let

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}\right|=\left|{\displaystyle \frac{\gamma q\varsigma D_q\left(w\left(\varsigma \right)\right)tanh\left(qw\left(\varsigma \right)\right)}{\left(A_q-B_q\right){\left(\mathrm{sech}\left(qw\left(\varsigma \right)\right)\right)}^{n-1}+q\gamma B_q\left(\varsigma D_{q}w\left(\varsigma \right)tanh\left(qw\left(\varsigma \right)\right)\right)}}\right|.$$

(40)

Let ${\varsigma }_0\in \mathcal{D}$, such that ${\displaystyle \underset{\left|\varsigma \right|\le \left|{\varsigma }_0\right|}{max}}\left|w\left(\varsigma \right)\right|=\left|w\left({\varsigma }_0\right)\right|=1$, then in view of Lemma 1, there exists $m\ge 1$, such that ${\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)=mw\left({\varsigma }_0\right)$. Let $w\left({\varsigma }_0\right)=e^{i\theta }$, $\theta \in \left[-\pi ,\pi \right]$ then

$$\begin{array}{ccc}\hfill \left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|& =& \left|{\displaystyle \frac{\gamma q{\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}{\left(A_q-B_q\right){\left(\mathrm{sech}\left(qw\left(\varsigma \right)\right)\right)}^{n-1}+q{B}_q\gamma {\varsigma }_0{D}_{q}w\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{\gamma qmw\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)}{\left(A_q-B_q\right){\left(\mathrm{sech}\left(qw\left(\varsigma \right)\right)\right)}^{n-1}+q{B}_q\left(\gamma mw\left({\varsigma }_0\right)tanh\left(qw\left({\varsigma }_0\right)\right)\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{q\gamma m{e}^{i\theta }tanh\left(q{e}^{i\theta }\right)}{\left(A_q-B_q\right){\left(\mathrm{sech}\left(qw\left(\varsigma \right)\right)\right)}^{n-1}+q{B}_q\gamma m{e}^{i\theta }tanh\left(q{e}^{i\theta }\right)}}\right|\hfill \\ & \ge & {\displaystyle \frac{qm\left|\gamma \right|\left|tanh\left(q{e}^{i\theta }\right)\right|}{\left(A_q-B_q\right){\left|\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}^{n-1}+qm\left|\gamma \right|\left|B_q\right|\left|tanh\left(q{e}^{i\theta }\right)\right|}}.\hfill \end{array}$$

(41)

Using (14) and (15) in (41), we have

$$\left|{\displaystyle \frac{p\left({\varsigma }_0\right)-1}{A_q-B_{q}p\left({\varsigma }_0\right)}}\right|\ge {\displaystyle \frac{qm\left|\gamma \right|tanh\left(q\right)}{\left(A_q-B_q\right){\left(sec\left(q\right)\right)}^{n-1}+qm\left|\gamma \right|\left|B_q\right|tan\left(q\right)}}=\psi \left(m\right).$$

It can easily be seen that $\psi \left(m\right)$ is an increasing function of $m\in \left[1,\infty \right)$, also by (33) we have

$$\left|{\displaystyle \frac{p\left(\varsigma \right)-1}{A_q-B_{q}p\left(\varsigma \right)}}\right|\ge {\displaystyle \frac{q\left|\gamma \right|tanh\left(q\right)}{\left(A_q-B_q\right){\left(sec\left(q\right)\right)}^{n-1}+q\left|\gamma \right|\left|B_q\right|tan\left(q\right)}}\ge 1.$$

This contradicts the assumption of the Theorem. Hence, the result is proved. □

Corollary 4.

Let $\chi \in \mathcal{A}$ and $p\left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}},$ satisfying the subordination

$$1+\gamma \varsigma {\left({\displaystyle \frac{\chi \left(\varsigma \right)}{\varsigma D_q\chi \left(\varsigma \right)}}\right)}^n{D}_q\left({\displaystyle \frac{\varsigma D_q\chi \left(\varsigma \right)}{\chi \left(\varsigma \right)}}\right)\prec {\displaystyle \frac{1+A_q\left(\varsigma \right)}{1+B_q\left(\varsigma \right)}},\varsigma \in \mathcal{D},$$

and

$$\left|\gamma \right|>{\displaystyle \frac{\left(A_q-B_q\right){\left(secq\right)}^{n-1}}{q\left(tanh\left(q\right)-\left|B_q\right|tan\left(q\right)\right)}}.$$

Then $\chi \in {\mathcal{S}}_{\mathrm{sech}q}^{*}$.

2.1. Convolution Results

Theorem 5.

Let $\mathcal{X}\in \mathcal{A}$, and of the form given in (1). Then, $\mathcal{X}\in {\mathcal{S}}_{\mathrm{sech}q}^{*}$ if and only if

$${\displaystyle \frac{1}{\varsigma }}\left(\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma -\mathcal{L}q{\varsigma }^2}{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)\ne 0,\varsigma \in \mathcal{D},$$

(42)

for $\mathcal{L}={\mathcal{L}}_{\theta }={\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}},$ $0\le \theta <2\pi$ as well as $\mathcal{L}=1.$

Proof. 

Let $\mathcal{X}\in {\mathcal{S}}_{\mathrm{sech}q}^{*}$ having the series representation given in (1). Then, $\mathcal{X}\left(\varsigma \right)$ is analytic in $\mathcal{D}$, and ${\displaystyle \frac{1}{\varsigma }}\mathcal{X}\left(\varsigma \right)\ne 0$ for all $\varsigma \in \mathcal{D}$, so the condition in (42) holds true for $\mathcal{L}=1$. On the other hand, by (3), we have

$${\displaystyle \frac{\varsigma D_q\mathcal{X}\left(\varsigma \right)}{\mathcal{X}\left(\varsigma \right)}}=\mathrm{sech}\left(qw\left(\varsigma \right)\right),$$

where $w\left(\varsigma \right)$ is known as Schwarz function. Let $w\left(\varsigma \right)=e^{i\theta }$, $0\le \theta <2\pi$, such that

$${\displaystyle \frac{\varsigma D_q\mathcal{X}\left(\varsigma \right)}{\mathcal{X}\left(\varsigma \right)}}\ne \mathrm{sech}\left(q{e}^{i\theta }\right),$$

(43)

From (43), we can write as

$${\displaystyle \frac{1}{\varsigma }}\left(\mathcal{X}\left(\varsigma \right)\mathrm{sech}\left(q{e}^{i\theta }\right)-\varsigma D_q\mathcal{X}\left(\varsigma \right)\right)\ne 0.$$

(44)

Using the convolution relations

$$\mathcal{X}\left(\varsigma \right)=\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{1-\varsigma }},\mathrm{and}\varsigma D_q\mathcal{X}\left(\varsigma \right)=\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{\left(1-\varsigma \right)\left(1-q\varsigma \right)}},$$

(45)

the expression in (44), becomes

$${\displaystyle \frac{1}{\varsigma }}\left(\left(\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{1-\varsigma }}\right)\mathrm{sech}\left(q{e}^{i\theta }\right)-\left(\mathcal{X}\left(\varsigma \right)*{\displaystyle \frac{\varsigma }{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)\right)\ne 0.$$

(46)

Some simple calculations result in

$${\displaystyle \frac{1}{\varsigma }}\left(\mathcal{X}\left(\varsigma \right)\ast \left({\displaystyle \frac{\left(\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right)\varsigma -\left(\mathrm{sech}\left(q{e}^{i\theta }\right)\right)q{\varsigma }^2}{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)\right)\ne 0,$$

this expression can be written as

$${\displaystyle \frac{\left(\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right)}{\varsigma }}\left(\mathcal{X}\left(\varsigma \right)\ast \left({\displaystyle \frac{\varsigma -\left(\frac{\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}\right)q{\varsigma }^2}{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)\right)\ne 0,$$

which leads to the required result.

Conversely, let the condition given in (42) hold. Let $\xi \left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\mathcal{X}\left(\varsigma \right)}{\mathcal{X}\left(\varsigma \right)}},$ which is analytic in $\mathcal{D}$ and $\xi \left(0\right)=1$. Further, suppose that $\zeta \left(\varsigma \right)=\mathrm{sech}\left(q{e}^{i\theta }\right)$, $\varsigma \in \mathcal{D},$ since (42) and (43) are identical, then by (43), it is clear that $\zeta (\partial \mathcal{D})\cap \xi \left(\varsigma \right)=\varphi .$ Therefore, the connected component $\mathbb{C}\setminus \zeta (\partial \mathcal{D})$contains the domain $\xi \left(\varsigma \right)$, which is also connected. Now, by the univalence of “$\zeta$” together with the assumption $\xi \left(0\right)=\zeta \left(0\right)=1,$ this shows that $\xi \prec \zeta$, which means $\mathcal{X}\in {\mathcal{S}}_{\mathrm{sech}q}^{\ast }.$ □

Corollary 5.

Let $\mathcal{X}\in \mathcal{A}$. Then, the necessary and sufficient condition for a function $\mathcal{X}\in {\mathcal{S}}_{\mathrm{sech}q}^{\ast }$ is

$$1-\sum _{t=2}^{\infty }\left({\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right)a_t{\varsigma }^{t-1}\ne 0,\varsigma \in \mathcal{D}.$$

(47)

Proof. 

Let $\mathcal{X}\in {\mathcal{S}}_{\mathrm{sech}q}^{\ast }$, then by Theorem $5,$ we have

$${\displaystyle \frac{1}{\varsigma }}\left(\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma -\mathcal{L}q{\varsigma }^2}{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)\ne 0,$$

which can be written as

$${\displaystyle \frac{1}{\varsigma }}\left(\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}-\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\mathcal{L}q{\varsigma }^2}{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)\ne 0,$$

simplification gives

$${\displaystyle \frac{1}{\varsigma }}\left[\left(\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}\right)-\mathcal{L}\left(\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{\left(1-\varsigma \right)\left(1-q\varsigma \right)}}-\mathcal{X}\left(\varsigma \right)\ast {\displaystyle \frac{\varsigma }{\left(1-\varsigma \right)}}\right)\right]\ne 0.$$

(48)

Using (45) in (48), we have

$${\displaystyle \frac{1}{\varsigma }}\left(\varsigma D_q\mathcal{X}\left(\varsigma \right)-\mathcal{L}\left[\varsigma D_q\mathcal{X}\left(\varsigma \right)-\mathcal{X}\left(\varsigma \right)\right]\right)\ne 0,$$

(49)

since

$$\mathcal{X}\left(\varsigma \right)=\varsigma +\sum _{t=2}^{\infty }{a}_t{\varsigma }^t\mathrm{and}{D}_q\mathcal{X}\left(\varsigma \right)=1+\sum _{t=2}^{\infty }{\left[t\right]}_q{a}_t{\varsigma }^{t-1},$$

then, putting these values in (49) and simplifying it, we obtain

$$1-\sum _{t=2}^{\infty }\left({\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right)a_t{\varsigma }^{t-1}\ne 0.$$

□

Remark 1.

From Corollary 5, it is clear that for $\mathcal{X}\in {\mathcal{S}}_{\mathrm{sech}q}^{\ast }$, either

$$\sum _{t=2}^{\infty }\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\left|a_t\right|<1,$$

(50)

or

$$\sum _{t=2}^{\infty }\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\left|a_t\right|>1,$$

(51)

For the next results, let us denote a subclass of ${\mathcal{S}}_{\mathrm{sech}q}^{\ast }$ satisfying (50) by $\mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$.

2.2. Growth and Distortion Results

Theorem 6.

Let $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$. Then, for $\left|\varsigma \right|=r<1,$ we have

$$r-{\displaystyle \frac{\left|\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right|}{\left|{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}}{r}^2\le \left|\mathcal{X}\left(\varsigma \right)\right|\le r+{\displaystyle \frac{\left|\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right|}{\left|{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}}{r}^2.$$

(52)

Proof. 

Let $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$ be given in (1). Then,

$$\left|\mathcal{X}\left(\varsigma \right)\right|=\left|\varsigma +\sum _{t=2}^{\infty }{a}_t{\varsigma }^t\right|\le r+\sum _{t=2}^{\infty }\left|a_t\right|r^t.$$

(53)

Since $r^t\le r^2$ for all $t\ge 2$, then we can write (53) as:

$$\left|\mathcal{X}\left(\varsigma \right)\right|\le r+r^2\sum _{t=2}^{\infty }\left|a_t\right|.$$

(54)

Similarly

$$\left|\mathcal{X}\left(\varsigma \right)\right|\ge r-r^2\sum _{t=2}^{\infty }\left|a_t\right|.$$

(55)

From (50), it can be easily seen that

$$\left|{\displaystyle \frac{{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\sum _{t=2}^{\infty }\left|a_t\right|\le \sum _{t=2}^{\infty }\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\left|a_t\right|<1.$$

So, we can write

$$\sum _{t=2}^{\infty }\left|a_t\right|\le \left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|.$$

(56)

By using (56) in (54) and (55), the required result can be obtained. □

Theorem 7.

Let $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$. Then, for $\left|\varsigma \right|=r<1,$ we have

$$1-{\left[2\right]}_q\left({\displaystyle \frac{\left|\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right|}{\left|{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}}\right)r\le \left|D_q\mathcal{X}\left(\varsigma \right)\right|\le 1+{\left[2\right]}_q\left({\displaystyle \frac{\left|\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right|}{\left|{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}}\right)r.$$

(57)

Proof. 

Let $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$ be given in (1). Then

$$\left|D_q\mathcal{X}\left(\varsigma \right)\right|=\left|1+\sum _{t=2}^{\infty }{\left[t\right]}_q{a}_t{\varsigma }^{t-1}\right|\le 1+\sum _{t=2}^{\infty }{\left[t\right]}_q\left|a_t\right|r^{t-1}.$$

(58)

As $r^{t-1}\le r$ for all $t\ge 2$, so from (58), we have

$$\left|D_q\mathcal{X}\left(\varsigma \right)\right|\le 1+r\sum _{t=2}^{\infty }{\left[t\right]}_q\left|a_t\right|,\mathrm{and}\left|D_q\mathcal{X}\left(\varsigma \right)\right|\ge 1-r\sum _{t=2}^{\infty }{\left[t\right]}_q\left|a_t\right|.$$

(59)

From the consequence of (56), we have

$$\sum _{t=2}^{\infty }{\left[t\right]}_q\left|a_t\right|\le {\left[2\right]}_q\left({\displaystyle \frac{\left|\mathrm{sech}\left(q{e}^{i\theta }\right)-1\right|}{\left|{\left[2\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)\right|}}\right),$$

(60)

Using (60) in (59), the required result can be obtained. □

2.3. Radius of Starlikeness and Linear Combination

Theorem 8.

Let $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$ be given in (1). Then, $\mathcal{X}$ is starlike of order α, $0\le \alpha <1$ in the disc $\left|\varsigma \right|<r_1$, where

$$r_1=inf{\left\{{\displaystyle \frac{1-\alpha }{{\left[t\right]}_q-\alpha }}.\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\right\}}^{{\displaystyle \frac{1}{t-1}}},t\ge 2.$$

(61)

Proof. 

In order to prove (61), it is enough to show that

$$\left|{\displaystyle \frac{\varsigma D_q\mathcal{X}\left(\varsigma \right)}{\mathcal{X}\left(\varsigma \right)}}-1\right|\le 1-\alpha .$$

(62)

Using (1) and after some calculations, we have

$$\left|{\displaystyle \frac{\varsigma D_q\mathcal{X}\left(\varsigma \right)}{\mathcal{X}\left(\varsigma \right)}}-1\right|=\left|{\displaystyle \frac{{\displaystyle \sum _{t=2}^{\infty }}{\left[t\right]}_q{a}_t{\varsigma }^{t-1}-{\displaystyle \sum _{t=2}^{\infty }}{a}_t{\varsigma }^{t-1}}{1+{\displaystyle \sum _{t=2}^{\infty }}{a}_t{\varsigma }^{t-1}}}\right|\le {\displaystyle \frac{{\displaystyle \sum _{t=2}^{\infty }}\left({\left[t\right]}_q-1\right)\left|a_t\right|{\left|\varsigma \right|}^{t-1}}{1-{\displaystyle \sum _{t=2}^{\infty }}\left|a_t\right|{\left|\varsigma \right|}^{t-1}}}.$$

(63)

The expression in (63) is bounded above by $1-\alpha$, if and only if

$$\sum _{t=2}^{\infty }\left({\left[t\right]}_q-1\right)\left|a_t\right|{\left|\varsigma \right|}^{t-1}\le \left(1-\alpha \right)\left(1-\sum _{t=2}^{\infty }\left|a_t\right|{\left|\varsigma \right|}^{t-1}\right),$$

(64)

which is equivalent to

$$\sum _{t=2}^{\infty }{\displaystyle \frac{\left({\left[t\right]}_q-\alpha \right)}{\left(1-\alpha \right)}}\left|a_t\right|{\left|\varsigma \right|}^{t-1}\le 1.$$

(65)

As $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$, so by (50), we can see that

$$\left|a_t\right|\le \left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|,\mathrm{for}\mathrm{some}\mathrm{fixed}t.$$

(66)

From (65) and (66), we have

$$\left({\displaystyle \frac{{\left[t\right]}_q-\alpha }{1-\alpha }}\right){\left|\varsigma \right|}^{t-1}\le \left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|.$$

This implies

$${\left|\varsigma \right|}^{t-1}\le \left({\displaystyle \frac{1-\alpha }{{\left[t\right]}_q-\alpha }}\right)\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|,$$

which gives $r_1$, as given in (61). □

Theorem 9.

Let ${\mathcal{X}}_1\left(\varsigma \right)=\varsigma$, and

$${\mathcal{X}}_t\left(\varsigma \right)=\varsigma +\left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|{\varsigma }^t,t\in \mathbb{N}\setminus \left\{1\right\}.$$

(67)

Then, $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$ if and only if $\mathcal{X}$ can be written as follows:

$$\mathcal{X}\left(\varsigma \right)=\sum _{t=1}^{\infty }{\rho }_t{\mathcal{X}}_t\left(\varsigma \right),{\rho }_t>0,\mathit{and}\sum _{t=1}^{\infty }{\rho }_t=1.$$

(68)

Proof. 

Let $\mathcal{X}$ be written as

$$\mathcal{X}\left(\varsigma \right)=\sum _{t=1}^{\infty }{\rho }_t{\mathcal{X}}_t\left(\varsigma \right)={\rho }_1{\mathcal{X}}_1\left(\varsigma \right)+\sum _{t=2}^{\infty }{\rho }_t{\mathcal{X}}_t\left(\varsigma \right),$$

(69)

then, by (67), we have

$$\begin{array}{ccc}\hfill \mathcal{X}\left(\varsigma \right)& =& {\rho }_1{\mathcal{X}}_1\left(\varsigma \right)+\sum _{t=2}^{\infty }{\rho }_t\left\{\varsigma +\left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|{\varsigma }^t\right\},\hfill \\ & =& {\rho }_1\varsigma +\sum _{t=2}^{\infty }{\rho }_t\varsigma +\sum _{t=2}^{\infty }{\rho }_t\left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|{\varsigma }^t.\hfill \end{array}$$

Some simple calculations lead to

$$\mathcal{X}\left(\varsigma \right)=\varsigma +\sum _{t=2}^{\infty }{\rho }_t\left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|{\varsigma }^t.$$

(70)

Then, consider

$$\begin{array}{ccc}& & \sum _{t=2}^{\infty }{\rho }_t\left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\hfill \\ & =& \sum _{t=2}^{\infty }{\rho }_t=\sum _{t=1}^{\infty }{\rho }_t-{\rho }_1=1-{\rho }_1\le 1,\hfill \end{array}$$

which means $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$.

Conversely, let $\mathcal{X}\in \mathcal{M}{S}_{\mathrm{sech}q}^{\ast }$, then by (66), we have

$$\left|a_t\right|\le \left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|,\mathrm{for}\mathrm{some}\mathrm{fixed}t.$$

Consider

$${\rho }_t=\left|{\displaystyle \frac{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}}\right|\left|a_t\right|,$$

and ${\rho }_1=1-{\displaystyle \sum _{t=2}^{\infty }}{\rho }_t$. Then

$$\mathcal{X}\left(\varsigma \right)=\varsigma +\sum _{t=2}^{\infty }{a}_t{\varsigma }^t=\varsigma +\sum _{t=2}^{\infty }{\rho }_t\left|{\displaystyle \frac{\mathrm{sech}\left(q{e}^{i\theta }\right)-1}{{\left[t\right]}_q-\mathrm{sech}\left(q{e}^{i\theta }\right)}}\right|{\varsigma }^t.$$

(71)

From (67), we can write (71) as

$$\mathcal{X}\left(\varsigma \right)=\varsigma +\sum _{t=2}^{\infty }{\rho }_t\left({\mathcal{X}}_t\left(\varsigma \right)-\varsigma \right).$$

After some simple calculations, we get

$$\mathcal{X}\left(\varsigma \right)={\rho }_1\varsigma +\sum _{t=2}^{\infty }{\rho }_t{\mathcal{X}}_t\left(\varsigma \right)=\sum _{t=1}^{\infty }{\rho }_t{\mathcal{X}}_t\left(\varsigma \right).$$

□

3. Conclusions

In this article, we introduce a novel subclass of analytic functions employing the q-difference operator in conjunction with a q-generalized form of the secant hyperbolic function. The newly defined class is the q-generalization of the class defined and studied by [32]. This approach is motivated by the significant applications of quantum hyperbolic functions in various scientific and engineering fields. In particular, quantum secant hyperbolic functions, which play a crucial role in quantum mechanics, where they help to describe particle behaviors in specific potential fields, and in signal processing, where they are used to reduce noise during signal transmission and improve clarity. Using the framework of geometric function theory, we establish several subordination results involving the q-analogue of Janowski-type functions. Furthermore, by utilizing the convolution operator techniques, we derived important results like coefficient estimates, growth and distortion bounds, and the radius of starlikeness. These findings not only reinforce the mathematical robustness of the proposed class, but also pave the way for further exploration of related function classes within the q-calculus framework.