Subclasses of q-Uniformly Starlike Functions Obtained via the q-Carlson–Shaffer Operator

Abstract

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class $S{T}_q(m,c,d,\beta )$. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties.

1. Introduction

Quantum calculus, sometimes called q-calculus, is a branch of mathematics that generalizes traditional calculus by introducing a parameter q to replace limits in differentiation and integration. In quantum calculus, instead of approaching infinitesimal changes, the calculus operates over discrete steps, where the parameter q controls the granularity of these steps. This approach is particularly useful in areas that naturally involve discrete structures, such as quantum theory, number theory, and combinatorics. The concepts of derivative and integral within the framework of q-calculus were formally introduced by Jackson—see [1,2]—establishing the foundational principles of q-calculus. For a more comprehensive understanding, please consult [3]. q-Calculus has extensive applications across various fields of mathematics and physics, including the calculus of variations, quantum groups, the theory of relativity, combinatorics, and quantum mechanics. This wide-ranging applicability underscores the importance of q-calculus as a formidable mathematical instrument. Recently, the use of q-calculus in Geometric Function Theory (GFT) has attracted considerable attention from scholars. A significant application is presented in [4], where Srivastava employed q-hypergeometric functions within the context of GFT. Furthermore, the modified q-derivative has been utilized to define q-starlike functions, as outlined in [5,6]. The exploration of the q-generalization of close-to-convex functions is discussed in [7], while [8,9] introduce generalized forms of q-starlike and q-close-to-convex functions, leading to the development of modified subclasses of analytic functions within the q-calculus framework. The categorization of q-convex functions with a complex order, which generalizes several traditional classes in GFT, is presented in [10]. Additionally, the q-extensions of starlike functions, particularly those associated with Janowski functions, are established in [11]. In the realm of GFT, linear operators have been constructed and utilized to characterize new subclasses of analytic functions, resulting in significant progress, especially with the recent introduction of q-special functions defined in terms of the q-Pochhammer symbol, q-shifted factorial, q-Gamma function, q-hypergeometric function, and deformed q-Lerch–Hurwitz function. The advancement of q-convoluted linear operators and their applications has become a central theme in contemporary research, as indicated in [12]. These investigations seek to formulate q-analogues of various subclasses of analytic functions, thereby providing fresh perspectives and methodologies for mathematical analysis.

This study is dedicated to defining the q-counterparts of subclasses of analytic functions via the generalized q-Carlson–Shaffer operator. Furthermore, it examines and establishes multiple characteristic properties of these functions, thereby enhancing the overall comprehension and application of q-calculus within geometric function theory. In summary, the investigation and advancement of q-calculus in the context of geometric function theory signify a dynamic and progressive area of research. The incorporation of q-special functions and the development of new linear operators continue to propel research forward, providing innovative solutions and broadening the theoretical underpinnings of both q-calculus and geometric function theory.

This section’s goal is to provide some fundamental ideas about geometric function theory so that readers can better comprehend the article’s key conclusions. For this, let A be the set of analytic functions $f\left(\xi \right)$ normalized under the conditions $f\left(0\right)=f^{\prime }\left(0\right)-1=0$ the analytic

$$f\left(\xi \right)=\xi +\sum _{k=2}^{\infty }{a}_k{\xi }^k,$$

(1)

in the open unit disc $E=\{\xi \in \mathbb{C}:|\xi |<1\}$. Furthermore, the set of normalized univalent functions is represented by the subset S of A. Köebe [13] first proposed this class in 1907, and it is now the mainstay of groundbreaking studies in this area. This idea sparked a lot of curiosity; however, Bieberbach [14] quickly published an article in which the well-known coefficient hypothesis was presented. According to this conjecture, for every $n\ge 2$, $\left|a_n\right|\le n$, if $f\left(z\right)\in S$ and has the series form (1). This remained a difficulty for function theorists for 69 years, despite the efforts of several mathematicians. This long-running rumor was resolved in 1985 by de-Branges [15]. Many papers were written about this conjecture and the associated coefficient difficulties throughout the course of the last 69 years. New subfamilies of S were defined and the problems with their coefficients were examined.

Let $f\left(\xi \right)$ be denoted by (1) and $g\left(\xi \right)$ defined by

$$g\left(\xi \right)=\xi +\sum _{k=2}^{\infty }{b}_k{\xi }^k.$$

The Hadamard product (or convolution) of f and g is defined and discussed in [16,17] as follows:

$$(f\ast g)\left(\xi \right)=\xi +\sum _{k=2}^{\infty }{a}_k{b}_k{\xi }^k.$$

Let $f,h\in A$. Then, f is subordinate to h, written as $f\prec h$ or $f\left(\xi \right)\prec h\left(\xi \right)$, $\xi \in E$, if there exists a Schwartz function $\omega \left(\xi \right)$ analytic in E with $\omega \left(0\right)=0$ and $\left|\omega \left(\xi \right)\right|<1$ for $\xi \in E$, such that $f\left(\xi \right)=h\left(\omega \right(\xi \left)\right)$.

A subset $B\subset \mathbb{C}$ is called q-geometric if $q\xi \in B$ whenever $\xi \in B$, and it contains all the geometric sequences ${\left\{\xi q^k\right\}}_0^{\infty }.$ In the context of geometric function theory, the q-derivative of $f\left(\xi \right)$ is defined in [5] as

$$D_{q}f\left(\xi \right)={\displaystyle \frac{f\left(\xi \right)-f\left(q\xi \right)}{(1-q)\xi }},q\in (0,1),(\xi \in B\setminus \left\{0\right\}),$$

(2)

and $D_{q}f\left(0\right)=f^{\prime }\left(0\right)$. For a function $g\left(\xi \right)={\xi }^k$, the q-derivative is

$$D_{q}g\left(\xi \right)=\left[k\right]{\xi }^{k-1},$$

where $\left[k\right]={\displaystyle \frac{1-q^k}{1-q}}=1+q+q^2+\dots .+q^{k-1}.$

We note that as $q\to 1^{-}$, $D_{q}f\left(\xi \right)\to f^{\prime }\left(\xi \right)$, which is the ordinary derivative. From (1) and (2), we can deduce that:

$$D_{q}f\left(\xi \right)=1+\sum _{k=2}^{\infty }\left[k\right]a_k{\xi }^k.$$

Let $f\left(\xi \right)$ and $g\left(\xi \right)$ be defined on a q-geometric set $B\subset \mathbb{C}$ such that q-derivatives of $f\left(\xi \right)$ and $g\left(\xi \right)$ exist $\forall \xi \in B$. Then, for complex numbers $b,c$, we have

$$D_q(bf\left(\xi \right)\pm cg\left(\xi \right))=b{D}_{q}f\left(\xi \right)\pm c{D}_{q}g\left(\xi \right).$$

$$D_q\left(f\left(\xi \right)g\left(\xi \right)\right)=f\left(q\xi \right)D_{q}g\left(\xi \right)+g\left(\xi \right)D_{q}f\left(\xi \right).$$

$$D_q\left(logf\left(\xi \right)\right)={\displaystyle \frac{ln{q}^{-1}}{1-q}}{\displaystyle \frac{D_{q}f\left(\xi \right)}{f\left(\xi \right)}}.$$

Jackson [2] introduced the q-integral of a function f using the following:

$${\int }_0^{\xi }f\left(t\right)D_{q}t=\xi (1-q)\sum _{k=0}^{\infty }{q}^{k}f\left(q^k\xi \right),$$

provided that the series converges.

In [18], the q–Carlson Shaffer operator $L_q(c,d)\left(\varsigma \right)$ is defined as

$$\begin{array}{ccc}\hfill L_q(c,d)f\left(\varsigma \right)& =& {\varphi }_q(c,d;\varsigma )\ast f\left(\varsigma \right)\hfill \\ & =& \varsigma +\stackrel{\infty }{\sum _{k=2}}{\displaystyle \frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}}{a}_k{\varsigma }^k,\hfill \end{array}$$

where $c\in \mathbb{R},d\in \mathbb{R}\setminus {\mathbb{Z}}_0,$

$${\varphi }_q(c,d;\varsigma )=\varsigma +\stackrel{\infty }{\sum _{k=2}}{\displaystyle \frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}}{a}_k{\varsigma }^k$$

and $f\left(\varsigma \right)$ is given by $\left(1\right)$. If $q\to 1^{-}$, it is the usual Carlson–Shaffer operator [19]. The q-Carlson-0Shaffer operator satisfies the following identity:

$$\varsigma D_q\left(L_q(c,d\right)f\left(\varsigma \right)={\displaystyle \frac{\left[c\right]}{q^{c-1}}}{L}_q(1+c,d)f\left(\varsigma \right)-{\displaystyle \frac{\left[c-1\right]}{q^{c-1}}}{L}_q(c,d)f\left(\varsigma \right).$$

(3)

Now, we define the class $S{T}_q(m,c,d,\beta )$ in the context of the q-Carlson–Shaffer operator as

$$\Re \left[{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\right]>m\left|{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}-1\right|+\beta ,$$

(4)

where $0\le \beta <1,m\in [0,\infty ),c,d\in {\mathbb{R}}^{+}$ and $f\left(\varsigma \right)$ is given by (1).

• Cases:

For different values of parameters $c,d,m$,$\beta$ and $q\to 1^{-}$, the class $S{T}_q(m,c,d,\beta )$ reduces to the following known and established classes as follows:

(i) For $c=1,d=1$ and $\beta =0,$ the class $S{T}_q(m,c,d,\beta )$ reduces to class m-$U{M}_q(0,0)$, as discussed in [20].

(ii) If $c=1,d=1$ and $m=0,$ it is $S_q\left(\beta \right)$—the class of q-starlike functions defined in [6].

(iii) If $c=\tau +1(\tau >0),d=1,$ it is the class $ST(m,\beta ,\tau ,q)$ discussed in [21].

We need the following Lemmas to obtain our results.

Lemma 1 

([22]). If $p\left(\varsigma \right)=1+c_1\varsigma +c_2{\varsigma }^2+\dots \in P\left(p_m\right)$ is an analytic function in $E,$ then

$$\left|c_2-\mu c_1^2\right|\le \left\{\begin{array}{cc}{P}_1-\mu P_1^2& \mu \le 0,\\ P_1& 0<\mu <1,\\ P_1-(1-\mu )P_1^2& 1>\mu .\end{array}\right.$$

(5)

The sharpness of the result is for the functions $p\left(\varsigma \right)={\displaystyle \frac{1+{\varsigma }^2}{1-{\varsigma }^2}}$ for $0<\mu <1$ and $p\left(\varsigma \right)={\displaystyle \frac{1+\varsigma }{1-\varsigma }}$ otherwise.

Lemma 2 

([23]). Let

$$\varphi \left(\varsigma \right)=\sum _{k=1}^{\infty }{c}_k{\varsigma }^k.$$

For any complex number $\lambda ,$

$$\left|c_2-\lambda c_1^2\right|\le max\left\{1,\left|\lambda \right|\right\}.$$

The result is sharp for $\varphi \left(\varsigma \right)={\varsigma }^2$ for $\left|\lambda \right|<1$ and $\varphi \left(\varsigma \right)=\varsigma$ for $\left|\lambda \right|\ge 1.$

2. Sufficiency Criteria

In first part of this section, the sufficient conditions for $f\in A$ to be in $S{T}_q(m,c,d,\beta )$ are established. In the second part, the salient features of the class $S{T}_q(m,c,d,\beta )$ are studied in the context of conic domains and the results, including coefficient bounds, the solution of the Feketo–Szegö problem, and the necessary and sufficient condition in terms of the Hadamard product, are proved.

Theorem 1.

Let $f\in A$ be given by (1). If the inequality

$$\sum _{k=2}^{\infty }\left\{\left[k\right](1+m)-k-\beta \right\}{\displaystyle \frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}}\le 1-\beta$$

(6)

is true for some $0\le \beta <1,m\in [0,\infty ),c,d\in {\mathbb{R}}^{+},$ then $f\in S{T}_q(m,c,d,\beta ).$ This bound is sharp and is attained using the following function:

$$f_k\left(\varsigma \right)=\varsigma -{\displaystyle \frac{(1-\beta ){\left(\left[d\right]\right)}_{k-1}}{\left\{\left[k\right](1+m)-k-\beta \right\}{\left(\left[c\right]\right)}_{k-1}}}{\varsigma }^k.$$

Proof. 

With the utilization of (3), it suffices to prove

$$m\left|{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}-1\right|-\Re \left\{{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}-1\right\}<1-\beta ,$$

under the condition (6).

Note that

$$\begin{array}{c}m\left|{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}-1\right|-\Re \left\{{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}-1\right\}\\ \le (m+1)\left|{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\right|=(m+1)\left|{\displaystyle \frac{{\sum }_{k=2}^{\infty }(\left[k\right]-1)\frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}{a}_k{\varsigma }^{k-1}}{1+{\sum }_{k=2}^{\infty }\frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}{a}_k{\varsigma }^{k-1}}}\right|\\ \le (m+1)\left\{{\displaystyle \frac{{\sum }_{k=2}^{\infty }(\left[k\right]-1)\frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}\left|a_k\right|}{1-{\sum }_{k=2}^{\infty }\frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}\left|a_k\right|}}\right\},\end{array}$$

which is bounded by $1-\beta$ if the condition (6) holds.

It is clear that the function $f_k$ satisfies (6), so the value cannot be replaced by a larger number. Now, to show $f_k\in S{T}_q(m,c,d,\beta ).$ Since

$$\begin{array}{ccc}\hfill m\left|{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f_k\left(\varsigma \right)\right)}{L_q(c,d)f_k\left(\varsigma \right)}}-1\right|& =& m\left|{\displaystyle \frac{(1-\beta )(1-\left[k\right]){\varsigma }^{k-1}}{\left(\left[k\right](m+1)-m-\beta \right)-(1-\beta ){\varsigma }^{k-1}}}\right|\hfill \\ & =& m(1-\beta )\left|{\displaystyle \frac{(1-\left[k\right]){\varsigma }^{k-1}}{\left(\left[k\right](m+1)-m-\beta \right)-(1-\beta ){\varsigma }^{k-1}}}\right|\hfill \\ & <& {\displaystyle \frac{m(1-\beta )(1+\left[k\right])}{\left(\left[k\right](m+1)-m-\beta \right)+(1-\beta )}}\hfill \\ & =& {\displaystyle \frac{m(1-\beta )}{m+1}}\hfill \end{array}$$

and

$$\Re \left\{{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f_k\left(\varsigma \right)\right)}{L_q(c,d)f_k\left(\varsigma \right)}}\right\}=\Re \left\{{\displaystyle \frac{\left[k\right](m+1)-m-\beta -\left[k\right](1-\beta ){\varsigma }^{k-1}}{\left[k\right](m+1)-m-\beta -(1-\beta ){\varsigma }^{k-1}}}\right\}>{\displaystyle \frac{m+\beta }{m+1}}.$$

Now,

$$\Re \left\{{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f_k\left(\varsigma \right)\right)}{L_q(c,d)f_k\left(\varsigma \right)}}\right\}-m\left|{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f_k\left(\varsigma \right)\right)}{L_q(c,d)f_k\left(\varsigma \right)}}-1\right|>{\displaystyle \frac{\beta +m}{1+m}}+\left({\displaystyle \frac{m(\beta -1)}{m+1}}\right)=\beta .$$

Thus, $f_k\in S{T}_q(m,c,d,\beta ).$  □

Corollary 1.

Let $f\left(\varsigma \right)=\varsigma +a_k{\varsigma }^k.$ If

$$\left|a_k\right|\le {\displaystyle \frac{\left(1-\alpha \right){\left(\left[d\right]\right)}_{k-1}}{\{\left[k\right](m+1)-m-\beta \}{\left(\left[c\right]\right)}_{k-1}}},k\ge 2,$$

then $f_k\in S{T}_q(m,c,d,\beta ).$

In this part of the main results, the class $S{T}_q(m,c,d,\beta )$ is discussed in the context of conic domains and then various results are proved. Consider

$$p\left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}.$$

The condition $\left(4\right)$ can be written as

$$\Re \left(p\left(\varsigma \right)\right)>m\left|p\left(\varsigma \right)-1\right|+\beta ,\varsigma \in E.$$

The range of $p\left(\varsigma \right)$ is a conic domain given as

$${\mathsf{\Omega }}_{m,\beta }=\left\{\omega \in \mathbb{C}:\Re \left(\omega \right)>m\left|\omega -1\right|+\beta \right\},$$

where $m\in [0,\infty )$ and $\beta \in [0,1).$ It is pertinent to mention that $1\in {\mathsf{\Omega }}_{m,\beta }$ and $\partial {\mathsf{\Omega }}_{m,\beta }$ is a curve defined as

$$\partial {\mathsf{\Omega }}_{m,\beta }=\left\{\omega ={\varsigma }_1+i{\varsigma }_2:{\left({\varsigma }_1-\beta \right)}^2=m^2{\left({\varsigma }_1-1\right)}^2+m^2{\varsigma }_2^2\right\}.$$

Simple computations show that $\partial {\mathsf{\Omega }}_{m,\beta }$ is a conic section that is symmetric about the real line. The domain ${\mathsf{\Omega }}_{m,\beta }$ is bordered by ellipses for $m>1,$ by a parabola for $m=1$, and by a hyperbola if $0<m<1.$ If $m=0,$ ${\mathsf{\Omega }}_{m,\beta }$ is a right half plane $\Re \left(\omega \right)>\beta .$

From $\left(4\right)$, $f\in S{T}_q(m,c,d,\beta )$ iff

$${\displaystyle \frac{\xi D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\in {\mathsf{\Omega }}_{m,\beta }.$$

(7)

Using the characteristics of the domain ${\mathsf{\Omega }}_{m,\beta }$ and (4.2.11), it follows that

$$\Re \left({\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\right)>{\displaystyle \frac{m+\beta }{m+1}}$$

and

$$\left|Arg\left({\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\right)\right|\le \left\{\begin{array}{cc}arctan\left({\displaystyle \frac{1-\beta }{\sqrt{\left|m^2-{\beta }^2\right|}}}\right)& 0\le \beta <1,m>0,\\ {\displaystyle \frac{\pi }{2}}& m=0.\end{array}\right.$$

Considering $p_{m,\beta }\in P$, the class of Caratheodory functions such that $p_{m,\beta }\left(E\right)={\mathsf{\Omega }}_{m,\beta }.$ Let $P\left(p_{m,\beta }\right)$ symbolizes the following class of functions:

$$\begin{array}{ccc}\hfill P\left(p_{m,\beta }\right)& =& \left\{p\in P:p\left(E\right)\subset {\mathsf{\Omega }}_{m,\beta }\right\}\hfill \\ & =& \left\{p\in P:p\prec p_{m,\beta }\mathrm{in}E\right\}.\hfill \end{array}$$

The extremal functions are defined as

$$p_{m,\beta }\left(\varsigma \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1+(1-2\beta )\varsigma }{1-\varsigma }}& m=0,\\ 1+{\displaystyle \frac{2(1-\beta )}{{\pi }^2}}log\left({\displaystyle \frac{1+\sqrt{\varsigma }}{1+\sqrt{\varsigma }}}\right)& m=1,\\ {\displaystyle \frac{1-\beta }{1-m^2}}cosC\left(m\right)ilog\left({\displaystyle \frac{1+\sqrt{\varsigma }}{1+\sqrt{\varsigma }}}\right)-{\displaystyle \frac{m^2-\beta }{1-m^2}}& 0<m<1,\\ {\displaystyle \frac{1-\beta }{1-m^2}}sin\left({\displaystyle \frac{\pi }{2Q\left(t\right)}}{\int }_0^{{\displaystyle \frac{u\left(\varsigma \right)}{\sqrt{t}}}}{\displaystyle \frac{dx}{\sqrt{1-x^2}\sqrt{1-t^2{x}^2}}}\right)& m>1,\end{array}\right.$$

where

$$C\left(m\right)={\displaystyle \frac{2}{\pi }}arccosm,$$

and

$$u\left(\varsigma \right)={\displaystyle \frac{\varsigma -\sqrt{t}}{1-\sqrt{t}\varsigma }}\left(0<t<1\right),$$

Here, t is considered using the following:

$$m=cosh\left({\displaystyle \frac{\pi Q^{\prime }\left(t\right)}{4Q\left(t\right)}}\right).$$

$Q\left(t\right)$ is Legendre’s complete elliptic integral of the first kind and $Q^{\prime }\left(t\right)$ is a complementary integral of $Q\left(t\right).$

If $m=0$, then

$$p_{0,\beta }\left(\varsigma \right)=1+2(1-\beta )\varsigma +2(1-\beta ){\varsigma }^2+\dots$$

For $m=1,$

$$p_{1,\beta }\left(\varsigma \right)=1+{\displaystyle \frac{8}{{\pi }^2}}(1-\beta )\varsigma +{\displaystyle \frac{16}{{\pi }^2}}(1-\beta ){\varsigma }^2+\dots$$

Using the Taylor expansion in [21,24], for $0<m<1,$

$$p_{m,\beta }\left(\varsigma \right)=1+{\displaystyle \frac{1-\beta }{1-m^2}}\sum _{k=1}^{\infty }\sum _{r=1}^{2k}\left(\begin{array}{c}C\left(m\right)\\ r\end{array}\right)\left(\begin{array}{c}2k-1\\ 2k-r\end{array}\right).$$

Finally, for $m>1$

$$p_{m,\beta }\left(\varsigma \right)=1+{\displaystyle \frac{{\pi }^2(1-\beta )}{4\sqrt{t}\left(m^2-1\right)Q^2\left(t\right)(1+t)}}\left\{\varsigma +{\displaystyle \frac{4Q^2\left(t\right)(t^2+6t+1)-{\pi }^2}{24\sqrt{t}{Q}^2\left(t\right)(1+t)}}{\varsigma }^2+\dots \right\},$$

so that, denoting $p_{m,\beta }\left(\varsigma \right)=1+P_1\varsigma +P_2{\varsigma }^2+\dots \left(P_j=P_j(m,\beta ),j=1,2,\dots \right),$ to obtain

$$P_1=\left\{\begin{array}{cc}{\displaystyle \frac{8(1-\beta ){\left(arccosm\right)}^2}{{\pi }^2(1-m^2)}}& 0\le m<1,\\ {\displaystyle \frac{8(1-\beta )}{{\pi }^2(1-m^2)}}& m=1,\\ {\displaystyle \frac{{\pi }^2(1-\beta )}{4\sqrt{t}{Q}^2\left(t\right)\left(m^2-1\right)}}& m>1.\end{array}\right.$$

Let $f_{m,\beta }=\varsigma +A_2{\varsigma }^2+A_3{\varsigma }^3+\dots$ be the extremal function in the class $S{T}_q(m,c,d,\beta ).$ Then, the relationship between the extremal functions in the classes $P\left(p_{m,\beta }\right)$ and $S{T}_q(m,c,d,\beta )$ is given by

$$p_{m,\beta }={\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f_{m,\beta }\left(\varsigma \right)\right)}{L_q(c,d)f_{m,\beta }\left(\varsigma \right)}}.$$

(8)

Now, $\left(6\right)$, $\left(8\right)$ and the construction of the q-Carlson–Shaffer operator can be used to obtain the following coefficient relation:

$${\displaystyle \frac{\left[k\right]{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}}{A}_k=\sum _{n=1}^{k-1}{\displaystyle \frac{{\left(\left[c\right]\right)}_{n-1}}{{\left(\left[d\right]\right)}_{n-1}}}{A}_n{P}_{k-n},A_1=1.$$

(9)

Now,

$$A_2={\displaystyle \frac{\left[d\right]}{\left[c\right]\left[2\right]}}{P}_1$$

(10)

and

$$A_3={\displaystyle \frac{\left(\left[2\right]{\left[d\right]}^2{P}_2+P_1^2\right){\left[d\right]}_2}{\left[3\right]\left[2\right]{\left[c\right]}_2{\left[d\right]}^2}}.$$

(11)

Since $q\in (0,1),c,d\in {\mathbb{R}}^{+}$ and $P_k$ are non-negative, it follows that $A_k$ are non-negative.

Theorem 2.

Let $m\in [0,\infty ),$$q\in (0,1)$ and $0\le \beta <1.$ If $f\in S{T}_q(m,c,d,\beta )$ then

$$\left|a_2\right|\le A_{2}and\left|a_3\right|\le A_3.$$

where f is given by $\left(1\right)$.

Proof. 

Let

$$p\left(\varsigma \right)={\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}.$$

We apply the q-Carlson–Shaffer operator to $p\left(\varsigma \right)=1+p_1\varsigma +p_2{\varsigma }^2+\dots$ to obtain

$${\displaystyle \frac{\left[k\right]{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}}{a}_k=\sum _{n=1}^{k-1}{\displaystyle \frac{{\left(\left[c\right]\right)}_{n-1}}{{\left(\left[d\right]\right)}_{n-1}}}{A}_n{p}_{k-n},A_1=1.$$

Since $p_{m,\beta }$ is univalent in $E$, the function

$$\begin{array}{ccc}\hfill q\left(\xi \right)& =& {\displaystyle \frac{1+p_{m,\beta }^{-1}\left(p\left(\xi \right)\right)}{1-p_{m,\beta }^{-1}(\left(p\left(\varsigma \right)\right)}}\hfill \\ & =& 1+p_1\varsigma +p_2{\varsigma }^2+\dots \hfill \end{array}$$

is analytic in E, with $\Re \left(q\left(\varsigma \right)\right)>0$.

$$\begin{array}{ccc}\hfill p\left(\varsigma \right)& =& p_{m,\beta }\left\{{\displaystyle \frac{q\left(\varsigma \right)-1}{q\left(\varsigma \right)+1}}\right\}\hfill \\ & =& 1+{\displaystyle \frac{1}{2}}{c}_1{P}_1\varsigma +\left\{{\displaystyle \frac{1}{2}}{c}_2{P}_1+{\displaystyle \frac{1}{4}}{c}_1(P_2-P_1)\right\}{\varsigma }^2+\dots \hfill \end{array}$$

Now

$$\left|a_2\right|={\displaystyle \frac{\left[d\right]}{\left[c\right]\left[2\right]}}\left|p_1\right|={\displaystyle \frac{\left[d\right]}{2\left[c\right]\left[2\right]}}\left|c_1{P}_1\right|\le {\displaystyle \frac{\left[d\right]}{\left[c\right]\left[2\right]}}{P}_1=A_2,$$

(12)

Here, the inequalities $\left|c_k\right|\le 2$ and $\left(9\right)$ are used. The relations ${\left|p_1\right|}^2+\left|p_2\right|\le P_1^2+P_2$ [24] and $\left(10\right)$ to can be used obtain

$$\begin{array}{ccc}\hfill \left|a_3\right|& =& \left|{\displaystyle \frac{\left\{\left[2\right]{\left[d\right]}^2{p}_2+p_1^2\right\}{\left[d\right]}_2}{\left[3\right]\left[2\right]{\left[c\right]}_2{\left[d\right]}^2}}\right|\hfill \\ & =& \left({\displaystyle \frac{\left[2\right]{\left[d\right]}^2\left\{{\left|p_1\right|}^2+\left|p_2\right|\right\}+1-\left[2\right]{\left[d\right]}^2{\left|p_1\right|}^2}{\left[3\right]\left[2\right]{\left[c\right]}_2{\left[d\right]}^2}}\right){\left[d\right]}_2\hfill \\ & \le & \left\{{\displaystyle \frac{\left[2\right]{\left[c\right]}^2{P}_2+P_1^2}{\left[3\right]\left[2\right]{\left[c\right]}_2{\left[d\right]}^2}}\right\}{\left[d\right]}_2=A_3.\hfill \end{array}$$

(13)

This concludes the proof.  □

Theorem 3.

Consider $m\in [0,\infty )$ and $\beta \in [0,1).$ If f of the form $\left(1\right)$ is in class $S{T}_q(m,c,d,\beta )$, then

$$\left|a_k\right|\le {\displaystyle \frac{{\left(\left[d\right]\right)}_{k-1}}{{\left(\left[c\right]\right)}_{k-1}}}{\displaystyle \frac{P_1(P_1+1)(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k\right]!}}.$$

(14)

Proof. 

The result holds for $k=2.$ Let $k>2$ and consider the inequality to be valid for $n\le k-1.$

$$\begin{array}{ccc}\hfill \left|a_k\right|& =& {\displaystyle \frac{{\left(\left[d\right]\right)}_{k-1}}{\left[k\right]{\left(\left[c\right]\right)}_{k-1}}}\left|p_{k-1}+\sum _{n=2}^{k-1}{\displaystyle \frac{{\left(\left[d\right]\right)}_{n-1}}{{\left(\left[c\right]\right)}_{n-1}}}{a}_n{p}_{k-n}\right|\hfill \\ & \le & {\displaystyle \frac{{\left(\left[d\right]\right)}_{k-1}}{\left[k\right]{\left(\left[c\right]\right)}_{k-1}}}{P}_1+\sum _{n=2}^{k-1}{\displaystyle \frac{{\left(\left[d\right]\right)}_{n-1}}{{\left(\left[c\right]\right)}_{n-1}}}\left|a_n\right|P_1\hfill \\ & =& {\displaystyle \frac{{\left(\left[d\right]\right)}_{k-1}}{\left[k\right]{\left(\left[c\right]\right)}_{k-1}}}{P}_1\left\{1+\sum _{n=2}^{k-1}{\displaystyle \frac{{\left(\left[d\right]\right)}_{n-1}}{{\left(\left[c\right]\right)}_{n-1}}}\left|a_n\right|\right\}\hfill \\ & \le & {\displaystyle \frac{{\left(\left[d\right]\right)}_{k-1}}{\left[k\right]{\left(\left[c\right]\right)}_{k-1}}}{P}_1\left\{1+\sum _{n=2}^{k-1}{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k-1\right]!}}\right\},\hfill \end{array}$$

Here $\left(9\right)$ is used and the hypothesis of induction to $\left|a_n\right|$ is applied, resulting in [25], $\left|p_k\right|\le P_1.$ Now, it is claimed that

$$1+\sum _{n=2}^{k-1}{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k-1\right]!}}={\displaystyle \frac{(P_1+1)(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k-1\right]!}}.$$

The claim is proved using the principle of Mathematical Induction. It is clearly valid for $k=2.$ Suppose it is true for $n=k-2$; that is,

$$1+\sum _{n=2}^{k-2}{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}={\displaystyle \frac{(P_1+1)(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}.$$

(15)

Now, we prove it is true for $n=k-1.$ Using $\left(15\right)$, we have

$$\begin{array}{c}1+\sum _{n=2}^{k-1}{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k-1\right]!}}\\ =1+\sum _{n=2}^{k-2}{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}+{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k-1\right]!}}\\ ={\displaystyle \frac{(P_1+1)(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}+{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-2])(P_1+[k-1])}{\left[k-1\right]!}}\\ ={\displaystyle \frac{(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}\left\{(P_1+1)+{\displaystyle \frac{P_1(P_1+[k-1])}{\left[k-1\right]}}\right\}\end{array}$$

Rearranging the terms, we have

$$\begin{array}{c}1+\sum _{n=2}^{k-1}{\displaystyle \frac{P_1(P_1+\left[2\right])\dots (P_1+[k-1])}{\left[k-1\right]!}}\\ ={\displaystyle \frac{(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}\left[{\displaystyle \frac{P_1^2+P_1[k-1]+P_1+[k-1]}{\left[k-1\right]}}\right]\\ ={\displaystyle \frac{(P_1+\left[2\right])\dots (P_1+[k-2])}{\left[k-2\right]!}}\left[P_1(P_1+[k-1])+1(P_1+[k-1])\right]\\ ={\displaystyle \frac{(P_1+1)(P_1+\left[2\right])\dots (P_1+[k-2])(P_1+[k-1])}{\left[k-1\right]!}}.\end{array}$$

Consequently, inequality $\left(14\right)$ follows.  □

3. Feketo-Szegö Inequality and Hadamard Product

Now, the solution of the Feketo–Szegö problem over the class $S{T}_q(m,c,d,\beta )$ is discussed using Lemmas $\left(1\right)$ and $\left(2\right)$.

Theorem 4.

Let $m\in [0,\infty )$ and $\beta \in [0,1).$ If f given by $\left(1\right)$ is in class $S{T}_q(m,c,d,\beta )$, then, for a complex number $\lambda ,$

$$\left|a_3-\lambda a_2^2\right|\le max\left\{1;\left|2{\displaystyle \frac{\left[3\right]\left[c+1\right]\lambda {\left[d\right]}^2-\left[c\right]\left[d+1\right]}{\left[2\right]\left[c\right]\left[d+1\right]{\left[d\right]}^2}}-1\right|\right\}.$$

Moreover, for a real parameter $\lambda ,$ we obtain the following bounds:

$$\left|a_3-\lambda a_2^2\right|\le \left\{\begin{array}{cc}{P}_1-\left\{{\displaystyle \frac{1}{\left[2\right]{\left[d\right]}^2}}-{\displaystyle \frac{\left[3\right]\left[d+1\right]\lambda }{\left[c\right]\left[2\right]\left[d+1\right]}}\right\}{P}_1^2& \lambda \ge {\displaystyle \frac{\left[c\right]\left[d+1\right]}{\left[3\right]\left[c+1\right]{\left[d\right]}^2}},\\ P_1& \lambda \in {\displaystyle \frac{\left[c\right]\left[d+1\right]}{\left[3\right]\left[c+1\right]{\left[d\right]}^2}}\left(1-2{\left[d\right]}^2,1\right),\\ P_1+\left({\displaystyle \frac{\left[c\right]\left[d+1\right]-\left[3\right]\left[c+1\right]\lambda {\left[d\right]}^2}{\left[2\right]\left[c\right]\left[d+1\right]{\left[d\right]}^2}}\right)P_1^2& \lambda \le \left({\displaystyle \frac{1-2{\left[d\right]}^2}{2{\left[d\right]}^2}}\right){\displaystyle \frac{\left[3\right]\left[c+1\right]}{\left[d\right]\left[2\right]\left[d+1\right]}}.\end{array}\right.$$

(16)

Proof. 

From $\left(12\right)$ and $\left(13\right)$, it follows

$$a_2={\displaystyle \frac{\left[d\right]}{\left[c\right]\left[2\right]}}{p}_1$$

(17)

and

$$a_3={\displaystyle \frac{\left\{\left[2\right]{\left[d\right]}^2{p}_2+p_1^2\right\}{\left[d\right]}_2}{\left[3\right]\left[2\right]{\left[c\right]}_2{\left[d\right]}^2}}.$$

(18)

In view of $\left(17\right)$ and $\left(18\right)$, for a complex number $\lambda$,

$$\begin{array}{ccc}\hfill a_3-\lambda a_2^2& =& {\displaystyle \frac{\left\{\left[2\right]{\left[d\right]}^2{p}_2+p_1^2\right\}{\left[d\right]}_2}{\left[3\right]\left[2\right]{\left[c\right]}_2{\left[d\right]}^2}}-{\displaystyle \frac{\lambda {\left[d\right]}^2}{{\left[c\right]}^2{\left[2\right]}^2}}{p}_1\hfill \\ & =& {\displaystyle \frac{{\left[d\right]}_2{p}_2}{\left[3\right]{\left[c\right]}_2}}+p_1^2\left\{{\displaystyle \frac{{\left[d\right]}_2}{\left[3\right]\left[2\right]\left[c\right]{\left[d\right]}^2}}-{\displaystyle \frac{\lambda {\left[d\right]}^2}{{\left[c\right]}^2{\left[2\right]}^2}}\right\}.\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill \left|a_3-\lambda a_2^2\right|& =& \left|{\displaystyle \frac{{\left[d\right]}_2{p}_2}{\left[3\right]{\left[c\right]}_2}}-p_1^2\left\{{\displaystyle \frac{\lambda {\left[d\right]}^2}{{\left[c\right]}^2{\left[2\right]}^2}}-{\displaystyle \frac{\left[d+1\right]}{\left[3\right]\left[2\right]\left[c\right]\left[d\right]}}\right\}\right|\hfill \\ & =& {\displaystyle \frac{{\left[d\right]}_2}{\left[3\right]{\left[c\right]}_2}}\left|p_2-\left\{{\displaystyle \frac{\left[3\right]\left[c+1\right]\lambda }{\left[c\right]\left[2\right]\left[d+1\right]}}-{\displaystyle \frac{1}{\left[2\right]{\left[d\right]}^2}}\right\}{p}_1^2\right|.\hfill \end{array}$$

Now, applying Lemma $\left(1\right)$, we can obtain

$$\left|a_3-\lambda a_2^2\right|\le max\left\{1;\left|2{\displaystyle \frac{\left[3\right]\left[c+1\right]\lambda {\left[d\right]}^2-\left[c\right]\left[d+1\right]}{\left[2\right]\left[c\right]\left[d+1\right]{\left[d\right]}^2}}-1\right|\right\}.$$

(19)

which is the requirement. The sharpness of $\left(5\right)$ implies the sharpness of $\left(19\right)$.

Let us suppose $\lambda$ is real:

$$\left|a_3-\lambda a_2^2\right|={\displaystyle \frac{{\left[d\right]}_2}{\left[3\right]{\left[c\right]}_2}}\left|p_2+\left\{{\displaystyle \frac{1}{\left[2\right]{\left[d\right]}^2}}-{\displaystyle \frac{\left[3\right]\left[c+1\right]\lambda }{\left[c\right]\left[2\right]\left[d+1\right]}}\right\}{p}_1^2\right|.$$

Now, using $\left(15\right)$ from Lemma $\left(1\right)$, the bounds for $\left|a_3-\lambda a_2^2\right|$ are obtained: $\left(16\right)$.  □

In the last part of the main results, the necessary and sufficient conditions for function $f\in A$ to be in class $S{T}_q(m,c,d,\beta )$, with reference to Hadamard product, are established.

Theorem 5.

Let $m\in [0,\infty )$ and $\beta \in [0,1).$ Then, a function is in $S{T}_q(m,c,d,\beta )$ if and only if

$${\displaystyle \frac{\left(f\ast H_{q,c,d}\right)\left(\varsigma \right)}{\varsigma }}\ne 0,$$

where,

$$H_{q,c,d}\left(\varsigma \right)={\varphi }_q(1+c,d;\varsigma )\left\{1-\left(1-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right){\displaystyle \frac{\left[c-1\right]+q^{c-1}w\left(t\right)}{q^{c-1}(w\left(t\right)-1)}}\right\},$$

(20)

where, ${\varphi }_q(c,d;\varsigma )=\varsigma +{\sum }_{k=2}^{\infty }{\displaystyle \frac{{\left(\left[c\right]\right)}_{k-1}}{{\left(\left[d\right]\right)}_{k-1}}}{\varsigma }^k,$ $w\left(t\right)=\left(mt+\beta \right)\pm i\sqrt{t^2-{\left(mt+\beta -1\right)}^2}$ and $t^2-{\left(mt+\beta -1\right)}^2\ge 0.$

Proof. 

From $\left(7\right)$, the values of ${\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}$ lie in ${\mathsf{\Omega }}_{m,\beta }.$ Therefore,

$${\displaystyle \frac{\xi D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\ne \left(mt+\beta \right)\pm i\sqrt{t^2-{\left(mt+\beta -1\right)}^2}=w\left(t\right),$$

where $t^2-{\left(mt+\beta -1\right)}^2\ge 0.$

Through the construction of $L_q(c,d)f\left(\varsigma \right)$ and the properties of Hadamard product, the condition $\left(20\right)$ holds if

$$f\left(\varsigma \right)\ast [\varsigma D_q{\varphi }_q(c,d;\varsigma )-w\left(t\right){\varphi }_q(c,d;\varsigma )]/\varsigma \ne 0.$$

(21)

Now, the identity $\left(3\right)$ and $\left(21\right)$ can be used to obtain

$$\begin{array}{c}\varsigma D_q{\varphi }_q(c,d;\varsigma )-w\left(t\right){\varphi }_q(c,d;\varsigma )\\ =\left\{{\displaystyle \frac{\left[c\right]}{q^{c-1}}}{\varphi }_q(1+c,d;\varsigma )-{\displaystyle \frac{\left[c-1\right]}{q^{c-1}}}{\varphi }_q(c,d;\varsigma )\right\}-w\left(t\right){\varphi }_q(c,d;\varsigma )\\ ={\varphi }_q(1+c,d;\varsigma )\left\{{\displaystyle \frac{\left[c\right]}{q^{c-1}}}-{\displaystyle \frac{\left[c-1\right]}{q^{c-1}}}{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}-w\left(t\right){\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right\}\\ ={\varphi }_q(1+c,d;\varsigma )\left[{\displaystyle \frac{\left[c\right]}{q^{c-1}}}-\left\{{\displaystyle \frac{\left[c-1\right]+q^{c-1}w\left(t\right)}{q^{c-1}}}\right\}{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right].\end{array}$$

Now, adding and subtracting ${\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}$, we can obtain the following:

$$\begin{array}{c}\varsigma D_q{\varphi }_q(c,d;\varsigma )-w\left(t\right){\varphi }_q(c,d;\varsigma )\\ ={\varphi }_q(1+c,d;\varsigma )\left[{\displaystyle \frac{\left[c\right]}{q^{c-1}}}-\left(1+{\displaystyle \frac{\left[c\right]}{q^{c-1}}}\right){\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}-\left\{w\left(t\right)-1\right\}{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right].\end{array}$$

As $1+{\displaystyle \frac{\left[c-1\right]}{q^{c-1}}}={\displaystyle \frac{\left[c\right]}{q^c}}$ and using $\left[c\right]+q^{c-1}=\left[c-1\right],$

$$\begin{array}{c}\varsigma D_q{\varphi }_q(c,d;\varsigma )-w\left(t\right){\varphi }_q(c,d;\varsigma )\\ ={\varphi }_q(1+c,d;\varsigma )\left[{\displaystyle \frac{\left[c\right]}{q^{c-1}}}-{\displaystyle \frac{\left[c\right]}{q^{c-1}}}{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}-\left\{w\left(t\right)-1\right\}{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right]\\ ={\varphi }_q(1+c,d;\varsigma )\left[\left(1-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right){\displaystyle \frac{\left[c\right]}{q^{c-1}}}-\left\{w\left(t\right)-1\right\}{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right]\\ =(w\left(t\right)-1){\varphi }_q(1+c,d;\varsigma )\left[\left(1-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right){\displaystyle \frac{\left[c\right]}{q^{c-1}(w\left(t\right)-1)}}-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right]\\ =(w\left(t\right)-1){\varphi }_q(1+c,d;\varsigma )\left[1-\left(1-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right)\left({\displaystyle \frac{\left[c\right]}{q^{c-1}(w\left(t\right)-1)}}+1\right)\right]\\ =(w\left(t\right)-1){\varphi }_q(1+c,d;\varsigma )\left[1-\left(1-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right)\left({\displaystyle \frac{\left[c\right]+q^{c-1}+q^{c-1}w\left(t\right)}{q^{c-1}(w\left(t\right)-1)}}\right)\right]\\ =(w\left(t\right)-1){\varphi }_q(1+c,d;\varsigma )\left[1-\left(1-{\displaystyle \frac{{\varphi }_q(c,d;\varsigma )}{{\varphi }_q(1+c,d;\varsigma )}}\right)\left({\displaystyle \frac{\left[c-1\right]+q^{a-1}w\left(t\right)}{q^{c-1}(w\left(t\right)-1)}}\right)\right].\end{array}$$

Therefore, ($f\ast H_{q,c,d})\left(\varsigma \right)/\varsigma \ne 0$ where $H_{q,c,d}\left(\varsigma \right)$ is given by $\left(20\right)$.

Conversely, $\left(f\ast H_{q,c,d}\right)\left(\varsigma \right)/\varsigma \ne 0$ in $E$, then the values of $\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)/L_q(c,d)f\left(\varsigma \right)$ lie entirely in ${\mathsf{\Omega }}_{m,\beta }$ or its complement, as follows:

$${\left.{\displaystyle \frac{\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)}{L_q(c,d)f\left(\varsigma \right)}}\right|}_{\xi =0}=1\in {\mathsf{\Omega }}_{m,\beta },$$

which implies $\varsigma D_q\left(L_q(c,d)f\left(\varsigma \right)\right)/L_q(c,d)f\left(\varsigma \right)\in {\mathsf{\Omega }}_{m,\beta }$, which concludes that f belongs to the class $S{T}_q(m,c,d,\beta ).$  □

4. Conclusions

In this investigation, we explored a novel class of q-uniformly starlike functions defined by the q-Carlson–Shaffer operator. We derived several significant analytical results, including necessary conditions, coefficient bounds, the Fekete–Szegö inequality, and the criteria established through the Hadamard product. These findings offer both necessary and sufficient conditions for these properties, contributing to a deeper understanding of this class of functions. Future work could focus on extending the results to other q-operators and exploring their applications in geometric function theory Future directions, such as q-convex functions, q-Bazilevic functions, bi-univalent functions, and multivalent functions, as well as exploring their implications for harmonic analysis.