Geometric Properties Connected with a Certain Multiplier Integral $\mathfrak{q}$−Analogue Operator

Abstract

The topic concerning the introduction and investigation of new classes of analytic functions using subordination techniques for obtaining certain geometric properties alongside coefficient estimates and inclusion relations is enriched by the results of the present investigation. The prolific tools of quantum calculus applied in geometric function theory are employed for the investigation of a new class of analytic functions introduced by applying a previously defined generalized $\mathfrak{q}-$integral operator and the concept of subordination. Investigations are conducted on the new class, including coefficient estimates, integral representation for the functions of the class, linear combinations, forms of the weighted and arithmetic means involving functions from the class, and the estimation of the integral means results.

1. Introduction and Preliminaries

The application of quantum calculus is frequent in mathematical sciences considering its many potential uses within number theory [1], combinatorics [2], orthogonal polynomials [3,4,5], and basic hypergeometric functions [6]. Some of the basic principles of q-calculus and how it is incorporated into mathematical theories are shown in [7,8,9].

The proper framework for applying the ideas of q-calculus within geometric function theory was provided by the publication of a book chapter authored by Srivastava in 1989 [10]. The q-calculus methods were first applied in geometric function theory for defining the concept of q-starlike function in 1990 [11]. After that, the development of this line of research included the introduction of numerous q-calculus operators applied using the means specific to geometric function theory for different studies, including the definition of new classes of analytic functions and obtaining various characteristics for them, among which geometric properties and coefficient estimates are the most popular ones.

Kanas and Răducanu initiated the line of research where the classical operators were adapted to the q-calculus aspects embedded into geometric function theory when they used the concept of convolution for defining the $\mathfrak{q}$-analogue of the Ruscheweyh differential operator [12]. Kanas and Răducanu proposed the investigation on its characteristics, and the idea was soon picked up by researchers. Aldweby and Darus [13], Mahmood and Sokol [14], and many other researchers investigated throughout time several classes of analytic functions defined using $\mathfrak{q}$-analogue of the Ruscheweyh differential operator. Analytic functions are investigated in conic domains using q-calculus aspects in [15]. Multiplier operators are introduced using $\mathfrak{q}$–calculus aspects in [16], and subordination techniques are applied for investigating subclasses of analytic functions introduced using the $\mathfrak{q}$-calculus multiplier operators. $\mathfrak{q}$-Hypergeometric function is applied for obtaining subordination and superordination results in [17], and the application of a $\mathfrak{q}$-integral operator results in new subclasses of bi-univalent functions in [18]. $\mathfrak{q}$-Choi–Saigo–Srivastava operator is used for the introduction of a new univalent family in [19].

A recent paper [20] highlights some aspects regarding the application of quantum calculus in geometric function theory, while a review article by Srivastava [21] shows other developments along with a multitude of q-operators derived by employing different kinds of operators that are specific to geometric function theory.

In view of the results presented above, the goal of the present research is to present a new application of the convolution-based $\mathfrak{q}$-analogue integral operator introduced in [22] involving the q-analogue multiplier operator and the q-analogue of Noor integral operator. The operator was used for the outcome presented in [22] considering the means of the fuzzy differential subordination theory. The research presented in this paper is completely different. This operator is being applied to introduce a new class of analytic functions using the concepts of the classical differential subordination theory.

For this investigation, the general environment is provided by the analytic functions in the unit disc $\mathbb{D}:=\left\{\zeta \in \mathbb{C}:\left|\zeta \right|<1\right\}$, seen from the geometric perspective given by the properties of starlikeness that certain classes of analytic functions possess.

Let $\mathcal{A}$ stand for the normalized analytic family comprised of functions $\mathfrak{f}$ written as:

$$\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{a}_{\kappa }{\zeta }^{\kappa },\zeta \in \mathbb{D}.$$

(1)

Let $S\subset \mathcal{A}$ represent the class of univalent functions in $\mathbb{D}$. Additionally, we let $\mathcal{T}$ denote the subclass of S, which consists of functions $\mathfrak{f}$ with the following power-series expansion:

$$\mathfrak{f}\left(\zeta \right)=\zeta -\sum _{\kappa =2}^{\infty }{a}_{\kappa }{\zeta }^{\kappa },(a_{\kappa }>0),\zeta \in \mathbb{D}.$$

(2)

This class was studied by Silverman [23].

The class denoted by $S^{*}\left(\delta \right)$ of starlike functions of order $\delta$ is said to include all function $\mathfrak{f}\in \mathcal{A}$ for which:

$$Re\left(\frac{\zeta {\mathfrak{f}}^{\prime }\left(\zeta \right)}{\mathfrak{f}\left(\zeta \right)}\right)>\delta ,(0\le \delta <1),\zeta \in \mathbb{D}.$$

Evidently, the well-known class of starlike functions is obtained when $\delta =0$, $S^{*}\left(0\right)=S^{*}.$

The concept of subordination initiated by Miller and Mocanu [24], synthesized in [25] and developed in [26], is applied in this research considering that if $\mathfrak{f}$ and $\hslash \in \mathcal{A}$ we say that $\mathfrak{f}$ is subordinate to ℏ, written as $\mathfrak{f}\left(\zeta \right)\prec \hslash \left(\zeta \right)$, if there exists $\varpi \in \mathcal{A}$, with $\varpi \left(0\right)=0$ and $\left|\varpi \left(\zeta \right)\right|<1$ for all $\zeta \in \mathbb{D}$, such that $\mathfrak{f}\left(\zeta \right)=\hslash \left(\varpi \right(\zeta \left)\right)$, $\zeta \in \mathbb{D}$. Furthermore, if the function $\hslash \in S$, then the equivalency shown below ([24,25,26]) is true:

$$\mathfrak{f}\left(\zeta \right)\prec \hslash \left(\zeta \right)\iff \mathfrak{f}\left(0\right)=\hslash \left(0\right)\mathrm{and}\mathfrak{f}\left(\mathbb{D}\right)\subset \hslash \left(\mathbb{D}\right).$$

Another fundamental concept in geometric function theory is applied in this investigation considering that for the function $\mathfrak{f}$ given by (1) and ℏ of the form

$$\hslash \left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{b}_{\kappa }{\zeta }^{\kappa },\zeta \in \mathbb{D},$$

the frequently used convolution product is described as:

$$(\mathfrak{f}\ast \hslash )\left(\zeta \right):=\zeta +\sum _{\kappa =2}^{\infty }{a}_{\kappa }{b}_{\kappa }{\zeta }^{\kappa },\zeta \in \mathbb{D}.$$

The notable class of Janowski functions is defined, stating that the function ℏ with $\hslash \left(0\right)=1$ belongs to this class denoted by $P[E,G]$, if and only if the following subordination is satisfied:

$$\hslash \left(\zeta \right)\prec \frac{1+E\zeta }{1+G\zeta }(-1\le G<E\le 1).$$

The class $P[E,G]$ was first presented and investigated by Janowski [27].

In this investigation, quantum calculus aspects are added to follow a prolific line of research since the $\mathfrak{q}$-derivative and $\mathfrak{q}$-integral introduced by Jackson [28,29] are functions with numerous applications in mathematics and other related fields. The basic q-calculus concepts applied for this research are next exposed.

The $\mathfrak{q}$-difference operator defined in [28], ${\mathcal{\partial }}_{\mathfrak{q}}:\mathcal{A}\to \mathcal{A}$, is used considering the following:

$${\mathcal{\partial }}_{\mathfrak{q}}\mathfrak{f}\left(\zeta \right):=\left\{\begin{array}{cc}\frac{\mathfrak{f}\left(\zeta \right)-\mathfrak{f}\left(\mathfrak{q}\zeta \right)}{(1-\mathfrak{q})\zeta }\hfill & \left(\zeta \ne 0;0<\mathfrak{q}<1\right)\hfill \\ \hfill & \\ {\mathfrak{f}}^{\prime }\left(0\right)\hfill & \left(\zeta =0\right).\hfill \end{array}\right.$$

(3)

If $\kappa \in \mathbb{N}$ and $\zeta \in \mathbb{D}$, it is known that:

$${\mathcal{\partial }}_{\mathfrak{q}}\left\{\sum _{\kappa =1}^{\infty }{a}_{\kappa }{\zeta }^{\kappa }\right\}=\sum _{\kappa =1}^{\infty }{\left[\kappa \right]}_{\mathfrak{q}}{a}_{\kappa }{\zeta }^{\kappa -1},$$

(4)

where

$$\begin{array}{ccc}\hfill {\left[\kappa \right]}_{\mathfrak{q}}& =& \frac{1-{\mathfrak{q}}^{\kappa }}{1-\mathfrak{q}}=1+\sum _{n=1}^{\kappa -1}{\mathfrak{q}}^n,{\left[0\right]}_{\mathfrak{q}}=0,\hfill \\ \hfill {\left[\kappa \right]}_{\mathfrak{q}}!& =& \left\{\begin{array}{ll}{\left[\kappa \right]}_{\mathfrak{q}}{\left[\kappa -1\right]}_{\mathfrak{q}}\cdots \cdots \cdots {\left[2\right]}_{\mathfrak{q}}{\left[1\right]}_{\mathfrak{q}}& \kappa =1,2,3,\cdots \\ 1& \kappa =0.\end{array}\right.\hfill \end{array}$$

(5)

Definition 1 ([28,29]).

Given $\mathrm{d},\kappa \in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},$ the definition of the $\mathfrak{q}$-shifted factorial is known to be:

$${(\mathrm{d};\mathfrak{q})}_0=1,{(\mathrm{d};\mathfrak{q})}_{\kappa }=\prod _{\mathcal{\ell }=0}^{\kappa -1}(1-\mathrm{d}{\mathfrak{q}}^{\mathcal{\ell }}),$$

written, considering the basic (or $\mathfrak{q}$-) gamma function, as:

$${({\mathfrak{q}}^{\mathrm{d}};\mathfrak{q})}_{\kappa }=\frac{(1-{\mathfrak{q}}^{\kappa }){\Gamma }_{\mathfrak{q}}(\mathrm{d}+\kappa )}{{\Gamma }_{\mathfrak{q}}\left(\mathrm{d}\right)}\kappa \in {\mathbb{N}}_0,$$

when the definition of the basic (or $\mathfrak{q}$-) gamma function is known as

$${\Gamma }_{\mathfrak{q}}\left(x\right)=\frac{{(1-\mathfrak{q})}^{1-x}{(\mathfrak{q};\mathfrak{q})}_{\infty }}{{({\mathfrak{q}}^x;\mathfrak{q})}_{\infty }}(\left|\mathfrak{q}\right|<1,x\in {\mathbb{N}}_0),$$

with

$${(\mathrm{d};\mathfrak{q})}_{\infty }=\prod _{\mathcal{\ell }=0}^{\infty }(1-\mathrm{d}{\mathfrak{q}}^{\mathcal{\ell }})\left|\mathfrak{q}\right|<1.$$

The $\mathfrak{q}$-difference operator is governed by the following basic rules.

$${\mathcal{\partial }}_{\mathfrak{q}}\left(c\mathfrak{f}\left(\zeta \right)\pm d\mathfrak{h}\left(\zeta \right)\right)=c{\mathcal{\partial }}_{\mathfrak{q}}\mathfrak{f}\left(\zeta \right)\pm d{\mathcal{\partial }}_{\mathfrak{q}}\mathfrak{h}\left(\zeta \right)$$

$${\mathcal{\partial }}_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\mathfrak{h}\left(\zeta \right)\right)=\mathfrak{f}\left(\mathfrak{q}\zeta \right){\mathcal{\partial }}_{\mathfrak{q}}\left(\mathfrak{h}\left(\zeta \right)\right)+\mathfrak{h}\left(\zeta \right){\mathcal{\partial }}_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\right)$$

$${\mathcal{\partial }}_{\mathfrak{q}}\left(\frac{\mathfrak{f}\left(\zeta \right)}{\mathfrak{h}\left(\zeta \right)}\right)=\frac{{\mathcal{\partial }}_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\right)\mathfrak{h}\left(\zeta \right)-\mathfrak{f}\left(\zeta \right){\mathcal{\partial }}_{\mathfrak{q}}\left(\mathfrak{h}\left(\zeta \right)\right)}{\mathfrak{h}\left(\mathfrak{q}\zeta \right)\mathfrak{h}\left(\zeta \right)},\mathfrak{h}\left(\mathfrak{q}\zeta \right)\mathfrak{h}\left(\zeta \right)\ne 0$$

$${\mathcal{\partial }}_{\mathfrak{q}}\left(log\mathfrak{f}\left(\zeta \right)\right)=\frac{ln\mathfrak{q}}{\mathfrak{q}-1}\frac{{\mathcal{\partial }}_{\mathfrak{q}}\left(\mathfrak{f}\left(\zeta \right)\right)}{\mathfrak{f}\left(\zeta \right)},$$

where $\mathfrak{f},\mathfrak{h}\in \mathcal{A}$, with c and d being real or complex numbers.

In view of the results presented above, the goal of the present research is to present a new application of the convolution-based $\mathfrak{q}$-analogue integral operator introduced in [22] involving the q-analogue multiplier operator and the q-analogue of Noor integral operator. Those operators are next recalled.

In [30], the multiplier $\mathfrak{q}$-analogue Cătaş operator $I_{\mathfrak{q}}^r(\lambda ,\mathcal{\ell }):\mathcal{A}\to \mathcal{A},(r\in {\mathbb{N}}_0,\mathcal{\ell },\lambda \ge 0,0<\mathfrak{q}<1)$ was considered to be:

$$I_{\mathfrak{q}}^r(\lambda ,\mathcal{\ell })\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\left(\frac{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}+\lambda ({\left[\kappa +\mathcal{\ell }\right]}_{\mathfrak{q}}-{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}})}{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}}\right)}^r{a}_{\kappa }{\zeta }^{\kappa }.$$

(6)

The operator $I_{\mathfrak{q}}^r(\lambda ,\mathcal{\ell })\mathfrak{f}\left(\zeta \right)$ is expressed in terms of convolution as:

$$I_{\mathfrak{q}}^r(\lambda ,\mathcal{\ell })\mathfrak{f}\left(\zeta \right)=\underset{{ }_{r-times}}{\underbrace{{\mathfrak{C}}_{\mathfrak{q}}(\lambda ,\mathcal{\ell })\ast \cdots \ast {\mathfrak{C}}_{\mathfrak{q}}(\lambda ,\mathcal{\ell })}}\ast \mathfrak{f}\left(\zeta \right)$$

(7)

where ${\mathfrak{C}}_{\mathfrak{q}}(\lambda ,\mathcal{\ell })$ is known from [31] as:

$${\mathfrak{C}}_{\mathfrak{q}}(\lambda ,\mathcal{\ell })=\frac{z-\left(1-\frac{{\mathfrak{q}}^{\mathcal{\ell }}}{{[1+\mathcal{\ell }]}_{\mathfrak{q}}}\lambda \right)\mathfrak{q}{z}^2}{(1-z)(1-\mathfrak{q}z)}.$$

In [32], the Noor integral operator $\mathfrak{q}$-analogue, ${\Im }_{\mathfrak{q}}^{\mu }\mathfrak{f}\left(\zeta \right):\mathcal{A}\to \mathcal{A}$, was given as:

$${\Im }_{\mathfrak{q}}^{\mu }\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }\frac{[\kappa ,\mathfrak{q}]!}{{[\mu +1,\mathfrak{q}]}_{\kappa -1}}{a}_{\kappa }{\zeta }^{\kappa },(\mu >-1,0<\mathfrak{q}<1).$$

(8)

Definition 2 ([22]).

Considering (7) and (8), the operator ${\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)$ is defined by the convolution of Noor integral operator $\mathfrak{q}$-analogue and the multiplier $\mathfrak{q}$-analogue Cătaş operator, as:

$$\begin{array}{ccc}\hfill {\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)& =& \underset{{ }_{r-times}}{\underbrace{{\mathfrak{C}}_{\mathfrak{q}}(\lambda ,\mathcal{\ell })\ast \cdots \ast {\mathfrak{C}}_{\mathfrak{q}}(\lambda ,\mathcal{\ell })}}\ast {\Im }_{\mathfrak{q}}^{\mu }\mathfrak{f}\left(\zeta \right)\hfill \\ & =& \zeta +\sum _{\kappa =2}^{\infty }{\left(\frac{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}+\lambda ({\left[\kappa +\mathcal{\ell }\right]}_{\mathfrak{q}}-{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}})}{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}}\right)}^r\frac{[\kappa ,\mathfrak{q}]!}{{[\mu +1,\mathfrak{q}]}_{\kappa -1}}{a}_{\kappa }{\zeta }^{\kappa },\hfill \\ \hfill r& \in & {\mathbb{N}}_0,\mathcal{\ell },\lambda \ge 0,\mu >-1,0<\mathfrak{q}<1.\hfill \end{array}$$

If we consider

$${\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu ):={\left(\frac{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}+\lambda ({\left[\kappa +\mathcal{\ell }\right]}_{\mathfrak{q}}-{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}})}{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}}\right)}^r\frac{[\kappa ,\mathfrak{q}]!}{{[\mu +1,\mathfrak{q}]}_{\kappa -1}},$$

then

$${\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }.$$

(9)

The new classes defined for this investigation using the operator shown in Definition 2 are next given.

Definition 3.

Let $-1\le G<E\le 1$ and $0<\mathfrak{q}<1.$ A function $\mathfrak{f}\in \mathcal{A}$ belongs to the class ${\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ if the following differential subordination involving it holds:

$$\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}\prec \frac{1+E\zeta }{1+G\zeta }.$$

Equivalently, $\mathfrak{f}\in {\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ if and only if:

$$\left|\frac{\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}-1}{E-G\left(\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}\right)}\right|<1.$$

(10)

Furthermore, in terms of the function class $\mathcal{T}$ of functions $\mathfrak{f}\left(\zeta \right)$ given by (3), we define the function class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ by

$$\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)={\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)\cap \mathcal{T}$$

We note that:

(i)

Put $\mu =0$, the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ reduce to the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(E,G)$;

(ii)

Put $r=0$, the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ reduce to the class $\mathcal{T}{\mathsf{\Theta }}^{\mathfrak{q}}(\mu ,E,G)$;

(iii)

Put $r=\mu =0$, the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ reduce to the class $\mathcal{T}{\mathsf{\Theta }}^{\mathfrak{q}}(E,G)$ [33];

(iv)

Put $q\to 1^{-},E=1-2\alpha (0\le \alpha <1)$ and $G=-1$, the class $\mathcal{T}{\mathsf{\Theta }}^q(E,G)$ reduce to the class $\mathcal{T}\mathsf{\Theta }\left(\alpha \right)$.

This study examines certain subordination results associated with the multiplier $\mathfrak{q}$-analogue integral operator ${\mathcal{CN}}_{\mathfrak{q},\lambda }^{r,\mu }\mathfrak{f}\left(\zeta \right)$ given in Definition 2. The goal of the research on the recently introduced class, ${\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ given by Definition 3, is to obtain coefficient estimates and integral representation and to investigate linear combination, weighted and arithmetic means, inclusion results, and other integral characteristics for functions that belong to this class by employing subordination techniques to univalent functions with a range that is symmetric with respect to the real axis.

2. Main Results

Assumptions made throughout the study are that $r\in {\mathbb{N}}_0,\mathcal{\ell },\lambda \ge 0,\mu >-1,0<\mathfrak{q}<1$ and $-1\le G<E\le 1.$ Furthermore, all coefficients of the functions $\mathfrak{f}\in \mathcal{A}$ are regarded as real positive values.

Theorem 1.

Suppose that $\mathfrak{f}\in \mathbf{A}$ be of the form (1) and $-1\le G<0$. Then $\mathfrak{f}\in {\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ if the following inequality is satisfied:

$$\sum _{\kappa =2}^{\infty }\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }<E-G.$$

(11)

Proof. 

Suppose that (11) holds. Then from (10) we have

$$\begin{array}{ccc}\hfill \left|\frac{\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}-1}{E-G\left(\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}\right)}\right|& =& \left|\frac{{\displaystyle \sum _{\kappa =2}^{\infty }}({\left[\kappa \right]}_{\mathfrak{q}}-1){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }}{(E-G)\zeta +{\displaystyle \sum _{\kappa =2}^{\infty }}(E-G{\left[\kappa \right]}_{\mathfrak{q}}){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }}\right|\hfill \\ & \le & \frac{{\displaystyle \sum _{\kappa =2}^{\infty }}({\left[\kappa \right]}_{\mathfrak{q}}-1){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }}{(E-G)-{\displaystyle \sum _{\kappa =2}^{\infty }}(E-G{\left[\kappa \right]}_{\mathfrak{q}}){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }}<1,\hfill \end{array}$$

and using (10) and (11), the proof is completed.  ☐

Theorem 2.

Given $\mathfrak{f}\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ and $-1\le G<0$, the following condition is satisfied:

$${\displaystyle \sum _{\kappa =2}^{\infty }}\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }<E-G.$$

(12)

Proof. 

Consider $\mathfrak{f}\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$. Then using (10) and (9), we have:

$$\left|\frac{\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}-1}{E-G\left(\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}\right)}\right|=\left|\frac{{\displaystyle \sum _{\kappa =2}^{\infty }}({\left[\kappa \right]}_{\mathfrak{q}}-1){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }}{(E-G)\zeta -{\displaystyle \sum _{\kappa =2}^{\infty }}(E-G{\left[\kappa \right]}_{\mathfrak{q}}){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }}\right|<1.$$

Since $\left|\Re \left(\zeta \right)\right|\le \left|\zeta \right|$, we obtain

$$\Re \left(\frac{{\displaystyle \sum _{\kappa =2}^{\infty }}({\left[\kappa \right]}_{\mathfrak{q}}-1){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }}{(E-G)-{\displaystyle \sum _{\kappa =2}^{\infty }}(E-G{\left[\kappa \right]}_{\mathfrak{q}}){\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }{\zeta }^{\kappa }}\right)<1.$$

(13)

We now select $\zeta$ values along the real axis such that $\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}$ is real. Upon clearing the denominator in (13) and letting $\zeta \to 1^{-}$, we obtain (12).  ☐

Corollary 1.

Let $\mathfrak{f}\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$, then

$$a_{\kappa }\le \frac{E-G}{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )}.$$

The result is sharp for the function

$$\mathfrak{f}\left(\zeta \right)=\zeta -\frac{E-G}{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )}{\zeta }^{\kappa }.$$

If we set $\mu =0$ in Theorem 1 we obtain:

Corollary 2.

$\mathfrak{f}\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(E,G)$ if and only if

$$\sum _{\kappa =2}^{\infty }\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\left(\frac{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}+\lambda ({\left[\kappa +\mathcal{\ell }\right]}_{\mathfrak{q}}-{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}})}{{\left[1+\mathcal{\ell }\right]}_{\mathfrak{q}}}\right)}^r{a}_{\kappa }<E-G.$$

If in Theorem 1 we set $r=0$, then we obtain:

Corollary 3.

$\mathfrak{f}\in \mathcal{T}{\mathsf{\Theta }}^{\mathfrak{q}}(\mu ,E,G)$ if and only if

$$\sum _{\kappa =2}^{\infty }\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]\frac{[\kappa ,\mathfrak{q}]!}{{[\mu +1,\mathfrak{q}]}_{\kappa -1}}{a}_{\kappa }<E-G.$$

Theorem 3.

Assume that $\mathfrak{f}\in {\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$. Then,

$${\mathcal{CN}}_{\mathfrak{q},\lambda }^{r,\mu }\mathfrak{f}\left(\zeta \right)=exp\left(\frac{lnq}{q-1}\underset{0}{\overset{\zeta }{\int }}\frac{1}{t}\left(\frac{1+E\phi \left(t\right)}{1+G\phi \left(t\right)}\right)d_{\mathfrak{q}}\left(t\right)\right),$$

where $\left|\phi \left(t\right)\right|<1.$

Proof. 

Let $\mathfrak{f}\in {\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ and putting

$$\frac{\zeta {\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}=\omega \left(\zeta \right)$$

with

$$\omega \left(\zeta \right)\prec \frac{1+E\zeta }{1+G\zeta },$$

equivalently, we can write

$$\left|\frac{\omega \left(\zeta \right)-1}{E-G\omega \left(\zeta \right)}\right|<1,$$

and,

$$\frac{\omega \left(\zeta \right)-1}{E-G\omega \left(\zeta \right)}=\phi \left(\zeta \right),$$

such that $\left|\phi \left(\zeta \right)\right|<1.$ Hence,

$$\frac{{\mathcal{\partial }}_{\mathfrak{q}}\left({\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)\right)}{{\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)}=\frac{1}{\zeta }\left(\frac{1+E\phi \left(\zeta \right)}{1+G\phi \left(\zeta \right)}\right).$$

Using a simple calculation and integration, we achieve the intended outcome.  ☐

Theorem 4.

Suppose that the functions ${\mathfrak{f}}_j\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ are given by:

$${\mathfrak{f}}_j\left(\zeta \right)=\zeta +\sum _{\iota =1}^{\infty }{a}_{\iota ,j}{\zeta }^{\iota },(j=1,2,3,\cdots ,\upsilon ).$$

Then $F\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G),$ where

$$F\left(\zeta \right)=\sum _{j=1}^{\upsilon }{c}_j{\mathfrak{f}}_j\left(\zeta \right)with\sum _{j=1}^{\upsilon }{c}_j=1.$$

Proof. 

From Theorem 2, we can write

$$\sum _{\kappa =2}^{\infty }\left\{\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )}{E-G}\right\}{a}_{\kappa ,j}<1.$$

Therefore, we obtain

$$F\left(\zeta \right)=\sum _{j=2}^{\upsilon }{c}_j(\zeta +\sum _{\kappa =2}^{\infty }{a}_{\kappa ,j}{\zeta }^{\kappa })=\zeta +\sum _{j=2}^{\upsilon }\sum _{\kappa =2}^{\infty }{c}_j{a}_{\kappa ,j}{\zeta }^{\kappa }=\zeta +\sum _{\kappa =2}^{\infty }\left(\sum _{j=2}^{\upsilon }{c}_j{a}_{\kappa ,j}\right){\zeta }^{\kappa }.$$

However,

$$\begin{array}{ccc}& & \sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )}{E-G}\left(\sum _{j=2}^{\upsilon }{c}_j{a}_{\kappa ,j}\right)\hfill \\ & =& \sum _{j=2}^{\upsilon }\left\{\sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )}{E-G}{a}_{\kappa ,j}\right\}{c}_j\le 1,\hfill \end{array}$$

then $F\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ and the proof is complete.  ☐

Theorem 5.

If $\mathfrak{f},\hslash \in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$, then ${\mathfrak{h}}_j(j\in \mathbb{N})$ is in $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$, with ${\mathfrak{h}}_j$ given by:

$${\mathfrak{h}}_j\left(\zeta \right)=\frac{(1-j)\mathfrak{f}\left(\zeta \right)+(1+j)\hslash \left(\zeta \right)}{2}$$

(14)

Proof. 

By applying (14), we write:

$${\mathfrak{h}}_j\left(\zeta \right)=\zeta +\sum _{\kappa =2}^{\infty }\left[\frac{(1-j)a_{\kappa }+(1+j)b_{\kappa }}{2}\right]{\zeta }^{\kappa }.$$

To prove that ${\mathfrak{h}}_j\left(\zeta \right)\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G),$ we must prove that

$$\sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]}{E-G}\left\{\frac{(1-j)a_{\kappa }+(1+j)b_{\kappa }}{2}\right\}{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )<1.$$

For that, consider

$$\begin{array}{ccc}& & \sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]}{E-G}\left\{\frac{(1-j)a_{\kappa }+(1+j)b_{\kappa }}{2}\right\}{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )\hfill \\ & =& \frac{(1-j)}{2}\sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]}{E-G}{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa }+\hfill \\ & & \frac{(1+j)}{2}\sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]}{E-G}{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )b_{\kappa }\hfill \\ & <& \frac{(1-j)}{2}+\frac{(1+j)}{2}=1,\hfill \end{array}$$

and using (12), the assertion of the theorem is proved.  ☐

Theorem 6.

Assume that ${\mathfrak{f}}_j\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ with $j=1,2,\cdots ..\alpha (\alpha \in \mathbb{N})$. Then, the function,

$$\mathfrak{h}\left(\zeta \right)=\frac{1}{\alpha }\sum _{j=1}^{\alpha }{\mathfrak{f}}_j\left(\zeta \right)$$

(15)

remains in the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$.

Proof. 

By applying (15), we obtain:

$$\mathfrak{h}\left(\zeta \right)=\frac{1}{\alpha }\sum _{j=1}^{\alpha }\left(\zeta +\sum _{\kappa =2}^{\infty }{a}_{\kappa ,j}{\zeta }^{\kappa }\right)=\zeta +\sum _{\kappa =2}^{\infty }\left(\frac{1}{\alpha }\sum _{j=1}^{\alpha }{a}_{\kappa ,j}\right){\zeta }^{\kappa }.$$

(16)

Since ${\mathfrak{f}}_j$$\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G),$ using (16) and (12), we write:

$$\begin{array}{ccc}& & \sum _{\kappa =2}^{\infty }\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )\left(\frac{1}{\alpha }\sum _{j=1}^{\alpha }{a}_{\kappa ,j}\right)\hfill \\ & =& \frac{1}{\alpha }\sum _{j=1}^{\alpha }\left(\sum _{\kappa =2}^{\infty }\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )a_{\kappa ,j}\right)\hfill \\ & \le & \frac{1}{\alpha }\sum _{j=1}^{\alpha }(E-G)=E-G,\hfill \end{array}$$

hence, the proof is completed.  ☐

Remark 1.

By setting $\mu =1+\lambda$ and $r=0$, we obtain the outcomes seen in [19].

3. Integral Means

In this section, integral means for functions belonging to the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ are obtained. In [23], Silverman found that the function ${\mathfrak{f}}_2\left(\zeta \right)=\zeta -\frac{{\zeta }^2}{2}$ is often extremal over the family $\mathcal{T}$. He applied this function to resolve his integral means inequality, conjectured in [34] and settled in [35], that

$${\int }_0^{2\pi }|\mathfrak{f}\left(\rho e^{i\theta }\right){|}^{\eta }d\theta \le {\int }_0^{2\pi }{\left|{\mathfrak{f}}_2\left(\rho e^{i\theta }\right)\right|}^{\eta }d\theta ,(\mathfrak{f}\in \mathcal{T},\eta >0,0<\rho <1).$$

In 1925, Littlewood [36] proved the following lemma.

Lemma 1.

If the functions $\mathfrak{f}$ and $\mathfrak{g}$ are analytic in $\mathbb{D}$ with $\mathfrak{g}\prec \mathfrak{f}$, then

$${\int }_0^{2\pi }|\mathfrak{g}\left(\rho e^{i\theta }\right){|}^{\eta }d\theta \le {\int }_0^{2\pi }{\left|\mathfrak{f}\left(\rho e^{i\theta }\right)\right|}^{\eta }d\theta ,(\eta >0,0<\rho <1).$$

Applying Lemma 1, Theorem 2 and Corollary 1, we prove the following theorem.

Theorem 7.

Suppose $\mathfrak{f}\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ and ${\mathfrak{f}}_2$ defined by

$${\mathfrak{f}}_2\left(\zeta \right)=\zeta -\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(2,\mu )}{\zeta }^2.$$

Then for $\zeta =\rho e^{i\theta },0<\rho <1,\eta >0,$ we have

$${\int }_0^{2\pi }{\left|\mathfrak{f}\left(\zeta \right)\right|}^{\eta }d\theta \le {\int }_0^{2\pi }{\left|{\mathfrak{f}}_2\left(\zeta \right)\right|}^{\eta }d\theta .$$

(17)

Proof. 

For $\mathfrak{f}\left(\zeta \right)=\zeta -{\sum }_{\kappa =2}^{\infty }{a}_{\kappa }{\zeta }^{\kappa }(a_{\kappa }\ge 0)$, (17) is equivalent to

$${\int }_0^{2\pi }|1-\sum _{\kappa =2}^{\infty }{a}_{\kappa }{\zeta }^{\kappa -1}{|}^{\eta }d\theta \le {\int }_0^{2\pi }|1-\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(2,\mu )}\zeta {|}^{\eta }d\theta .$$

Using Lemma 1, it suffices to show that

$$1-\sum _{\kappa =2}^{\infty }{a}_{\kappa }{\zeta }^{\kappa -1}\prec 1-\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(2,\mu )}\zeta .$$

Setting

$$1-\sum _{\kappa =2}^{\infty }{a}_{\kappa }{\zeta }^{\kappa -1}=1-\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(2,\mu )}w\left(\zeta \right),$$

and using Theorem 2, we obtain

$$\begin{array}{ccc}\hfill \left|w\right(\zeta \left)\right|& =& \left|\sum _{\kappa =2}^{\infty }\frac{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(2,\mu )}{E-G}{a}_{\kappa }{\zeta }^{\kappa -1}\right|\hfill \\ & \le & \left|\zeta \right|\sum _{\kappa =2}^{\infty }\frac{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(2,\mu )}{E-G}{a}_{\kappa }\hfill \\ & \le & \left|\zeta \right|\sum _{\kappa =2}^{\infty }\frac{\left[({\left[\kappa \right]}_{\mathfrak{q}}+E)-(G{\left[\kappa \right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\kappa ,\mu )}{E-G}{a}_{\kappa }\hfill \\ & \le & \left|\zeta \right|.\hfill \end{array}$$

This completes the proof of Theorem 7.  ☐

Corollary 4.

If $\mathfrak{f}\left(\zeta \right)\in \mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(E,G)$, then the assertion (17) holds true where

$${\mathfrak{f}}_2\left(\zeta \right)=\zeta -\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}\left(2\right)}{\zeta }^2.$$

Corollary 5.

If $\mathfrak{f}\left(\zeta \right)\in \mathcal{T}{\mathsf{\Theta }}^{\mathfrak{q}}(\mu ,E,G)$, then the assertion (17) holds true where

$${\mathfrak{f}}_2\left(\zeta \right)=\zeta -\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]{\mathfrak{K}}^{\mathfrak{q}}(2,\mu )}{\zeta }^2.$$

Corollary 6.

If $\mathfrak{f}\left(\zeta \right)\in \mathcal{T}{\mathsf{\Theta }}^{\mathfrak{q}}(E,G)$, then the assertion (17) holds true where

$${\mathfrak{f}}_2\left(\zeta \right)=\zeta -\frac{E-G}{\left[({\left[2\right]}_{\mathfrak{q}}+E)-(G{\left[2\right]}_{\mathfrak{q}}+1)\right]}{\zeta }^2.$$

Remark 2.

(i) Let $q\to 1^{-},E=1-2\alpha (0\le \alpha <1)$ and $G=-1$, Corollary 6 gives the result obtained in ([37], Remark 3.3);

(ii) Put $\alpha =0$ in (i) we obtained the result obtained in ([35], Theorem 2.2).

4. Conclusions

The results discussed in this work further advance the topic of introducing and investigating new classes of analytic functions with the help of quantum calculus operators. Motivated by the encouraging outcomes of integrating components of quantum calculus into the research regarding geometric function theory, this study uses a previously defined convolution-based $\mathfrak{q}$-analogue integral operator ${\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)$, given in Definition 2, introduced in [22] involving the q-analogue multiplier operator and the q-analogue of Noor integral operator, to introduce the subclasses ${\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ and $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$ of analytic functions presented in Definition 3. The first two theorems proved during this investigation reveal coefficient estimates followed by a theorem that provides the integral representation for functions belonging to the class ${\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G).$ It is further proved that a linear combination, weighted and arithmetic means of functions from the class $\mathcal{T}{\mathsf{\Theta }}_{\mathcal{\ell },\lambda }^{r,\mathfrak{q}}(\mu ,E,G)$, preserve the characteristics of the class, hence remaining members of it. The integral means are also established for this class.

For future investigations, the newly defined class can be considered using the means of the dual theory of differential superordination to connect the outcome of that study with the present results by means of sandwich-type results seen, for example, in [17]. Considering that the operator ${\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)$ was first used for applications in the theory of fuzzy differential subordination, and that the present investigation considers the classical theory of differential subordination, it is expected that the methods specific to strong differential subordination and superordination theories could be used to introduce new classes following the pattern set by the present investigation involving the $\mathfrak{q}$-analogue integral operator ${\mathcal{CN}}_{\mathfrak{q},\lambda ,\mathcal{\ell }}^{r,\mu }\mathfrak{f}\left(\zeta \right)$ and inspired by the recent publication [38].