Symmetric Properties of Janowski-Type q-Harmonic Close-to-Convex Functions

Abstract

We introduce and study a new subclass of Janowski-type harmonic close-to-convex functions in the open unit disk defined via the Jackson q-derivative operator. The structure of the operator naturally reflects certain symmetric properties in the analytic representation of the considered harmonic mappings. By applying subordination techniques, we establish sufficient conditions for sense-preserving close-to-convexity and distortion estimates. The extreme points of the class are determined, and its topological properties are examined, showing that the class is convex and compact. We further obtain the radius of starlikeness and prove that the class is closed under convolution. Moreover, as $q\to 1^{-}$, the operator reduces to the classical derivative, and our results recover several known results in the existing literature, demonstrating that the present work extends and generalizes earlier findings.

1. Introduction

A complex valued function, $\varkappa \left(z\right)=u\left(x,y\right)+iv\left(x,y\right)$, is known as harmonic function if both u and v are real harmonic. Denote by H, the class of all harmonic functions in a domain, $\mathbf{U}=\left\{z:z\in \mathbb{C},\left|z\right|<1\right\}$. It is well known that every $\varkappa \in H$ can be uniquely written in the form

$$\varkappa \left(z\right)=h\left(z\right)+\overline{g\left(z\right)},$$

(1)

where

$$\begin{array}{ccc}\hfill h\left(z\right)& =& \sum _{n=0}^{\infty }{a}_n{z}^n,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill g\left(z\right)& =& \sum _{n=1}^{\infty }{b}_n{z}^n.\hfill \end{array}$$

(3)

Let $H_0$ be a subclass of H containing functions $\varkappa \left(z\right)$, satisfying the conditions $\varkappa \left(0\right)=0,{{\varkappa }_z}^{´}\left(0\right)=1,$${{\varkappa }_{\overline{z}}}^{´}\left(0\right)=0$. Therefore, these functions have the representation

$$\varkappa \left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n+\sum _{n=2}^{\infty }\overline{b_n{z}^n}.$$

(4)

The study of harmonic functions has long been a central topic in complex analysis and its applications. These functions exhibit a rich structure that exhibits deep connections to various areas such as potential theory, conformal mapping, and fluid dynamics. Clunie and Sheil Small [1] began studying harmonic functions in 1984. They observed that a harmonic function, $\varkappa \left(z\right)$, in the open unit disk can be expressed as in the representation mentioned in (4). They introduced the notion that this representation serves as a foundation for analyzing harmonic mappings and also worked on the conditions under which these functions are univalent and sense-preserving. The necessary and sufficient condition for $\varkappa \left(z\right)$ to be locally univalent and sense-preserving is

$$|h^{\prime }\left(z\right)|>|g^{\prime }\left(z\right)|,$$

(5)

and provided that analytic univalent functions automatically satisfy these conditions, the presence of the conjugate term in harmonic mappings creates new challenges, making the theory more intricate [2]. Also, Duren [3] made a significant contribution by presenting an extensive study on harmonic mappings in the complex plane; he explored the functions of form (4) and examined the geometric and analytic conditions under which such functions remain univalent and sense-preserving and also provided detailed results concerning coefficient estimates, growth and distortion theorems and geometric subclasses, such as starlike and convex harmonic mappings. Their foundational work continues to serve as a basis for further developments in harmonic function theory. Also, the contribution of Hengartner and Schober plays a crucial role in the analysis of harmonic function; see [4].

Now, we denote by $S_h$ a subclass of $H_0$, which contains univalent functions; the geometric properties of $S_h$ were studied in [1,5,6,7,8]. A domain, $\mathbf{D}\subset \mathbb{C}$, is said to be starlike with respect to the point $z_0$ if the line segment joining $z_0$ to any other other point, z, in $\mathbf{D},$ remains in $\mathbf{D}.$ Then, the set $\mathbf{D}$ is known as starlike. In particular, if $z_0=0$, and $f\left(\mathbf{U}\right)$ is starlike, then a function, $f\in S_h$, is considered harmonic starlike, and a class of such functions is denoted by $S_h^{\ast }$ and is said to be close to convex if $f\left(\mathbf{U}\right)$ is close to convex, and the collection of such functions is denoted by $K_h.$

The derivative for the $S_h$ is given as

$$D_h\varkappa \left(z\right):=z{h}^{\prime }\left(z\right)-\overline{z{g}^{\prime }\left(z\right)},\left(z\in \mathbf{U}\right).$$

(6)

In recent years, q-calculus has attracted considerable attention due to its applications in geometric function theory, as well as in quantum analysis. In particular, the q-derivative operator provides an obvious extension of classical differential operators by specifying the parameter $q\in \left(0,1\right)$, which allows an enhanced influence over the analytic behavior and geometric properties of functions. By definition, in [9], the q-derivative operator for an analytic function, $f\left(z\right)$, is defined as follows:

$$\left(D^q\varkappa \right)\left(z\right)=\left\{\begin{array}{c}{\displaystyle \frac{\varkappa \left(z\right)-\varkappa \left(qz\right)}{\left(1-q\right)z}},z\ne 0,\\ {\varkappa }^{´}\left(0\right),z=0,\end{array}\right.$$

(7)

also ${\lim }_{q\to 1^{-}}{D}^q\varkappa \left(z\right)={\varkappa }^{´}\left(0\right).$ In Maclaurin series form, $D_q\varkappa \left(z\right)$ is defined as

$$D_{q}f\left(z\right)=\sum _{n=1}^{\infty }{\left[n\right]}_q{a}_n{z}^{n-1},$$

(8)

where ${\left[n\right]}_q$ is defined as

$${\left[n\right]}_q=\left\{\begin{array}{c}{\displaystyle \frac{1-q^n}{1-q}}\mathrm{if}n\in \mathbb{C},\\ \sum _{n=0}^{n-1}{q}^n\mathrm{if}n\in \mathbb{N}.\end{array}\right.$$

(9)

Along with the q-derivative operator, another central concept that helps in defining new subclasses is the investigation of various functions associated with the Janowski class. In $1973,$ Janowski [10] made a classical contribution by introducing and studying a family of analytic functions defined through the Janowski functions. These functions are the consequences of subordination over the Möbius transformation, and this idea not only generalized several well-known subclasses of univalent and starlike functions but also provided extremal problems, coefficient bounds, and sharp inequalities. Subsequently, further types of differential and convolution operators, and even new subclasses of analytic and harmonic mappings, have been developed using the concept of Janowski. Recent studies have focused on the investigation of various subclasses of harmonic functions through the use of different analytic operators and techniques. In particular, Dziok [11] examined harmonic functions satisfying a Montel-type normalization and derived several coefficient estimates, together with certain geometric characteristics within the unit disk. In another contribution, Dziok [12] introduced new subclasses of harmonic functions associated with the Carlson–Shaffer operator and analyzed their fundamental analytic properties. Furthermore, Khan et al. [13,14] utilized symmetric q-calculus operators to define additional families of harmonic functions and obtained results concerning coefficient bounds and inclusion relationships among these classes.

To introduce our subclass, we employ the principle of subordination, which say to let $\Omega , \mathrm{\Psi }$ be analytic functions. We say that $\Omega$ is subordinate to $\mathrm{\Psi },$ written as $\Omega \left(z\right)\prec \mathrm{\Psi }\left(z\right)$, if there exists a Schwarz function, $\varpi \left(z\right)$, satisfying $\varpi \left(0\right)=0,\left|\varpi \left(z\right)\right|<1$ for all z in $\mathbf{U}$, such that $\Omega \left(z\right)$ can be expressed as $\Omega \left(z\right)=\mathrm{\Psi }\left(\varpi \left(z\right)\right),z\in \mathbf{U}$; for more details, see [15].

Consider the function

$${\varkappa }_m\left(z\right)=z+\sum _{n=2}^{\infty }\left(a_{m,n}{z}^n+\overline{b_{m,n}{z}^n}\right),\left(z\in \mathbf{U},m=1,2,\dots \right)$$

and the convolution of ${\varkappa }_1$, ${\varkappa }_2$ is defined as

$$\left({\varkappa }_1\ast {\varkappa }_2\right)\left(z\right)=z+\sum _{n=2}^{\infty }\left(a_{1,n}{a}_{2,n}{z}^n+\overline{b_{1,n}{b}_{2,n}{z}^n}\right),\left(z\in \mathbf{U}\right).$$

A harmonic function, $\varkappa \left(z\right)=h\left(z\right)+\overline{g\left(z\right)}$, defined by (4), is said to be q-harmonic, locally univalent and sense-preserving in $\mathbf{U}$, if and only if the second dilatation $w_q$ satisfies the condition

$$\left|w_q\left(z\right)\right|=\left|{\displaystyle \frac{D^{q}g\left(z\right)}{D^{q}h\left(z\right)}}\right|<1,$$

(10)

where $q\in \left(0,1\right)$ and $z\in \mathbf{U}$; for det details, see [16]. The class $S_h$ with q-harmonic functions is denoted by $S_h^q$.

Recent work by Arif et al. [17] on Janowski close-to-convex harmonic mappings provides important geometric properties and coefficient bounds for such classes. Inspired by this, we introduce a q analogue of close-to-convex functions associated with Janowski functions [10], which extends the classical theory and yields new results via q-calculus. We now define a new subclass, $K_h^q(\zeta ;$$E,$$F)$, of the class $S_h$, which fulfills

$${\displaystyle \frac{D_h^q\varkappa \left(z\right)}{\zeta \left(z\right)}}\prec {\displaystyle \frac{1+Ez}{1+Fz}},0\le E<F\le 1,$$

(11)

where $\varkappa \left(z\right)\in S_h^q$ and $\zeta \left(z\right)\in S_h^{\ast }(E,F)$.

From (6) and (8), we define

$$D_h^q\varkappa \left(z\right)=z{D}^{q}h\left(z\right)-\overline{z{D}^{q}g\left(z\right)},$$

(12)

where

$$\begin{array}{ccc}\hfill D^{q}h\left(z\right)& =& 1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{z}^{n-1},\hfill \\ \hfill D^{q}g\left(z\right)& =& \sum _{n=2}^{\infty }{\left[n\right]}_q{b}_n{z}^{n-1}.\hfill \end{array}$$

(13)

From Equation (12), it follows that

$$D_h^q\varkappa \left(z\right)=z+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{z}^n-\sum _{n=2}^{\infty }{\left[n\right]}_q\overline{b_n{z}^n}.$$

(14)

We aim to explore the subclass $K_h^q(\zeta ;$$E,$$F)$ by investigating certain geometric properties, which are included in the Main Results section below. This study develops a new family of harmonic mappings by incorporating a q-deformation into the framework of close-to-convex functions associated with prescribed parameter conditions. The analysis combines coefficient techniques with operator-based arguments in order to obtain sufficient conditions for univalence and geometric behavior in the unit disk. Particular attention is given to how the presence of the q-parameter modifies classical bounds and how the established results reduce to known cases as the parameter approaches unity. The obtained inequalities provide a clear description of the structural constraints governing the analytic and co-analytic components of the mappings. In addition, the extremal structure and convex characteristics of the class are examined, offering insight into the sharpness of the derived estimates. Overall, the results contribute to the ongoing development of harmonic function theory by presenting a coherent extension that connects discrete operator methods with geometric function properties. In recent years, significant attention has been given to the study of harmonic and Janowski-type functions due to their geometric properties and applications in complex analysis. Mahmood et al. [18] investigated Janowski-type close-to-convex functions associated with conic regions, providing important results on their geometric characterization. Polatoğlu et al. [19] studied harmonic mappings for which the co-analytic part is a close-to-convex function of order b, highlighting the conditions under which these mappings preserve close-to-convexity. More recently, Arif et al. [20] introduced some Janowski-type harmonic q-starlike functions associated with symmetrical points, extending the study of harmonic mappings to the q-calculus framework and exploring their starlike properties. Collectively, these studies provide a strong foundation for further investigation into the geometric behavior of harmonic and Janowski-type functions and motivate the results presented in the current work.

2. Necessary and Sufficient Condition

Lemma 1

([21]). A function,

$$\zeta \left(z\right)=z+\sum _{n=2}^{\infty }{c}_n{z}^n+\sum _{n=2}^{\infty }{d}_n\overline{z^n}\in S_h^{\ast }(E,F)$$

(15)

if

$$\sum _{n=2}^{\infty }\left[\left(n(1+F)-(1+E)\right)|c_n|+\left(n(1+F)+(1+E)\right)\left|d_n\right|\right]\le F-E$$

(16)

holds.

Remark 2.

Since

$$\left.\begin{array}{c}1+E\le n\left(1+F\right)-\left(1+E\right),\\ 1+E\le n\left(1+F\right)+\left(1+E\right)\end{array}\right\}.$$

(17)

Using (17) in the expression (16), we deduce that

$$\mathrm{\Gamma }:=\sum _{n=2}^{\infty }\left(1+E\right)\left(|c_n|+|d_n|\right)\le F-E.$$

(18)

Theorem 3.

If the function $\varkappa \left(z\right)\in S_h$ possesses the form (4) and satisfies the condition

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}},$$

(19)

then $\varkappa \left(z\right)\in K_h^q(\zeta ;$$E,$$F)$.

Proof. 

Clearly,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\left[n\right]}_q\left(F+1\right)}{F-E-\mathrm{\Gamma }}}& =& {\displaystyle \frac{{\left[n\right]}_q\left(F-E-\mathrm{\Gamma }\right)+{\left[n\right]}_q+{\left[n\right]}_q\left(E+\mathrm{\Gamma }\right)}{F-E-\mathrm{\Gamma }}}\hfill \\ & =& {\left[n\right]}_q+{\displaystyle \frac{{\left[n\right]}_q\left(1+E+\mathrm{\Gamma }\right)}{F-E-\mathrm{\Gamma }}}\hfill \end{array}$$

Since $E,F>0,$ from (18) $\mathrm{\Gamma }>0$, and so $\left(1+E+\mathrm{\Gamma }\right)>0$, and also from (18), it is obvious that $\left(F-E-\mathrm{\Gamma }\right)>0.$

Hence,

$${\displaystyle \frac{{\left[n\right]}_q\left(F+1\right)}{F-E-\mathrm{\Gamma }}}\ge {\left[n\right]}_q,$$

which implies that

$${\displaystyle \frac{\left(F+1\right)}{F-E-\mathrm{\Gamma }}}\ge 1,{\left[n\right]}_q>1,$$

or

$${\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{\left(F+1\right)}}\le 1,$$

(20)

$$\left(F-E-\mathrm{\Gamma }\right)>0,1+F>0.$$

(21)

It is obvious that, from (20) and (21), it is concluded that

$$0<{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{\left(F+1\right)}}\le 1.$$

(22)

From (19) and (22), we have

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)\le 1.$$

(23)

Since $\varkappa \left(z\right)\in S_h$, we have to show that $\varkappa \left(z\right)\in K_h^q(\zeta ;$$E,$$F)$; firstly, we need to prove that $\varkappa \left(z\right)\in S_h^q.$ From the definition (10), a function, $\varkappa \left(z\right)\in S_h^q$, is obtained if it satisfies $\left|{\displaystyle \frac{D^{q}g\left(z\right)}{D^{q}h\left(z\right)}}\right|<1,$ or equivalently, we have to show that $\left|D^{q}h\left(z\right)\right|-\left|D^{q}g\left(z\right)\right|>0.$ Using (13), we have

$$\begin{array}{ccc}\hfill \left|D^{q}h\left(z\right)\right|-\left|D^{q}g\left(z\right)\right|& \ge & 1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|\left|z^{n-1}\right|-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|\left|z^{n-1}\right|\hfill \\ & =& 1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left(\left|a_n\right|+\left|b_n\right|\right)\left|z^{n-1}\right|\hfill \\ & =& 1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left(\left|a_n\right|+\left|b_n\right|\right)r^{n-1}.\hfill \end{array}$$

Since $0<r<1,r^{n-1}<1,$ it is implied that

$$\left|D^{q}h\left(z\right)\right|-\left|D^{q}g\left(z\right)\right|\ge 1-\sum _{n=2}^{\infty }{\left[n\right]}_q\left(\left|a_n\right|+\left|b_n\right|\right).$$

Using (23), we get

$$\left|D^{q}h\left(z\right)\right|-\left|D^{q}g\left(z\right)\right|\ge 0,$$

which implies that $\varkappa \left(z\right)\in S_h^q.$ From the expression (11), a function, $\varkappa \left(z\right)\in K_h^q(\zeta ;$$E,$$F)$, is obtained if

$${\displaystyle \frac{D_h^q\varkappa \left(z\right)}{\zeta \left(z\right)}}\prec {\displaystyle \frac{1+Ez}{1+Fz}},0\le E<F\le 1;$$

equivalently,

$$\left|{\displaystyle \frac{D_h^q\varkappa \left(z\right)-\zeta \left(z\right)}{F\left(D_h^q\varkappa \left(z\right)\right)-E\zeta \left(z\right)}}\right|<1.$$

(24)

Thus, in order to show that $\varkappa \left(z\right)\in K_h^q(\zeta ;$ $E,$$F)$, we must demonstrate that

$$\left|D_h^q\varkappa \left(z\right)-\zeta \left(z\right)\right|-\left|F\left(D_h^q\varkappa \left(z\right)\right)-E\zeta \left(z\right)\right|<0.$$

(25)

Using (14) and (15) in (25) and using $\left|a+b\right|\le \left|a\right|+\left|b\right|$ and $-\left|c+d+e\right|\le -\left|c\right|+\left|d\right|+\left|e\right|,$ we get

$$\begin{array}{cc}\hfill & \left|D_h^q\varkappa \left(z\right)-\zeta \left(z\right)\right|-\left|F\left(D_h^q\varkappa \left(z\right)\right)-E\zeta \left(z\right)\right|\hfill \\ \hfill & \le \left|\sum _{n=2}^{\infty }\left({\left[n\right]}_q{a}_n-c_n\right)z^n\right|+\left|\sum _{n=2}^{\infty }\left({\left[n\right]}_q\overline{b_n}+d_n\right)\overline{z^n}\right|-\left|(F-E)z\right|\hfill \\ \hfill & +\left|\sum _{n=2}^{\infty }\left(F{\left[n\right]}_q{a}_n-E{c}_n\right)z^n\right|+\left|\sum _{n=2}^{\infty }\left(F{\left[n\right]}_q\overline{b_n}+E{d}_n\right)\overline{z^n}\right|\hfill \\ \hfill & \le \sum _{n=2}^{\infty }\left|{\left[n\right]}_q{a}_n-c_n\right||z^n|+\sum _{n=2}^{\infty }\left|{\left[n\right]}_q\overline{b_n}+d_n\right||\overline{z^n}|-\left|F-E\right|\left|z\right|\hfill \\ \hfill & +\sum _{n=2}^{\infty }\left|\left(F{\left[n\right]}_q{a}_n-E{c}_n\right)\right||z^n|+\sum _{n=2}^{\infty }\left|\left(F{\left[n\right]}_q\overline{b_n}+E{d}_n\right)\right||\overline{z^n}|\hfill \\ \hfill & \le \sum _{n=2}^{\infty }\left({\left[n\right]}_q\left|a_n\right|+\left|c_n\right|\right)|z^n|+\sum _{n=2}^{\infty }\left({\left[n\right]}_q\left|\overline{b_n}\right|+\left|d_n\right|\right)|\overline{z^n}|-\left|F-E\right|\left|z\right|\hfill \\ \hfill & +\sum _{n=2}^{\infty }\left(F{\left[n\right]}_q\left|a_n\right|+E\left|c_n\right|\right)|z^n|+\sum _{n=2}^{\infty }\left(F{\left[n\right]}_q\left|\overline{b_n}\right|+E\left|d_n\right|\right)\left|\overline{z^n}\right|\hfill \\ \hfill & \le r\left\{\sum _{n=2}^{\infty }{\left[n\right]}_q\left(F+1\right)\left(\left|a_n\right|+\left|b_n\right|\right)+\sum _{n=2}^{\infty }\left(1+E\right)\left(\left|c_n\right|+\left|d_n\right|\right)-\left(F-E\right)\right\}.\hfill \end{array}$$

With the help of remark 2, we deduce that

$$\begin{array}{cc}\hfill & \left|D_h\varkappa \left(z\right)-\zeta \left(z\right)\right|-\left|F{D}_h\varkappa \left(z\right)-E\zeta \left(z\right)\right|\hfill \\ \hfill & \le r\left\{\sum _{n=2}^{\infty }{\left[n\right]}_q(F+1)\left(|a_n|+|b_n|\right)+\mathrm{\Gamma }-(F-E)\right\}\hfill \\ \hfill & =r(F+1)\left\{\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)-{\displaystyle \frac{F-E-\mathrm{\Gamma }}{F+1}}\right\}.\hfill \end{array}$$

Using (19), we get

$$\left|D_h^q\varkappa \left(z\right)-\zeta \left(z\right)\right|-\left|F{D}_h^q\varkappa \left(z\right)-E\zeta \left(z\right)\right|\le 0.$$

Hence, the condition (25) is proven, which results in $\varkappa \left(z\right)\in K_h^q(\zeta ;$$E,$$F).\hspace{1em}\square$

Taking $q\to 1^{-},$ we get the following result, which is given in [22].

Corollary 4.

Let $\varkappa \left(z\right)\in S_h$ be a function expressed as in (4) and fulfilling the condition

$$\sum _{n=2}^{\infty }n\left(|a_n|+|b_n|\right)\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}},$$

(26)

and then, $\varkappa \left(z\right)\in K_h(\zeta ;$$E,$$F)$.

Silverman [21] introduced the class G of functions $\varkappa \left(z\right)\in S_h$ of the form (4), such that $a_n=-|a_n|,b_n=\left|b_n\right|,$ that is

$$\varkappa \left(z\right)=z-\sum _{n=2}^{\infty }\left|a_n\right|z^n+\sum _{n=2}^{\infty }\left|b_n\right|\overline{z^n},$$

(27)

and furthermore, let $K_G^q(k;$$E,$$F):=G\cap K_h^q(\zeta ;$$E,$$F)$. The sufficient coefficient bound given in Theorem 3 is also necessary for function $\varkappa \left(z\right)\in G$ to be in the class $K_G^q(\zeta ;$$E,$$F)$, which is clarified in the next theorem.

Theorem 5.

If $\varkappa \left(z\right)\in G$ is of form (27), then $\varkappa \in K_G^q(k;$$E,$$F)$ if and only if condition (19) holds.

Proof. 

From the perspective of Theorem 3, we only need to show that, if $\varkappa \in K_G^q(\zeta ; E, F)$, then condition (19) holds.

For this, let $\varkappa \in K_G^q(\zeta ; E, F)$, and then, through (24), we get

$$\left|{\displaystyle \frac{D_h^q\varkappa \left(z\right)-\zeta \left(z\right)}{F{D}_h^q\varkappa \left(z\right)-E\zeta \left(z\right)}}\right|<1.$$

(28)

From (27) and (15), we have

$$\left|{\displaystyle \frac{-{\sum }_{n=2}^{\infty }\left({\left[n\right]}_q\left|a_n\right|+c_n\right)z^n-{\sum }_{n=2}^{\infty }\left({\left[n\right]}_q\left|b_n\right|+d_n\right)\overline{z^n}}{(F-E)z-{\sum }_{n=2}^{\infty }\left(F{\left[n\right]}_q\left|a_n\right|+E{c}_n\right)z^n-{\sum }_{n=2}^{\infty }\left(F{\left[n\right]}_q\left|b_n\right|+E{d}_n\right)\overline{z^n}}}\right|<1.$$

By inputting $z=r$, $0<r<1$, we obtain

$$\left|{\displaystyle \frac{-{\sum }_{n=2}^{\infty }\left({\left[n\right]}_q\left|a_n\right|+c_n\right)r^{n-1}-{\sum }_{n=2}^{\infty }\left({\left[n\right]}_q\left|b_n\right|+d_n\right)r^{n-1}}{(F-E)-{\sum }_{n=2}^{\infty }\left(F{\left[n\right]}_q\left|a_n\right|+E{c}_n\right)r^{n-1}-{\sum }_{n=2}^{\infty }\left(F{\left[n\right]}_q\left|b_n\right|+E{d}_n\right)r^{n-1}}}\right|<1.$$

Since the denominator is positive, we get

$$\begin{array}{c}\sum _{n=2}^{\infty }\left({\left[n\right]}_q\left|a_n\right|+c_n\right)r^{n-1}+\sum _{n=2}^{\infty }\left({\left[n\right]}_q\left|b_n\right|+d_n\right)r^{n-1}<(F-E)-\sum _{n=2}^{\infty }\left(F{\left[n\right]}_q\left|a_n\right|+E{c}_n\right)r^{n-1}-\sum _{n=2}^{\infty }\left(F{\left[n\right]}_q\left|b_n\right|+E{d}_n\right)r^{n-1}.\end{array}$$

Upon simplification,

$$\begin{array}{c}\sum _{n=2}^{\infty }(1+F){\left[n\right]}_q\left(\right|a_n|+|b_n\left|\right)r^{n-1}+\sum _{n=2}^{\infty }(1+E)(c_n+d_n)r^{n-1}<(F-E),\end{array}$$

and let $r\to 1^{-}$,

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left(\right|a_n|+|b_n\left|\right)<{\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}.\hspace{1em}\square$$

Taking $q\to 1^{-},$ we get the following result, which is given in [22].

Corollary 6.

If $\varkappa \left(z\right)\in G$ be of form (27), then $\varkappa \in K_G(\zeta ;$$E,$$F)$ if and only if condition (19) holds.

3. Distortion

Theorem 7.

If $\varkappa \in K_G^q(\zeta ;$$E,$$F)$, then

$$r-{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(F+1\right)}}{r}^2\le \left|D_h^q\varkappa \left(z\right)\right|\le r+{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(F+1\right)}}{r}^2.$$

Proof. 

Consider

$$\begin{array}{cc}\hfill \left|D_{\mathfrak{h}}^q\varkappa \left(z\right)\right|& \le \left|z\right|+\sum _{n=2}^{\infty }\left|{\left[n\right]}_q\left|a_n\right|z^n\right|+\sum _{n=2}^{\infty }\left|{\left[n\right]}_q\left|b_n\right|\overline{z^n}\right|\hfill \\ \hfill & =\left|z\right|+\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|\left|z^n\right|+\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|\left|\overline{z^n}\right|\hfill \\ \hfill & =r+\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|r^n+\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|r^n\hfill \\ \hfill & =r+\sum _{n=2}^{\infty }{\left[n\right]}_q\left(\left|a_n\right|+\left|b_n\right|\right)r^n.\hfill \end{array}$$

Using Theorem 5,

$$\left|D_{\mathfrak{h}}^q\varkappa \left(z\right)\right|\le r+{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(1+F\right)}}{r}^n.$$

For $0<r<1,$ we have $r^n\le r^2,$$n=2,3,4\cdots .$

$$\left|D_{\mathfrak{h}}^q\varkappa \left(z\right)\right|\le r+{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(1+F\right)}}{r}^2.$$

Now, consider

$$\begin{array}{cc}\hfill \left|D_{\mathfrak{h}}^q\varkappa \left(z\right)\right|& \ge \left|z\right|-\left|-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|z^n\right|-\left|-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|\overline{z^n}\right|\hfill \\ \hfill & =\left|z\right|-\left|\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|z^n\right|-\left|\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|\overline{z^n}\right|\hfill \\ \hfill & =\left|z\right|-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|\left|z^n\right|-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|\left|\overline{z^n}\right|\hfill \\ \hfill & =r-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|a_n\right|r^n-\sum _{n=2}^{\infty }{\left[n\right]}_q\left|b_n\right|r^n\hfill \\ \hfill & =r-\sum _{n=2}^{\infty }{\left[n\right]}_q\left(\left|a_n\right|+\left|b_n\right|\right)r^n.\hfill \end{array}$$

Using Theorem 5,

$$\left|D_{\mathfrak{h}}^q\varkappa \left(z\right)\right|\ge r-{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(1+F\right)}}{r}^n.$$

For $0<r<1,$ we have $-r^n\ge -r^2,$$n=2,3,4\cdots .$

$$\left|D_{\mathfrak{h}}^q\varkappa \left(z\right)\right|\ge r-{\displaystyle \frac{F-(E+\mathrm{\Gamma })}{1+F}}{r}^2.\hspace{1em}\square$$

Taking $q\to 1^{-},$ we get the following result, which is given in [22].

Corollary 8.

If $\varkappa \in K_G(\zeta ;$$E,$$F)$, then

$$r-{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(F+1\right)}}{r}^2\le \left|D_h\varkappa \left(z\right)\right|\le r+{\displaystyle \frac{F-\left(E+\mathrm{\Gamma }\right)}{\left(F+1\right)}}{r}^2.$$

4. Topological Properties

The standard topology on H is established by a metric, such that a sequence, ${\varkappa }_n$ in H, converges to $\varkappa$ if and only if it converges uniformly on each compact subset of $\mathbf{U}$. This topological space is complete, as stated by Weierstrass and Montel; see [23]. For $\mathbf{B}\subset H$, if

$$\varkappa =\gamma {\varkappa }_1\left(z\right)+(1-\gamma ){\varkappa }_2\left(z\right),\left({\varkappa }_1,{\varkappa }_2\in \mathbf{B},0<\gamma <1\right)$$

implies that ${\varkappa }_1={\varkappa }_2=\varkappa$ holds, then a function, $\varkappa$ in $\mathbf{B}$, is termed an extreme point of $\mathbf{B}$, where ${\mathrm{E}}_{\mathbf{B}}$ represents the set of all extreme points.

Moreover, it is clear that every $\varkappa$ in $\mathbf{B}$ is locally uniformly bounded. Specifically, there exists a positive constant, $L:=L\left(r\right)$, corresponding to each $0<r<1$, such that

$$\left|\varkappa \left(z\right)\right|\le L,\left(\varkappa \in \mathbf{B},\left|z\right|\le r\right).$$

(29)

The class $\mathbf{B}$ is said to be convex if

$$\gamma {\varkappa }_1\left(z\right)+(1-\gamma ){\varkappa }_2\left(z\right)\in \mathbf{B},\left({\varkappa }_1,{\varkappa }_2\in \mathbf{B},0\le \gamma \le 1\right).$$

(30)

The closed convex hull of $\mathbf{B}\subset H$, represented by $\overline{\mathrm{co}}\mathbf{B}$, is the smallest closed convex set in H that contains $\mathbf{B}$.

Moreover, a functional $\mathsf{F}:H\to \mathbb{R}$ is convex on class $\mathbf{B}$ if

$$\mathsf{F}\left(\gamma {\varkappa }_1+(1-\gamma ){\varkappa }_2\right)\le \gamma \mathsf{F}\left({\varkappa }_1\right)+(1-\gamma )\mathsf{F}\left({\varkappa }_2\right),\left({\varkappa }_1,{\varkappa }_2\in \mathbf{B},0\le \gamma \le 1\right).$$

The Krein–Milman [5] theorem plays a central role in the study of extreme points and leads directly to the following results.

Lemma 9.

If a class, $\mathbf{B}\subset \mathbf{H}$, is a nonempty compact set, then ${\mathrm{E}}_{\mathbf{B}}$ is nonempty, and $\overline{co}\mathbf{B}=\overline{co}$${\mathrm{E}}_{\mathbf{B}}$.

The following result is derived through Montel’s theorem [23].

Lemma 10.

If a class, $\mathbf{B}\subset \mathbf{H}$, is locally uniformly bounded and closed, then it is a compact.

Theorem 11.

The class $K_G^q(\zeta ;$$E,$$F)$ is convex and a compact subclass of H.

Proof. 

Let $0\le \gamma \le 1$ and ${\varkappa }_1,{\varkappa }_2\in K_G^q(\zeta ;$$E,$$F),$ where

$${\varkappa }_m\left(z\right)=z+\sum _{n=2}^{\infty }\left(-|a_{m,n}|z^n+\left|b_{m,n}\right|\overline{z^n}\right),z\in \mathbf{U},m=1,2,\dots$$

(31)

Consider

$$\begin{array}{ccc}\hfill \gamma {\varkappa }_1\left(z\right)+\left(1-\gamma \right){\varkappa }_2\left(z\right)& =& \gamma \left\{z+\sum _{n=2}^{\infty }\left(-\left|a_{1,n}\right|z^n+\left|b_{1,n}\right|\overline{z^n}\right)\right\}\hfill \\ & & +\left(1-\gamma \right)\left\{z+\sum _{n=2}^{\infty }\left(-\left|a_{2,n}\right|z^n+\left|b_{2,n}\right|\overline{z^n}\right)\right\}\hfill \\ & =& z+\sum _{n=2}^{\infty }\left\{\begin{array}{c}\left(-\gamma \left|a_{1,n}\right|-\left(1-\gamma \right)\left|a_{2,n}\right|\right)z^n\\ +\left(\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right)\overline{z^n}\end{array}\right\}\hfill \\ & =& z+\sum _{n=2}^{\infty }\left\{\begin{array}{c}-\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right)z^n\\ +\left(\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right)\overline{z^n}\end{array}\right\}.\hfill \end{array}$$

We need to show that $K_G^q(\zeta ;$$E,$$F)$ is convex. From expression (30), we need to show that $\gamma {\varkappa }_1\left(z\right)+\left(1-\gamma \right){\varkappa }_2\left(z\right)\in K_G^q(\zeta ;$$E,$$F)$, and so, by Theorem 5, $\gamma {\varkappa }_1\left(z\right)+\left(1-\gamma \right){\varkappa }_2\left(z\right)\in K_G^q(\zeta ;$$E,$$F)$ if

$$\begin{array}{c}\hfill \sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left|-\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right)\right|+\left|\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right|\right\}\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{(1+F)}}.\end{array}$$

For this, consider

$$\begin{array}{c}\sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left|-\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right)\right|+\left|\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right|\right\}\hfill \\ \hspace{1em}\hspace{1em}=\sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left|\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right)\right|+\left|\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right|\right\}\\ \hspace{1em} \le \sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left(\left|\gamma \left|a_{1,n}\right|\right|+\left|\left(1-\gamma \right)\left|a_{2,n}\right|\right|\right)+\left|\gamma \left|b_{1,n}\right|\right|+\left|\left(1-\gamma \right)\left|b_{2,n}\right|\right|\right\}\\ \hspace{1em}\hspace{1em} \le \sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|+\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right\}\\ \hspace{1em}\hspace{1em} =\sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\gamma \left(\left|a_{1,n}\right|+\left|b_{1,n}\right|\right)+\left(1-\gamma \right)\left(\left|a_{2,n}\right|+\left|b_{2,n}\right|\right)\right\}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\sum _{n=2}^{\infty }\gamma {\left[n\right]}_q\left\{\left(\left|a_{1,n}\right|+\left|b_{1,n}\right|\right)\right\}+\left(1-\gamma \right){\left[n\right]}_q\left\{\left(\left|a_{2,n}\right|+\left|b_{2,n}\right|\right)\right\}.\end{array}$$

Since ${\varkappa }_1,{\varkappa }_2\in K_G^q(\zeta ;$$E,$$F),$ Theorem 5 implies that

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left(\left|a_{1,n}\right|+\left|b_{1,n}\right|\right)\right\}\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{(1+F)}},\sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left(\left|a_{2,n}\right|+\left|b_{2,n}\right|\right)\right\}\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{(1+F)}},$$

which implies that

$$\begin{array}{ccc}\hfill \sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left|-\left(\gamma \left|a_{1,n}\right|+\left(1-\gamma \right)\left|a_{2,n}\right|\right)\right|+\left|\gamma \left|b_{1,n}\right|+\left(1-\gamma \right)\left|b_{2,n}\right|\right|\right\}& \le & \gamma \left({\displaystyle \frac{F-E-\mathrm{\Gamma }}{(1+F)}}\right)\hfill \\ & & +\left(1-\gamma \right)\left({\displaystyle \frac{F-E-\mathrm{\Gamma }}{(1+F)}}\right)\hfill \\ & =& {\displaystyle \frac{F-E-\mathrm{\Gamma }}{(1+F)}}.\hfill \end{array}$$

Therefore, the function $\gamma {\varkappa }_1\left(z\right)+\left(1-\gamma \right){\varkappa }_2\left(z\right)\in K_G^q(k;$$E,$$F)$ implies that this class is convex.

Let $\varkappa \in$$K_G^q(\zeta ;$$E,$$F)$ of form (27),$\left|z\right|=r<1,$ and consider

$$\begin{array}{ccc}\hfill \left|\varkappa \right(z\left)\right|& =& \left|z+\sum _{n=2}^{\infty }\left(-\left|a_n\right|z^n+\left|b_n\right|\overline{z^n}\right)\right|\hfill \\ & \le & r+\sum _{n=2}^{\infty }\left(|a_n|+|b_n|\right)r^n\hfill \\ & \le & r+\sum _{n=2}^{\infty }\left(|a_n|+|b_n|\right)\hfill \\ & \le & r+\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)\hfill \\ & \le & r+{\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}:=L\hfill \end{array}$$

From the expression (29), we found that the class $K_G^q(\zeta ;$$E,$$F)$ is locally uniformly bounded. Now, we have to show that this class is closed. For closeness, if ${\varkappa }_m\in K_G^q(\zeta ;$$E,$$F)$ and ${\varkappa }_m$→$\varkappa ,$ then $\varkappa \in K_G^q(\zeta ;$$E,$$F).$ Assume that ${\varkappa }_m$ and $\varkappa$ are given by (31) and (34), respectively. According to Theorem 5, we can write

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_{m,n}|+|b_{m,n}|\right)\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}.$$

(32)

Since ${\varkappa }_m$→$\varkappa$, it follows that $a_{m,n}$ $\to a_n$ and $b_{m,n}$$\to b_n$ as $n\to \infty (n\in \mathbf{N}).$ The sequence of partial sums $\left\{S_n\right\}$, related to the series ${\sum }_{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)$, is a nondecreasing sequence; also, according to the expression (32), it is bounded by ${\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}$. Therefore, the sequence $\left\{S_n\right\}$ is convergent, and ${\sum }_{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)={lim}_{n\to \infty }{S}_n\le$${\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}$. This gives the condition (19), and in consequence, $\varkappa \in$$K_G^q(\zeta ;$$E,$$F)$, which implies that the class $K_G^q(\zeta ;$ $E,$$F)$ is closed. From Lemma 10, the class $K_G^q(\zeta ;$$E,$$F)$ is compact. Hence, the theorem is proven. □

Theorem 12.

The collection of the extreme point of the class $K_G^q(\zeta ;$$E,$$F)$ is given as

$$E_{K_G^q(\zeta ;E,F)}=\left\{h_n:n\in \mathbb{N}\right\}\cup \left\{g_n:n\in \left\{2,3\dots \right\}\right\},$$

where

$$h_1=z,h_n=z-{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}{z}^n;g_n=z+{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\overline{z^n},n=2,3\dots .$$

(33)

Proof. 

Assume that $0<\gamma <1$ and

$$h_n=\gamma {\varkappa }_1\left(z\right)+\left(1-\gamma \right){\varkappa }_2\left(z\right),$$

where ${\varkappa }_1,{\varkappa }_2\in K_G^q(\zeta ; E, F)$ of form (31). From expression (19)$,$ we have

$$\left|b_{1,n}\right|=\left|b_{2,n}\right|={\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}},$$

as a result $a_{1,m}=a_{2,m}=0$ for $m\in \left\{2,3\dots \right\}$ and $b_{1,m}=b_{2,m}=0$ for $m\in \left\{2,3\dots \right\}\setminus \left\{n\right\},$ and this implies that $h_n={\varkappa }_1={\varkappa }_2$, and in consequence, $h_n\in E_{K_G^q(\zeta ;E,F)}.$ Similarly, we show that $g_n\in E_{K_G^q(\zeta ;E,F)}.$ Now, let $\varkappa \in E_{K_G^q(\zeta ;E,F)}$ be not of the form (33), and then there exists $m\in \left\{2,3\dots \right\}$, such that

$$0<\left|a_m\right|<{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[m\right]}_q\left(F+1\right)}},$$

or

$$0<\left|b_m\right|<{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[m\right]}_q\left(F+1\right)}}.$$

If $0<\left|a_m\right|<{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[m\right]}_q\left(F+1\right)}},$ then putting

$$\gamma ={\displaystyle \frac{\left|a_m\right|{\left[m\right]}_q\left(F+1\right)}{F-E-\mathrm{\Gamma }}},\mathrm{{\rm Y}}={\displaystyle \frac{1}{1-\gamma }}\left(\varkappa -\gamma h_n\right),$$

where $0<\gamma <1$ and $h_n,\mathrm{{\rm Y}}\in K_G^q(\zeta ; E, F)$ with $h_n\ne \mathrm{{\rm Y}}$ and

$$\varkappa =\gamma h_n+\left(1-\gamma \right)\mathrm{{\rm Y}}.$$

It implies that $\varkappa \notin E_{K_G^q(\zeta ;E,F)}.$ Likewise, if $0<\left|b_m\right|<{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[m\right]}_q\left(F+1\right)}},$ then putting

$$\gamma ={\displaystyle \frac{\left|b_m\right|{\left[m\right]}_q\left(F+1\right)}{F-E-\mathrm{\Gamma }}},\mathrm{\Phi }={\displaystyle \frac{1}{1-\gamma }}\left(\varkappa -\gamma g_n\right),$$

where $0<\gamma <1$ and $g_n,\mathrm{{\rm Y}}\in K_G^q(\zeta ; E, F)$ with $g_n\ne \mathrm{\Phi }$ and

$$\varkappa =\gamma g_n+\left(1-\gamma \right)\mathrm{\Phi }.$$

It implies that $\varkappa \notin E_{K_G^q(\zeta ;E,F)}.$ Hence, the result is complete. □

We see that the class $\mathbf{B}=\left\{{\varkappa }_n\in \mathbf{H}:n\in \mathbb{N}\right\}$ is locally uniformly bounded, and then

$$\overline{co}\mathbf{B}=\left\{\sum _{n=1}^{\infty }{\mu }_n{\varkappa }_n:\sum _{n=1}^{\infty }{\mu }_n=1,{\mu }_n\ge 0,n\in \mathbb{N}\right\}.$$

Now, we have following result deduced from Theorems 11 and 12.

Theorem 13.

A function, $\varkappa \left(z\right)\in$$K_G^q(\zeta ;$$E,$$F)$, is obtained if and only if

$$\varkappa \left(z\right)=\sum _{n=1}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right),$$

where

$$\begin{array}{c}\hfill \left.\begin{array}{c}{h}_1=z,g_1=z,\\ h_n=z-{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}{z}^n,g_n=z+{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\overline{z^n},n=2,3\dots \\ \sum _{n=1}^{\infty }\left({\mu }_n+{\nu }_n\right)=1,{\mu }_n,{\nu }_n\ge 0.\end{array}\right\}.\end{array}$$

(34)

Proof. 

Let

$$\begin{array}{ccc}\hfill \varkappa \left(z\right)& =& \sum _{n=1}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right)\hfill \\ & =& {\mu }_1{h}_1+{\nu }_1{g}_1+\sum _{n=2}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right)\hfill \\ & =& {\mu }_{1}z+{\nu }_{1}z+\sum _{n=2}^{\infty }{\mu }_n\left(z-{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}{z}^n\right)+\sum _{n=2}^{\infty }{\nu }_n\left(z+{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\overline{z^n}\right)\hfill \\ & =& \sum _{n=1}^{\infty }\left({\mu }_n+{\nu }_n\right)z+\sum _{n=2}^{\infty }\left(-{\mu }_n{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\right)z^n+\sum _{n=2}^{\infty }\left({\nu }_n{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\right)\overline{z^n}.\hfill \end{array}$$

We need to show that $\varkappa \left(z\right)\in K_G^q(\zeta ;$$E,$$F)$. For this, we show that

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left\{\left|-{\mu }_n{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\right|+\left|{\nu }_n{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\right|\right\}\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{\left(F+1\right)}}.$$

(35)

For this, we consider the following for ${\mu }_n,{\nu }_n\ge 0,$

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\left[n\right]}_q\left\{\left|-{\mu }_n{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\right|+\left|{\nu }_n{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q\left(F+1\right)}}\right|\right\}\hfill \\ \hfill & ={\displaystyle \frac{F-E-\mathrm{\Gamma }}{\left(F+1\right)}}\sum _{n=1}^{\infty }\left({\mu }_n+{\nu }_n\right)\hfill \\ \hfill & ={\displaystyle \frac{F-E-\mathrm{\Gamma }}{\left(F+1\right)}}\left(1-\left({\mu }_1+{\nu }_1\right)\right)\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{\left(F+1\right)}}.\hfill \end{array}$$

From the perspective of Theorem 5, it is concluded that $\varkappa \left(z\right)={\sum }_{n=1}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right)\in K_G^q(\zeta ;$$E,$$F)$.

Conversely, suppose that $\varkappa \left(z\right)\in K_G^q(\zeta ;$$E,$$F)$; we show that $\varkappa \left(z\right)$ can be written as

$$\varkappa \left(z\right)=\sum _{n=1}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right).$$

(36)

For this set,

$${\mu }_n={\displaystyle \frac{{\left[n\right]}_q\left(F+1\right)}{F-E-\mathrm{\Gamma }}}\left|a_n\right|,{\nu }_n={\displaystyle \frac{{\left[n\right]}_q\left(F+1\right)}{F-E-\mathrm{\Gamma }}}\left|b_n\right|.$$

(37)

Since $\varkappa \left(z\right)\in$$K_G^q(\zeta ;$$E,$$F)$, so by (27)

$$\varkappa \left(z\right)=z-\sum _{n=2}^{\infty }\left|a_n\right|z^n+\sum _{n=2}^{\infty }\left|b_n\right|\overline{z^n.}$$

From the expression (37), we have

$$\left|a_n\right|={\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{\mu }_n,\left|b_n\right|={\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{\nu }_n.$$

Using the above expressions in (27), we have

$$\varkappa \left(z\right)=z-\sum _{n=2}^{\infty }{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{\mu }_n{z}^n+\sum _{n=2}^{\infty }{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{\nu }_n\overline{z^n}.$$

Moreover, ${\sum }_{n=1}^{\infty }\left({\mu }_n+{\nu }_n\right)=1$, which implies

$$\begin{array}{ccc}\hfill \varkappa \left(z\right)& =& \sum _{n=1}^{\infty }\left({\mu }_n+{\nu }_n\right)z-\sum _{n=2}^{\infty }{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{\mu }_n{z}^n+\sum _{n=2}^{\infty }{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{\nu }_n\overline{z^n}\hfill \\ & =& \left({\mu }_1+{\nu }_1\right)z+\sum _{n=2}^{\infty }\left(z-{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}{z}^n\right){\mu }_n+\sum _{n=2}^{\infty }\left(z+{\displaystyle \frac{\left(F-E-\mathrm{\Gamma }\right)}{{\left[n\right]}_q\left(F+1\right)}}\overline{z^n}\right){\nu }_n.\hfill \end{array}$$

Upon using (34), we have

$$\begin{array}{ccc}\hfill \varkappa \left(z\right)& =& {\mu }_1{h}_1+{\nu }_1{g}_1+\sum _{n=2}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right)\hfill \\ \hfill \varkappa \left(z\right)& =& \sum _{n=1}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right).\hfill \end{array}$$

□

Taking $q\to 1^{-},$ we get the following conclusion, which is given in [22].

Corollary 14.

A function, $\varkappa \left(z\right)\in$$K_G(\zeta ;$$E,$$F)$, is obtained if and only if

$$\varkappa \left(z\right)=\sum _{n=1}^{\infty }\left({\mu }_n{h}_n+{\nu }_n{g}_n\right),$$

where

$$\begin{array}{ccc}\hfill h_1& =& z,g_1=z,\hfill \\ \hfill h_n& =& z-{\displaystyle \frac{F-E-\mathrm{\Gamma }}{n\left(F+1\right)}}{z}^n;g_n=z+{\displaystyle \frac{F-E-\mathrm{\Gamma }}{n\left(F+1\right)}}\overline{z^n},n=2,3\dots \hfill \\ \hfill \mathit{and}\sum _{n=1}^{\infty }\left({\mu }_n+{\nu }_n\right)& =& 1,{\mu }_n,{\nu }_n\ge 0.\hfill \end{array}$$

(38)

5. Radius of Starlikeness

A function, $\varkappa \in K_G^q(\zeta ;$$E,$$F)$, is regarded as starlike and of order A in $\mathbf{U}\left(\mathbf{r}\right)$ if

$$\Re {\displaystyle \frac{D_h\varkappa \left(z\right)}{\varkappa \left(z\right)}}>A,z\in \mathbf{U},$$

(39)

or, equivalently,

$$\left|{\displaystyle \frac{D_h\varkappa \left(z\right)-\varkappa \left(z\right)}{D_h\varkappa \left(z\right)-\left(2A-1\right)\varkappa \left(z\right)}}\right|<1,$$

(40)

For more details, see [16]. Let B be a subclass of the class $K_G^q(\zeta ;$$E,$$F).$ We define the radius of starlikeness for the class B by

$$R_A^{\ast }\left(B\right):=\underset{\varkappa \in B}{inf}\left(Sup\left\{r\in (0,1]:\varkappa \mathrm{is}\mathrm{starlike}\mathrm{of}\mathrm{order}A\mathrm{in}\mathbf{U}\right\}\right).$$

(41)

Specifically, for the subclass $B\subset K_G^q\left(\zeta :E,F\right)$, the radius of starlikeness $R_A^{\ast }\left(B\right)$ is defined as the largest radius r for which all functions in B remain starlike and of order A in the disk $\mathbf{U}.$ This concept illustrates the extent to which the functions retain starlikeness and serves as a foundation for the applications and results presented in the following section.

Theorem 15.

The radius of starlikeness of the class $K_G^q(\zeta ;$$E,$$F)$ is given as

$$R_A^{\ast }\left(K_G^q(k; E, F)\right)=\underset{n\in \mathbf{N}}{inf}{\left({\displaystyle \frac{{\left[n\right]}_q(1+F)(1-A)}{\left(F-E-\mathrm{\Gamma }\right)}}min\left\{{\displaystyle \frac{1}{\left(n-A\right)}},{\displaystyle \frac{1}{\left(n+A\right)}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}}.$$

(42)

Proof. 

From expression (41), we have

$$R_A^{\ast }\left(K_G^q(k; E, F)\right):=\underset{\varkappa \in K_G^q(k;E,F)}{inf}\left(Sup\left\{r\in (0,1]:\varkappa \mathrm{is}\mathrm{starlike}\mathrm{of}\mathrm{order}A\mathrm{in}\mathbf{U}\right\}\right).$$

The expression (40) implies that $\varkappa$ is starlike and of order A if

$$\left|{\displaystyle \frac{D_{\mathfrak{h}}\varkappa \left(z\right)-\varkappa \left(z\right)}{D_{\mathfrak{h}}\varkappa \left(z\right)-\left(2A-1\right)\varkappa \left(z\right)}}\right|<1,$$

(43)

Using (27) and (6) in (43), we get

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{D_{\mathfrak{h}}\varkappa \left(z\right)-\varkappa \left(z\right)}{D_{\mathfrak{h}}\varkappa \left(z\right)-\left(2A-1\right)\varkappa \left(z\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{z-{\sum }_{n=2}^{\infty }n\left|a_n\right|z^n-{\sum }_{n=2}^{\infty }n\left|b_n\right|\overline{z^n}-\left(z-{\sum }_{n=2}^{\infty }\left|a_n\right|z^n+{\sum }_{n=2}^{\infty }\left|b_n\right|\overline{z^n}\right)}{z-{\sum }_{n=2}^{\infty }n\left|a_n\right|z^n-{\sum }_{n=2}^{\infty }n\left|b_n\right|\overline{z^n}-\left(2A-1\right)\left(z-{\sum }_{n=2}^{\infty }\left|a_n\right|z^n+{\sum }_{n=2}^{\infty }\left|b_n\right|\overline{z^n}\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{-{\sum }_{n=2}^{\infty }\left(n-1\right)\left|a_n\right|z^n-{\sum }_{n=2}^{\infty }\left(n+1\right)\left|b_n\right|\overline{z^n}}{2(1-A)z-{\sum }_{n=2}^{\infty }\left(n-\left(2A-1\right)\right)|a_n|z^n-{\sum }_{n=2}^{\infty }\left(n+2A-1\right)\left|b_n\right|\overline{z^n}}}\right|.\hfill \end{array}$$

Putting $z=r$, $\left(0<r<1\right),$ and after simplification, we get

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{D_{\mathfrak{h}}\varkappa \left(z\right)-\varkappa \left(z\right)}{D_{\mathfrak{h}}\varkappa \left(z\right)-\left(2A-1\right)\varkappa \left(z\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{-{\sum }_{n=2}^{\infty }\left(n-1\right)\left|a_n\right|r^{n-1}-{\sum }_{n=2}^{\infty }\left(n-1\right)\left|b_n\right|r^{n-1}}{2(1-A)-{\sum }_{n=2}^{\infty }\left(n-\left(2A-1\right)\right)|a_n|r^{n-1}-{\sum }_{n=2}^{\infty }\left(n+2A-1\right)\left|b_n\right|r^{n-1}}}\right|.\hfill \end{array}$$

For $r\in \left(0,1\right),$ it is clear that the denominator of the left side cannot be zero. Furthermore, it is $2(1-A)>0$ that implies that the expression in the modulus of the denominator is positive, which yields

$$\begin{array}{ccc}\hfill \sum _{n=2}^{\infty }\left(n-1\right)|a_n|r^{n-1}+\sum _{n=2}^{\infty }\left(n-1\right)\left|b_n\right|r^{n-1}& <& 2(1-A)-\sum _{n=2}^{\infty }\left(n-2A+1\right)\left|a_n\right|r^{n-1}\hfill \\ & & -\sum _{n=2}^{\infty }\left(n+2A-1\right)\left|b_n\right|r^{n-1},\hfill \end{array}$$

or

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }\left(n-1\right)|a_n|r^{n-1}+\sum _{n=2}^{\infty }\left(n-1\right)|b_n|r^{n-1}+\sum _{n=2}^{\infty }\left(n-2A+1\right)|a_n|r^{n-1}+\sum _{n=2}^{\infty }\left(n+2A-1\right)|b_n|r^{n-1}\hfill \\ \hfill & =\sum _{n=2}^{\infty }2\left(\left(n-A\right)|a_n|+\left(n+A\right)\left|b_n\right|\right)r^{n-1}<2(1-A)\hfill \\ \hfill & =\sum _{n=2}^{\infty }\left(\left(n-A\right)|a_n|+\left(n+A\right)\left|b_n\right|\right)r^{n-1}<(1-A).\hfill \end{array}$$

From the above, we get the case in which $\varkappa$ is starlike and of order A in $\mathbf{U}$, if

$$\sum _{n=2}^{\infty }\left({\displaystyle \frac{\left(n-A\right)}{(1-A)}}|a_n|+{\displaystyle \frac{\left(n+A\right)}{(1-A)}}\left|b_n\right|\right)r^{n-1}<1.$$

(44)

According to Theorem 5, the inequality (44) holds only if

$${\displaystyle \frac{n-A}{1-A}}{r}^{n-1}\le {\displaystyle \frac{{\left[n\right]}_q(1+F)}{F-E-\mathrm{\Gamma }}},{\displaystyle \frac{n+A}{1-A}}{r}^{n-1}\le {\displaystyle \frac{{\left[n\right]}_q(1+F)}{F-E-\mathrm{\Gamma }}},$$

and that is, if

$$r\le {\left({\displaystyle \frac{{\left[n\right]}_q(1+F)\left(1-A\right)}{\left(F-E-\mathrm{\Gamma }\right)\left(n-A\right)}}\right)}^{{\displaystyle \frac{1}{n-1}}},r\le {\left({\displaystyle \frac{{\left[n\right]}_q(1+F)\left(1-A\right)}{\left(F-E-\mathrm{\Gamma }\right)\left(n+A\right)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$$

or

$$r\le {\left({\displaystyle \frac{{\left[n\right]}_q(1+F)(1-A)}{\left(F-E-\mathrm{\Gamma }\right)}}min\left\{{\displaystyle \frac{1}{\left(n-A\right)}},{\displaystyle \frac{1}{\left(n+A\right)}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}}.$$

Thus, it follows that the function is starlike and of order A in $\mathbf{U}$, where

$$r^{\ast }:=\underset{n\in \mathbb{N}}{inf}{\left({\displaystyle \frac{{\left[n\right]}_q(1+F)(1-A)}{\left(F-E-\mathrm{\Gamma }\right)}}min\left\{{\displaystyle \frac{1}{\left(n-A\right)}},{\displaystyle \frac{1}{\left(n+A\right)}}\right\}\right)}^{{\displaystyle \frac{1}{n+1}}}.$$

(45)

The starlikeness radii $r^{\ast }\left(h_n\right),r^{\ast }\left(g_n\right)$ of the functions $h_n,g_n$ admit the form

$$h_n\left(z\right)=z-{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q(1+F)}}{z}^n,g_n\left(z\right)=z+{\displaystyle \frac{F-E-\mathrm{\Gamma }}{{\left[n\right]}_q(1+F)}}\overline{z^n},\left(n\in \mathbb{N},z\in \mathbf{U}\right),$$

and they are given by

$$r^{\ast }\left(h_n\right)={\left({\displaystyle \frac{{\left[n\right]}_q(1+F)(1-A)}{\left(F-E-\mathrm{\Gamma }\right)\left(n-A\right)}}\right)}^{{\displaystyle \frac{1}{n+1}}},r^{\ast }\left(g_n\right)={\left({\displaystyle \frac{{\left[n\right]}_q(1+F)(1-A)}{\left(F-E-\mathrm{\Gamma }\right)\left(n+A\right)}}\right)}^{{\displaystyle \frac{1}{n+1}}}.$$

Consequently, the given radius, $r^{\ast }$, cannot be exceeded by (45). Thus, we have (42). □

6. Convolution

The following theorem comprises the findings that demonstrate that a class, $K_G^q(\zeta ;$$E,$$F)$, is closed under convolution; this means that combining two functions from this class via convolution produces another function that remains in the class. The constraints on the coefficients ensure that the series remains well behaved, preventing the function from exceeding the bounds defined by the class. This result is important because it guarantees the stability of the class under convolution, allowing one to construct new functions while preserving the desired properties.

Theorem 16.

Let $\varkappa ,\chi \in$$K_G^q(\zeta ;$$E,$$F)$, where $\varkappa \left(z\right)$ has the form (4), and $\chi \left(z\right)$ has the form such that

$$\chi \left(z\right)=z-\sum _{n=2}^{\infty }\left|A_n\right|z^n+\sum _{n=2}^{\infty }\left|B_n\right|\overline{z^n},|A_n|,|B_n|\le 1,$$

and then $\varkappa \ast \chi \in$$K_G^q(\zeta ;$$E,$$F).$

Proof. 

Consider

$$\varkappa \ast \chi =z-\sum _{n=2}^{\infty }\left|a_n\right|\left|A_n\right|z^n+\sum _{n=2}^{\infty }\left|b_n\right|\left|B_n\right|\overline{z^n}.$$

We have to show that $\varkappa \ast \chi \in$$K_G^q(\zeta ;$$E,$$F)$. From Theorem 5, we need to prove that ${\sum }_{n=2}^{\infty }{\left[n\right]}_q\left(|a_n{A}_n|+|b_n{B}_n|\right)\le$${\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}$. For this, consider

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n{A}_n|+\left|b_n{B}_n\right|\right)=\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n\left|\right|A_n|+|b_n\left|\right|B_n|\right).$$

As $|A_n|,|B_n|\le 1$, and with the use of Theorem 5,

$$\sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n{A}_n|+|b_n{B}_n|\right)\le \sum _{n=2}^{\infty }{\left[n\right]}_q\left(|a_n|+|b_n|\right)\le {\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}}.$$

Hence, it is proven that ${\sum }_{n=2}^{\infty }{\left[n\right]}_q\left(|a_n{A}_n|+|b_n{B}_n|\right)\le$${\displaystyle \frac{F-E-\mathrm{\Gamma }}{1+F}},$ which implies $\varkappa \ast \chi \in$$K_G^q(\zeta ;$$E,$$F).$ □

7. Conclusions

In the present work, we have introduced a new subclass of Janowski-type harmonic close-to-convex functions in the open unit disk by employing the Jackson q-derivative operator. The formulation of the class through subordination provides a unified framework that connects harmonic mapping theory with tools from q-calculus. Within this setting, we derived growth and distortion bounds, and we established sufficient and necessary conditions, ensuring that the functions are sense-preserving and close to convex.

A detailed geometric analysis of the class was carried out. In particular, we determined its extreme points and examined its topological structure, proving that the class forms a convex and compact family under the topology of uniform convergence on compact subsets of the unit disk. The radius of starlikeness was obtained, providing further insight into the geometric behavior of functions in this class. Moreover, we showed that the class is closed under convolution, which highlights its structural stability and algebraic compatibility within the theory of harmonic functions. An important feature of the present investigation is the limiting transition $q\to 1^{-}$. In this case, the Jackson q-derivative reduces to the classical derivative, and several known results available in the literature are recovered, demonstrating that the present study extends and generalizes earlier findings while revealing structural symmetry in the considered harmonic mappings. This demonstrates that the proposed class not only generalizes earlier Janowski-type harmonic close-to-convex functions but also provides a broader and more flexible framework for studying geometric properties through q-calculus techniques. The methods developed in this paper open the door to further research directions, including the study of Fekete–Szegö-type inequalities, Hankel determinant problems, partial sums, integral transforms, and radius problems for related harmonic subclasses defined via other generalized q-operators. Such extensions may contribute significantly to the ongoing development of harmonic mappings and their connections with modern operator theory.