On Certain Subclasses of Analytic Functions Associated with a Symmetric q-Differential Operator

Abstract

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities involving their coefficients. Additionally, we establish several inclusion relations between these subclasses, obtained by varying the defining parameters. In the second part, we focus on differential subordination and superordination for functions transformed by the operator. We provide sufficient conditions under which such functions are subordinate or superordinate to univalent functions, and we determine the best dominant and best subordinant in specific cases. These results are complemented by several corollaries that highlight particular instances of the main theorems. Furthermore, we present a sandwich-type result that brings together the subordination and superordination frameworks in a unified analytic statement.

1. Introduction and Preliminaries

Over the past decade, symmetric quantum calculus has garnered increasing interest due to its wide-ranging usefulness throughout various fields of mathematics and physics. Its flexible framework is especially effective in tackling problems in non-integer order calculus, geometric complex analysis, mathematical physics, and quantum theory, where traditional analytical methods may fall short [1]. A central motivation behind this growing interest lies in the symmetric q-derivative’s capacity to approximate the classical derivative more effectively than the standard q-derivative in many cases, often exhibiting superior convergence behavior, though this remains an area of ongoing research.

Conceptually, symmetric q-calculus is a natural extension of classical q-calculus, a theory with roots in the early 20th century. The classical framework originated with F.H. Jackson [2], who formulated q-analogues of operators involving differentiation and integration. By introducing enhanced operator symmetry, the symmetric variant expands on this foundation and becomes particularly effective for studying function classes that possess intrinsic geometric symmetry. A major breakthrough in this field occurred in 1990, when Ismail and collaborators [3] introduced the concept of q-starlike functions, laying the groundwork for extensive future research. Since then, the theory has evolved to encompass a wide variety of research concerning the structure and geometric properties of q-starlike and related function classes. Subsequent developments included the work of Arif et al. [4], who utilized convolution methods to establish the Noor-type q-integral operator and applied it to derive novel subsets of holomorphic functions. Simultaneously, other researchers such as those in [5] explored q-analogues of differential operators, extending their application to broader families of functions. In another advancement, Srivastava et al. [6] proposed a category of k-symmetric harmonic mappings using derivatives defined in q-calculus, further enriching the theoretical landscape.

This sustained activity has led to the identification and study of numerous new analytic subfamilies using both q-fractional integral and differential operators, highlighting the depth and adaptability of this approach. For a comprehensive view of the current developments in this field, readers may refer to [7,8,9,10,11].

Numerous significant developments have arisen through the use of symmetrized q-calculus in the context of complex function theory with a geometric focus, underscoring its growing relevance and theoretical depth. A key contribution in this direction was made by Kanas et al. [12], who proposed q-extensions of well-known tools used in classical analysis. These operators were used to define new families of functions characterized by specific geometric behavior, thereby establishing a foundational framework that has since supported and inspired a wide range of subsequent investigations in this field.

Building on this foundation, Khan et al. [13] revisited the concept of generalized symmetric conic domains, enriching the theory through the adoption of symmetrized q-calculus principles. The work they carried out culminated in the definition of a novel class of symmetric q-starlike functions, which expanded the theoretical landscape of this subject. In particular, their study deepened the understanding of the behavior of symmetric q-operators and provided structural insights into various subclasses of functions characterized by q-starlikeness and q-convexity. Additionally, their approaches introduced new methodologies for broadening and analyzing mappings characterized by particular geometric features within the unit disk.

In a related line of inquiry, Sabir et al. [14] conducted a detailed and systematic analysis of m-fold symmetric bi-univalent functions using symmetric q-calculus tools. By applying the symmetric q-operator, they derived several key results that further enriched the theoretical framework of GFT, particularly in a multivalent setting.

Further extending the scope of its applications, Khan et al. [15] employed the symmetric q-derivative to study multivalent analytic functions, establishing new classes and revealing significant potential for further generalizations. In a more recent contribution, Khan et al. [16] synthesized ideas derived from q-calculus with symmetry considerations, and geometric principles related to conic shapes were employed to build an extended class of balanced domains influenced by conic structures. The investigation culminated in the definition of a new category of functions with symmetric properties, considered within a conventional analytic framework, offering both theoretical novelty and practical tools for exploring analytic functions governed by geometric constraints.

These cumulative efforts have substantially broadened the analytical reach of symmetric q-calculus, enabling new developments in geometric function theory, particularly in relation to sharp bounds, functional inclusions, and operator theory. For more recent and detailed contributions to this dynamically expanding area of research, readers are also encouraged to consult [17,18,19,20,21].

To ensure a clearer and more rigorous understanding of the results presented in this paper, our initial step is to provide a few key definitions, mathematical symbols, and properties based on the principles of symmetrized q-difference analysis. These foundational concepts carry great significance for the proper formulation and progression of the main theorems presented in the subsequent sections.

Unless stated otherwise, we assume throughout the manuscript that the base parameter satisfies $0<q<1$ and $z\in U$ and that $\mathbb{N}$ refers to the following sequence of positive counting numbers: $\left\{1,2,3\dots .\right\}$.

The open unit disk in the complex plane is

$$U=\{z\in \mathbb{C}:|z|<1\}.$$

Consider $\mathcal{H}\left(U\right)$ the collection of analytic functions taking complex values and defined on U. Among these, we consider the subclass $\mathcal{H}\left(a,p\right)$, which includes all the functions in $\mathcal{H}\left(U\right)$ that can be represented by the power series $f\left(z\right)=a+a_p{z}^p+a_{p+1}{z}^{p+1}+\dots .$, $z\in U$, where $a\in \mathbb{C}$ and $p\in \mathbb{N}=\left\{1,2,3\dots .\right\}$.

Moreover, ${\mathcal{A}}_p$ refers to a normalized subfamily of $\mathcal{H}\left(U\right)$ characterized by the condition described below:

$${\mathcal{A}}_p=\{f\in \mathcal{H}\left(U\right):f\left(z\right)=z^p+a_{p+1}{z}^{p+1}+\dots ,z\in U\}.$$

Here and throughout the entire paper, $p\in \mathbb{N}$ denotes a positive integer. When $p=1$, ${\mathcal{A}}_p$ corresponds to the standard set of analytic functions with conventional normalization, denoted by $\mathcal{A}={\mathcal{A}}_1$.

For each $j\in \mathbb{N}$, the expression $\widetilde{{\left[j\right]}_q}$, referred to as the symmetric q-analogue of a number, is introduced below:

$$\widetilde{{\left[j\right]}_q}={\displaystyle \frac{q^{-j}-q^j}{q^{-1}-q}}\mathrm{with}\widetilde{{\left[0\right]}_q}=0.$$

(1)

It is important to emphasize that the symmetric q-number differs from the classical (non-symmetric) q-number typically used within the framework of quantum harmonic oscillators under q-deformation [22]. This distinction is significant, especially in applications involving symmetry and operator algebra.

The symmetric q-shifted factorial, denoted by $\widetilde{{\left[j\right]}_q}!$, is defined recursively in analogy with the classical factorial, but using symmetric q-numbers:

$$\widetilde{{\left[j\right]}_q}!=\left\{\begin{array}{c}\widetilde{{\left[j\right]}_q}·\widetilde{{\left[j-1\right]}_q}·\widetilde{{\left[j-2\right]}_q}··\widetilde{{\left[1\right]}_q},\mathrm{for}j=1,2,\dots ;\\ 1,\mathrm{for}j=0.\end{array}\right.$$

(2)

A notable limiting property is

$$\underset{q\to 1^{-}}{lim}\widetilde{{\left[j\right]}_q}=j,$$

which shows that the symmetric q-factorial converges to the classical factorial in the standard limit⁻.

We now define the symmetric q-derivative operator [23], denoted as $\widetilde{D_q}f\left(z\right)$, which acts on functions $f\in {\mathcal{A}}_p$. The operator is defined by

$$\widetilde{D_q}f\left(z\right)=\left\{\begin{array}{c}{\displaystyle \frac{f\left(qz\right)-f\left(q^{-1}z\right)}{z(q-q^{-1})}},z\ne 0,q\ne 1,z\in U;\hfill \\ f^{\prime }\left(0\right),\mathrm{for}z=0.\hfill \end{array}\right.$$

(3)

As $q\to 1^{-}$, the operator converges to the classical derivative:

$$\underset{q\to 1^{-}}{lim}\widetilde{D_q}f\left(z\right)=f^{\prime }\left(z\right).$$

A direct computation confirms that the symmetric q-derivative of the monomial $z^p$ yields

$$\widetilde{D_q}{z}^p=\widetilde{{\left[p\right]}_q}{z}^{p-1},$$

and for a general function, $f\left(z\right)$, its symmetric q-derivative has the following power series expansion:

$$z\widetilde{D_q}f\left(z\right)={\widetilde{\left[p\right]}}_q{z}^p+\sum _{j=1}^{\infty }\widetilde{{[j+p]}_q}{a}_{j+p}{z}^{j+p}.$$

(4)

The operator $\widetilde{D_q}$ satisfies several essential identities [12], which mirror the properties of the classical derivative but reflect the underlying symmetric q-structure:

$$\begin{array}{ccc}\hfill \widetilde{D_q}\left(f\left(z\right)+g\left(z\right)\right)& =& \widetilde{D_q}f\left(z\right)+\widetilde{D_q}g\left(z\right),\hfill \\ \hfill \widetilde{D_q}\left(f\left(z\right)g\left(z\right)\right)& =& g\left(q^{-1}z\right)\widetilde{D_q}f\left(z\right)+f\left(qz\right)\widetilde{D_q}g\left(z\right)=g\left(qz\right)\widetilde{D_q}f\left(z\right)+f\left(q^{-1}z\right)\widetilde{D_q}g\left(z\right),\hfill \\ \hfill \widetilde{D_q}\left({\displaystyle \frac{f\left(z\right)}{g\left(z\right)}}\right)& =& {\displaystyle \frac{f\left(qz\right)\widetilde{D_q}g\left(z\right)-g\left(q^{-1}z\right)\widetilde{D_q}f\left(z\right)}{g\left(q^{-1}z\right)g\left(qz\right)}},\hfill \\ \hfill \widetilde{D_q}f\left(z\right)& =& \widetilde{D_{q^2}}f\left(q^{-1}z\right).\hfill \end{array}$$

These properties illustrate the rich algebraic structure of symmetric q-calculus and lay the foundation for advanced developments in geometric function theory, quantum calculus, and operator theory.

Building on these foundational properties, we now turn our attention to a geometric perspective on analytic functions involving $\widetilde{D_q}$, examined through the framework of differential subordination and superordination.

Differential subordination and superordination are two fundamental and deeply interconnected principles within the framework of geometric function theory (GFT). They provide a rigorous analytical foundation for establishing relationships between analytic functions based on how these functions respond to various differential operators. Rather than merely examining functions in isolation, these tools allow for a comparative analysis, where one function is studied in relation to another based on the structure of differential inequalities.

This methodology is not only central to understanding the geometric properties of analytic functions—such as their univalence, convexity, and starlikeness—but it also enables us to obtain inclusion results and develop dominance principles and sandwich-type theorems. As such, differential subordination and superordination serve as essential instruments for function classification and transformation within the unit disk and beyond.

The origins of subordination theory can be traced back to the early 20th century, with its initial formulation introduced by Ernst Lindelöf in 1909. This foundational idea was later extended and refined by J.E. Littlewood [24] and other prominent mathematicians, laying the groundwork for what would become a central concept in geometric function theory.

A significant leap in the formal development of the framework took place in the latter part of the 1970s and the initial years of the 1980s, when Sanford S. Miller and Petru T. Mocanu [25,26] established a comprehensive and rigorous framework for differential subordination. Their pioneering work introduced systematic methods for investigating analytic functions through the lens of differential inequalities, fundamentally shaping the direction of subsequent research in the field.

In 2003, Miller and Mocanu expanded the theoretical scope of the subject by formulating the concept of differential superordination. This complementary idea not only enriched the mathematical structure of the theory but also led to the formulation of sandwich-type theorems, which establish precise relationships between subordination and superordination, thereby offering a more unified and versatile approach to functional analysis within the unit disk.

In recent years, the theory of differential subordination and superordination has witnessed remarkable progress, with ongoing research pushing the boundaries of classical formulations. Notably, scholars have extended the theory to encompass higher-order cases, including third- and fourth-order differential inequalities. These generalizations have provided deeper and more refined insights into the structural behavior of analytic functions under increasingly complex differential constraints (see, e.g., [27,28]).

An especially dynamic area of development has emerged from the integration of q-calculus, where researchers have begun investigating subordination and superordination phenomena within the framework of q-analogues of classical differential operators. This shift has opened up rich new avenues of exploration, particularly in the context of symmetric functions and the derivation of sandwich-type theorems that are uniquely adapted to q-deformed environments (see, e.g., [29,30]).

As a result of this continued evolution, differential subordination and superordination remain active and fertile fields of research within geometric function theory. To obtain a detailed perspective on modern developments, some of the latest contributions can be found in [31,32].

Differential subordination is a technique used to determine the conditions under which one analytic function, say, $f\left(z\right)$, is subordinate to another analytic function, $g\left(z\right)$. This relationship, denoted as $f\prec g$, holds if one can identify a Schwarz function, $w\left(z\right)$, analytic in the domain U, for which

$$w\left(0\right)=0,\left|w\right(z\left)\right|<1\mathrm{and}f\left(z\right)=g\left(w\right(z\left)\right).$$

Assuming that g is also a one-to-one mapping in U, the relation $f\prec g$ holds if and only if the conditions below are satisfied:

$$f\left(0\right)=g\left(0\right)\mathrm{and}f\left(U\right)\subset g\left(U\right).$$

Now, let $\psi$ be a given mapping from ${\mathbb{C}}^3\times U$ to $\mathbb{C}$ and h an injective function on U. A function, s, holomorphic in U is said to satisfy a second-order differential subordination if it fulfills the condition

$$\psi (s\left(z\right),z{s}^{\prime }\left(z\right),z^2{s}^{″}\left(z\right);z)\prec h\left(z\right).$$

(5)

In this context, a function, $\upsilon$, univalent in U, is called a dominant of all the solutions of (5) if

$$s\prec \upsilon ,$$

for all s satisfying (5). Furthermore, if $\upsilon$ is the smallest such dominant in a subordination sense—meaning that for every other dominant, v, of (5), we have

$$\upsilon \left(z\right)\prec \nu \left(z\right),$$

then $\upsilon$ is referred to as the optimal dominant. Except for a rotational transformation of the unit disk U, the leading dominant remains singular, illustrating the inherent rotational symmetry present in the class of Schwarz functions.

Let $\psi :{\mathbb{C}}^3\times U\to \mathbb{C}$ be a given function, and let h be an analytic function defined in U. Suppose that s is a holomorphic mapping in U such that both s and $\psi \left(s\left(z\right),z{s}^{\prime }\left(z\right),z^2{s}^{″}\left(z\right);z\right)$ are univalent in U. If the differential superordination condition

$$h\left(z\right)\prec \psi (s\left(z\right),z{s}^{\prime }\left(z\right),z^2{s}^{″}\left(z\right);z)$$

(6)

is satisfied, then the function s constitutes the fulfillment of the second-order differential inequality condition defined by (6) (see [25]). In this context, the function $\upsilon$, being analytic, is regarded as the subordinate entity under the differential superordination condition (6) if

$$\upsilon \prec s,$$

for all s satisfying (6). If $\upsilon$ is the greatest such function in the sense of subordination —i.e., if

$$\nu \left(z\right)\prec \upsilon \left(z\right),$$

for every subordinant v of (6) then $\upsilon$ is called the best subordinant. This function represents the maximal element (with respect to subordination) among all the admissible subordinants and is unique up to rotation in U.

In their foundational work, Miller and Mocanu [26] proposed that the implication below is valid under appropriate constraints on the functions h, $\upsilon$, and $\psi$:

$$h\left(z\right)\prec \psi (s\left(z\right),z{s}^{\prime }\left(z\right),z^2{s}^{″}\left(z\right);z)\Rightarrow \upsilon \left(z\right)\prec s\left(z\right).$$

Under these conditions, the function $\upsilon$ serves as the best subordinant for the given differential superordination problem.

Under the established framework of differential subordination and superordination, one is often interested not only in identifying extremal functions like the best subordinant, but also in exploring structural operations that preserve or reveal the subordination properties. One such operation, which plays a central role in geometric function theory, is the Hadamard product (or convolution). This operation allows us to combine analytic functions in a coefficient-wise manner and examine how subordination relationships behave under such combinations.

Let us consider two functions that are holomorphic in U, given by

$$f\left(z\right)=z^p+\sum _{j=2}^{\infty }{a}_{j+p}{z}^{j+p}\mathrm{and}g\left(z\right)=z^p+\sum _{j=2}^{\infty }{b}_{j+p}{z}^{j+p}.$$

The function $\left(f\ast g\right)\left(z\right)$, representing the Hadamard (convolution) product of f and g, is expressed as

$$f\left(z\right)\ast g\left(z\right)=\left(f\ast g\right)\left(z\right)=z^p+\sum _{j=2}^{\infty }{a}_{j+p}{b}_{j+p}{z}^{j+p}.$$

Below, we present the definition of the Al-Oboudi operator, which will play a key role in obtaining our forthcoming results.

Definition 1 

(Al Oboudi [33]). Let f be an element of the class $\mathcal{A}$, β a non-negative real number, and m a natural number; the operator $D_{\beta }^m$ is described as $D_{\beta }^m:\mathcal{A}\to \mathcal{A}$:

$$\begin{array}{ccc}\hfill D_{\beta }^{0}f\left(z\right)& =& f\left(z\right),\hfill \\ \hfill D_{\beta }^{1}f\left(z\right)& =& \left(1-\beta \right)f\left(z\right)+\beta z{f}^{\prime }\left(z\right)=D_{\beta }f\left(z\right),\hfill \\ & & ...\hfill \\ \hfill D_{\beta }^{m}f\left(z\right)& =& \left(1-\beta \right)D_{\beta }^{m-1}f\left(z\right)+\beta z{\left(D_{\beta }^{m}f\left(z\right)\right)}^{\prime }=D_{\beta }\left(D_{\beta }^{m-1}f\left(z\right)\right),\mathrm{for}z\in U.\hfill \end{array}$$

Remark 1 

(Al Oboudi [33]). If $f\in \mathcal{A}$ and $f\left(z\right)=z+{\sum }_{j=2}^{\infty }{a}_j{z}^j$, then

$$D_{\beta }^{m}f\left(z\right)=z+\sum _{j=2}^{\infty }{\left[1+\left(j-1\right)\beta \right]}^m{a}_j{z}^j,\mathrm{for}z\in U.$$

(7)

Remark 2. 

The earlier formulation reduces to the Sălăgean differential operator upon setting $\beta =1$ [16].

The following classical results are required to establish our main findings.

Definition 2 

([25]). Denote by Q the set of all the functions, f, that are analytic and injective on $\overline{U}\setminus F\left(f\right)$, where $F\left(f\right)=\{\zeta \in \partial U:{\displaystyle \underset{z\to \zeta }{lim}}f\left(z\right)=\infty \},$ and are such that $f^{\prime }\left(\zeta \right)\ne 0$ for $\zeta \in \partial U\setminus F\left(f\right)$.

Lemma 1 

([25]). Assume that the function υ is one-to-one in U and that ϑ and φ are holomorphic in a domain, D, which contains the image of $\upsilon \left(U\right)$. Further suppose that $\phi \left(w\right)\ne 0$ for every $w\in \upsilon \left(U\right)$. Define the auxiliary functions $\Theta \left(z\right)=z{\upsilon }^{\prime }\left(z\right)\phi \left(\upsilon \left(z\right)\right)$ and $h\left(z\right)=\vartheta \left(\upsilon \left(z\right)\right)+\Theta \left(z\right)$. Let the following conditions be satisfied:

1. Θ is starlike and injective in U;

2. $Re\left({\displaystyle \frac{s{h}^{\prime }\left(z\right)}{\Theta \left(z\right)}}\right)>0$, for all $z\in U$.

Now, consider a function, s, that is analytic in U, such that

$$\begin{array}{ccc}\hfill s\left(0\right)& =& \upsilon \left(0\right),\hfill \\ \hfill s\left(U\right)& \subseteq & D,\hfill \end{array}$$

and

$$\vartheta \left(s\left(z\right)\right)+z{s}^{\prime }\left(z\right)\phi \left(s\left(z\right)\right)\prec \vartheta \left(\upsilon \left(z\right)\right)+z{\upsilon }^{\prime }\left(z\right)\phi \left(\upsilon \left(z\right)\right).$$

Then it follows that

$$s\left(z\right)\prec \upsilon \left(z\right),$$

and υ is the extremal function in this subordination framework.

Lemma 2 

([34]). Let υ be a convex and injective function within U, and suppose that ν and φ are holomorphic in a region, D, that includes the image of $\upsilon \left(U\right)$. Assume that the following conditions hold:

1. $Re\left({\displaystyle \frac{{\nu }^{\prime }\left(\upsilon \left(z\right)\right)}{\phi \left(\upsilon \left(z\right)\right)}}\right)>0$, for $z\in U$;

2. $\psi \left(z\right)=z{\upsilon }^{\prime }\left(z\right)\phi \left(\upsilon \left(z\right)\right)$ is univalent and starlike in U.

Now consider a function, $s\left(z\right)$, satisfying

$$\begin{array}{ccc}& & s\left(z\right)\in \mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q,\hfill \\ & & s\left(U\right)\subseteq D,\hfill \\ & & \nu \left(s\left(z\right)\right)+z{s}^{\prime }\left(z\right)\phi \left(s\left(z\right)\right)\mathit{is}\mathit{injective}\mathit{in}U,\hfill \end{array}$$

and

$$\nu \left(\upsilon \left(z\right)\right)+z{u}^{\prime }\left(z\right)\phi \left(\upsilon \left(z\right)\right)\prec \nu \left(s\left(z\right)\right)+z{s}^{\prime }\left(z\right)\phi \left(s\left(z\right)\right).$$

Then it follows that

$$\upsilon \left(z\right)\prec s\left(z\right),$$

and υ is the best subordinant.

The present work is driven by the increasing interest in generalizing classical differential operators within the framework of symmetric q-calculus—a setting that offers a more balanced and adaptable analytical structure compared to its traditional counterparts. In particular, symmetric q-calculus provides a natural extension for studying the differential properties of analytic functions, capturing both classical behavior and quantum deformations in a unified approach.

This article is organized into two main sections, each addressing a distinct aspect of the analysis of analytic functions within the framework of symmetric q-calculus. In the Section 1, we investigate a special class of analytic functions denoted by ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$, which is defined in terms of the previously introduced symmetric q-differential operator ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$. The focus is placed on establishing coefficient-based conditions for membership in this class, along with inclusion results that relate different instances of the class, for varying values of the parameter i. The analysis leverages the properties of convex and monotonic sequences and exploits the geometric characteristics of the associated functions to derive sharp bounds and inclusion criteria. In the Section 2, we turn to the study of differential subordination and superordination involving functions transformed by the operator ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$. We provide sufficient conditions under which such functions are subordinate or superordinate to given univalent functions, and we explicitly identify the best dominant and best subordinant in each case. To unify these dual perspectives, we further derive a sandwich-type theorem, providing two-sided constraints that simultaneously reflect both subordination and superordination properties. Collectively, these results demonstrate the robustness and versatility of the proposed symmetric q-operator as a powerful tool in analyzing differential inequalities, identifying sharp bounds, and classifying subclasses of analytic and multivalent functions.

Let $f\in {\mathcal{A}}_p$, where $f\left(z\right)=z^p+{\sum }_{j=1}^{\infty }{a}_{j+p}{z}^{j+p}$. Now define the symmetric q-differential operator ${\tilde{\mathcal{V}}}_{\eta ,q}:{\mathcal{A}}_p\to {\mathcal{A}}_p$, depending on the parameters $\eta \ge 0$, $0<q<1$, and $p\in \mathbb{N}$, using the expression

$${\tilde{\mathcal{V}}}_{\eta ,q,p}f\left(z\right)=\left(1-\eta {\widetilde{\left[p\right]}}_q\right)f\left(z\right)+\eta z{\tilde{D}}_{q}f\left(z\right),$$

where ${\tilde{D}}_q$ denotes the symmetric q-derivative. Using the series expansion of f, we obtain

$${\tilde{\mathcal{V}}}_{\eta ,q,p}f\left(z\right)=z^p+\sum _{j=1}^{\infty }\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta \widetilde{{\left[j+p\right]}_q}\right)a_{j+p}{z}^{j+p}.$$

(8)

We further define the iterated forms of the operator as follows:

$${\tilde{\mathcal{V}}}_{\eta ,q,p}^{0}f\left(z\right)=f\left(z\right),$$

$${\tilde{\mathcal{V}}}_{\eta ,q,p}^{2}f\left(z\right)={\tilde{\mathcal{V}}}_{\eta ,q,p}\left({\tilde{\mathcal{V}}}_{\eta ,q,p}f\left(z\right)\right)=z^p+\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^2{a}_{j+p}{z}^{j+p}.$$

Proceeding by induction, the general formula for the ith iterate of the operator is given by

$${\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)={\tilde{\mathcal{V}}}_{\eta ,q,p}\left({\tilde{\mathcal{V}}}^{i-1}{V}_{\eta ,q}^{i-1}f\left(z\right)\right)=z^p+\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{a}_{j+p}{z}^{j+p}.$$

(9)

Remark 3. 

In the limiting case $q\to 1^{-}$ and $p=1$, the symmetric q-derivative, as outlined in (9), becomes equivalent to the well-known operator attributed to Al-Oboudi (7) [33]. Furthermore, under the specific choices $q\to 1^{-}$, $p=1$, and $\eta =1$, the operator defined in (9) coincides with Salagean’s well-known differential operator [35].

Remark 4. 

The operator ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$ considered in this paper is closely related to the operator ${\tilde{\mathcal{L}}}_{j,q}$ previously introduced by the author in [36]. Both are constructed using the symmetric q-derivative, but they differ in their structure and scope. Specifically, while ${\tilde{\mathcal{L}}}_{\tau ,q}$ acts on meromorphic multivalent functions in the punctured disk $U^{\ast }$, the present operator ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$ allows us to explore a broader family of analytic functions. Thus, while both operators are symmetric q-differentials in nature, the previous study addressed meromorphic multivalent geometry (coefficients, growth, radii) in $U^{\ast }$, whereas the current work focuses on analytic subclasses in U and on sharp subordination–superordination structures. Together, the two approaches are complementary rather than overlapping.

Differentiating ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$ using the symmetric q-derivative operator gives

$$\left({\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\right)\left(z\right)={\widetilde{\left[p\right]}}_q{z}^{p-1}+\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\widetilde{\left[j+p\right]}}_q{a}_{j+p}{z}^{j+p-1}.$$

(10)

A direct computation leads to the following identity for $\eta \ge 0$.

$$\eta z\left({\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\right)\left(z\right)+(1-\eta {\widetilde{\left[p\right]}}_q){\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)={\tilde{\mathcal{V}}}_{\eta ,q,p}^{i+1}f\left(z\right).$$

(11)

In the limiting case where $q\to 1^{-}$ and $p=1$, the identity in Equation (11) reduces to the classical relation associated with the Al-Oboudi differential operator:

$$z{\left({\tilde{\mathcal{V}}}_{\eta ,1,1}^{i}f\right)}^{\prime }\left(z\right)={\displaystyle \frac{1}{\eta }}{\tilde{\mathcal{V}}}_{\eta ,1,1}^{i+1}f\left(z\right)-{\displaystyle \frac{1-\eta }{\eta }}{\tilde{\mathcal{V}}}_{\eta ,1,1}^{i}f\left(z\right).$$

(12)

2. Coefficient Inequalities and Inclusion Results

Our aim in this section is to prove a subordination theorem that characterizes membership in the class ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$. Our approach relies on analytic subordination and the properties of convex decreasing sequences.

Definition 3. 

Let $f\in$${\mathcal{A}}_p$ denote a function from the analytic class ${\mathcal{A}}_p$. The function f is said to be a member of the class ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$ if it fulfills the required subordination condition:

$${\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left({\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}-\beta \right)\prec {\displaystyle \frac{1+Rz}{1+Sz}},z\in U,$$

(13)

where $-1\le R<S\le 1;$ $0<S\le 1$ and $0\le \beta <{\widetilde{\left[p\right]}}_q$. According to the subordination principle, $f\in {\tilde{\mathcal{P}}}_q^i\left(R,S,\eta ,\beta ,p\right)$ if and only if the following inequality holds:

$$\left|{\displaystyle \frac{\frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}-{\widetilde{\left[p\right]}}_q}{S\frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}+\beta \left(S-R\right)-{\widetilde{\left[p\right]}}_{q}R}}\right|<1.$$

(14)

Remark 5. 

In the limiting scenario where $q\to 1^{-}$, it is observed that by assigning specific values to the parameters i, p, R, and S, the class ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$ simplifies to various established subclasses of analytic functions. Such cases include the following:

• 1. The class

  $${\tilde{\mathcal{J}}}_1^0\left(R,S,\eta ,\beta ,p\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{1}{p-\beta }}\left({\displaystyle \frac{z{\left(V_{\eta ,p}^{0}f\left(z\right)\right)}^{\prime }}{z^p}}-\beta \right)\prec {\displaystyle \frac{1+Rz}{1+Sz}}\right\}$$
  was investigated in detail by Aouf (see [37]);
• 2. The subclasses

  $${\tilde{\mathcal{J}}}_1^0\left(-1,1,\eta ,\beta ,p\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{1}{p-\beta }}\left({\displaystyle \frac{z{\left(V_{\eta ,p}^{0}f\left(z\right)\right)}^{\prime }}{z^p}}-\beta \right)\prec {\displaystyle \frac{1-z}{1+z}}\right\}$$
  and

  $${\tilde{\mathcal{J}}}_1^0\left(\left(2\xi -1\right)m,m,\eta ,0,p\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,p}^{0}f\left(z\right)\right)}^{\prime }}{p{z}^p}}\prec {\displaystyle \frac{1+\left(2\xi -1\right)mz}{1+mz}}\right\}$$
  were explored by Owa (see [38,39]);
• 3. The class

  $${\tilde{\mathcal{P}}}_1^0\left(R,S,\eta ,0,p\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,p}^{0}f\left(z\right)\right)}^{\prime }}{p{z}^p}}\prec {\displaystyle \frac{1+Rz}{1+Sz}}\right\}$$
  was analyzed by Chen (see [40]);
• 4. The class

  $${\tilde{\mathcal{J}}}_1^0\left(\left(\xi -1\right)n,mn,\eta ,0,p\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,p}^{0}f\left(z\right)\right)}^{\prime }}{p{z}^p}}\prec {\displaystyle \frac{1+\left(\xi -1\right)nz}{1+mnz}}\right\}$$
  was the subject of analysis by Kim and Lee (see [41]);
• 5. The class

  $${\tilde{\mathcal{J}}}_1^0\left(\left(2\xi -1\right)m,m,\eta ,0,1\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,1}^{0}f\left(z\right)\right)}^{\prime }}{z}}\prec {\displaystyle \frac{1+\left(2\xi -1\right)mz}{1+mz}}\right\}$$
  was considered by Juneja and Mogra (see [42]);
• 6. The class

  $${\tilde{\mathcal{J}}}_1^0\left(R,S,\eta ,0,1\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,1}^{0}f\left(z\right)\right)}^{\prime }}{z}}\prec {\displaystyle \frac{1+Rz}{1+Sz}}\right\}$$
  was examined by Mehrok (see [43]);
• 7. The class

  $${\tilde{\mathcal{J}}}_1^0\left(-1,1,\eta ,0,1\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,1}^{0}f\left(z\right)\right)}^{\prime }}{z}}\prec {\displaystyle \frac{1-z}{1+z}}\right\}$$
  was considered by Mac-Gregor (see [44]);
• 8. The class

  $${\tilde{\mathcal{J}}}_1^0\left(-\xi ,\xi ,\eta ,0,1\right)=\left\{f\in {\mathcal{A}}_p:{\displaystyle \frac{z{\left(V_{\eta ,1}^{0}f\left(z\right)\right)}^{\prime }}{z}}\prec {\displaystyle \frac{1-\xi z}{1+\xi z}}\right\}$$
  was investigated by Padmanabhan (see [45]) and by Caplinger and Causey (see [46]).

The following preliminary lemmas are essential to establish our results for this section.

Lemma 3 

(see [44]). If the constants $R_2$, $R_1$, $S_1$, and $S_2$ are real and satisfy $-1\le R_2\le R_1<S_1\le S_2\le 1$, then ${\displaystyle \frac{1+S_{1}z}{1+R_{1}z}}$ is subordinate to ${\displaystyle \frac{1+S_{2}z}{1+R_{2}z}}$.

Lemma 4 

(see [47]). Let $a_1=1$ and suppose that $a_j\ge 0$ for all $j\ge 2$, with the sequence $\left\{a_j\right\}$ being convex and non-increasing; that is, $a_j-2a_{j+1}+a_{j+2}\ge 0$ and $a_{j+1}+a_{j+2}\ge 0$ for all $j\in \mathbb{N}$. Consequently, the following relation is satisfied throughout the unit disk U:

$$Re\left\{\sum _{j=1}^{\infty }{a}_j{z}^{j-1}\right\}>{\displaystyle \frac{1}{2}}.$$

Theorem 1. 

Consider $f\in$${\mathcal{A}}_p$. The function f is a member of the subclass ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$ provided that it fulfills the associated condition:

$$\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\displaystyle \frac{{\widetilde{\left[j+p\right]}}_q}{{\widetilde{\left[p\right]}}_q-\beta }}\left|a_{j+p}\right|<{\displaystyle \frac{S-R}{1+S}}.$$

(15)

Proof. 

Under the assumption that (15) holds true, it is sufficient to prove that

$$\left|{\displaystyle \frac{\frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}-{\widetilde{\left[p\right]}}_q}{S\frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}+\beta \left(S-R\right)-{\widetilde{\left[p\right]}}_{q}R}}\right|<1.$$

Indeed,

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{\frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}-{\widetilde{\left[p\right]}}_q}{S\frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}+\beta \left(S-R\right)-{\widetilde{\left[p\right]}}_{q}R}}\right|\hfill \\ & =& \left|{\displaystyle \frac{{\sum }_{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\widetilde{\left[j+p\right]}}_q{a}_{j+p}{z}^{j+p}}{\left({\widetilde{\left[p\right]}}_q-\beta \right)\left(S-R\right)z^p+S{\sum }_{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\widetilde{\left[j+p\right]}}_q{a}_{j+p}{z}^{j+p}}}\right|\hfill \\ & <& {\displaystyle \frac{{\sum }_{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\widetilde{\left[j+p\right]}}_q\left|a_{j+p}\right|}{\left({\widetilde{\left[p\right]}}_q-\beta \right)\left(S-R\right)-S{\sum }_{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\widetilde{\left[j+p\right]}}_q\left|a_{j+p}\right|}}.\hfill \end{array}$$

Provided that condition (15) is fulfilled, the preceding inequality is bounded above by 1. As a result, the function f qualifies as a member of the subclass ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$. □

For $S=1$ and $R=-1$, the subsequent statement is true.

Corollary 1. 

Assume that $f\in$${\mathcal{A}}_p$. The function f lies in the subclass ${\tilde{\mathcal{J}}}_q^i\left(-1,1,\eta ,\beta ,p\right)$ whenever the related condition holds:

$$\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\displaystyle \frac{{\widetilde{\left[j+p\right]}}_q}{{\widetilde{\left[p\right]}}_q-\beta }}\left|a_{j+p}\right|<1.$$

Remark 6. 

In connection with Corollary 1, it is worth emphasizing that the special choice of $R=-1$ and $S=1$ leads to the subordination condition

$${\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left({\displaystyle \frac{z{\tilde{D}}_q{\tilde{V}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}-\beta \right)\prec {\displaystyle \frac{1-z}{1+z}},z\in U,$$

which corresponds to the Janowski function mapping the unit disk onto the right half-plane. This case highlights a well-known subclass of q- starlike functions associated with the right half-plane, extending and generalizing earlier results obtained in a classical setting. Hence, Corollary 1 not only simplifies the analytic condition but also establishes a direct bridge between our generalized symmetric q-framework and these classical subclasses.

In order to prove the upcoming theorem, we begin by demonstrating the following lemma.

Lemma 5. 

Provided that η satisfies $\eta \ge {\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q}}$, the following inequality holds for all $z\in U:$

$$Re\left\{1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j\right\}>{\displaystyle \frac{1}{2}}.$$

(16)

Proof. 

Let

$$r\left(z\right)=1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j=1+\sum _{j=1}^{\infty }{Y}_j{z}^j,$$

with

$$Y_j={\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}},$$

given any $j\ge 1,\eta \ge 0$, $0<q<1$, and $p\in \mathbb{N}$. Clearly, $Y_j\ge 0$ for any $j\ge 1$.

A simple calculation shows that

$$Y_{j+1}={\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p+1\right]}}_q}},$$

(17)

and

$$Y_{j+2}={\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p+2\right]}}_q}},$$

(18)

so

$$Y_{j+1}-Y_{j+2}={\displaystyle \frac{\eta \left({\widetilde{\left[j+p+2\right]}}_q-{\widetilde{\left[j+p+1\right]}}_q\right)}{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p+1\right]}}_q\right)\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p+2\right]}}_q\right)}}\ge 0.$$

(19)

By applying (17), (18), and (19), we deduce that

$$\begin{array}{ccc}& & Y_j-2Y_{j+1}+Y_{j+2}\hfill \\ & =& {\displaystyle \frac{\eta X_1+{\eta }^2{X}_2}{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p+1\right]}}_q\right)\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p+2\right]}}_q\right)}},\hfill \end{array}$$

where $X_1=\left(1-\eta {\widetilde{\left[p\right]}}_q\right)\left(2{\widetilde{\left[j+p+1\right]}}_q-{\widetilde{\left[j+p\right]}}_q-{\widetilde{\left[j+p+2\right]}}_q\right)$ and $X_2={\widetilde{\left[j+p+1\right]}}_q{\widetilde{\left[j+p+2\right]}}_q-2{\widetilde{\left[j+p\right]}}_q{\widetilde{\left[j+p+2\right]}}_q+{\widetilde{\left[j+p\right]}}_q{\widetilde{\left[j+p+1\right]}}_q$. Based on the assumption that $\eta \ge {\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q}}$, we conclude that $Y_j-2Y_{j+1}-Y_{j+2}\ge 0$ for any $j\ge 1$. Consequently, since $\left\{Y_j\right\}$ is a monotonically decreasing convex sequence, Lemma 4 yields

$$Re\left\{r\left(z\right)\right\}=Re\left\{1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j\right\}>{\displaystyle \frac{1}{2}}.$$

Thus, the proof is established. □

Theorem 2. 

If $\eta \ge {\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q}}$,

$${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(R,S,\eta ,\beta ,p\right).$$

(20)

Proof. 

Assume that $f\in {\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$. Invoking the definition of the class ${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)$, it follows that

$${\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left({\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}-\beta \right)\prec {\displaystyle \frac{1+Rz}{1+Sz}}.$$

Through the application of ${\tilde{\mathcal{J}}}_q^{i-1}\left(R,S,\eta ,\beta ,p\right)$ and a convolution operation, we arrive at

$$\begin{array}{ccc}& & {\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i-1}f\left(z\right)}{{\widetilde{\left[p\right]}}_q{z}^p}}={\displaystyle \frac{{\widetilde{\left[p\right]}}_q{z}^p+{\sum }_{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^{i-1}{\widetilde{\left[j+p\right]}}_q{a}_{j+p}{z}^{j+p}}{{\widetilde{\left[p\right]}}_q{z}^p}}\hfill \\ & =& 1+\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^{i-1}{\displaystyle \frac{{\widetilde{\left[j+p\right]}}_q}{{\widetilde{\left[p\right]}}_q}}{a}_{j+p}{z}^j\hfill \\ & =& \left(1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j\right)\ast \left(1+\sum _{j=1}^{\infty }{\left(1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q\right)}^i{\displaystyle \frac{{\widetilde{\left[j+p\right]}}_q}{{\widetilde{\left[p\right]}}_q}}{a}_{j+p}{z}^j\right)\hfill \\ & =& \left(1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j\right)\ast {\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{{\widetilde{\left[p\right]}}_q{z}^p}}.\hfill \end{array}$$

Hence, combining the above expression with (13), it follows that

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left({\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i-1}f\left(z\right)}{z^p}}-\beta \right)\hfill \\ & =& {\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left\{{\widetilde{\left[p\right]}}_q\left[\left(1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j\right)\ast {\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{{\widetilde{\left[p\right]}}_q{z}^p}}\right]-\beta \right\}\hfill \\ & =& \left(1+\sum _{j=1}^{\infty }{\displaystyle \frac{1}{1-\eta {\widetilde{\left[p\right]}}_q+\eta {\widetilde{\left[j+p\right]}}_q}}{z}^j\right)\ast {\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left({\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}-\beta \right)\hfill \\ & =& r\left(z\right)\ast {\displaystyle \frac{1+R\nu \left(z\right)}{1+S\nu \left(z\right)}},\hfill \end{array}$$

with $\nu \left(0\right)=0$ and $\left|\nu \left(z\right)\right|<1$. The application of the Herglotz theorem and the previous lemma yields

$${\displaystyle \frac{1}{{\widetilde{\left[p\right]}}_q-\beta }}\left({\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i-1}f\left(z\right)}{z^p}}-\beta \right)\prec {\displaystyle \frac{1+Rz}{1+Sz}},$$

as ${\displaystyle \frac{1+Rz}{1+Sz}}$ possesses convexity and univalence in U. Accordingly, this implies that

$${\tilde{\mathcal{J}}}_q^i\left(R,S,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(R,S,\eta ,\beta ,p\right).$$

This establishes the claim. □

Theorem 3. 

If the constants $R_2$, $R_1$, $S_1$, and $S_2$ are real and satisfy $-1\le R_2\le R_1<S_1\le S_2\le 1$ and $i\ge 0$, it follows that

$${\tilde{\mathcal{J}}}_q^i\left(S_1,R_1,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(S_2,R_2,\eta ,\beta ,p\right).$$

Proof. 

Employing Lemma 3, it follows that

$${\tilde{\mathcal{J}}}_q^{i-1}\left(S_1,R_1,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(S_2,R_2,\eta ,\beta ,p\right).$$

(21)

In view of (20) and (21), it follows that

$${\tilde{\mathcal{J}}}_q^i\left(S_1,R_1,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(S_1,R_1,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(S_2,R_2,\eta ,\beta ,p\right).$$

Hence, we get

$${\tilde{\mathcal{J}}}_q^i\left(S_1,R_1,\eta ,\beta ,p\right)\subseteq {\tilde{\mathcal{J}}}_q^{i-1}\left(S_2,R_2,\eta ,\beta ,p\right).$$

Therefore, the desired result is proved. □

3. Subordination and Superordination Results Obtained Using the Symmetric q-Differential Operator

In this section, we investigate differential subordination and superordination properties associated with the symmetric $q$-differential operator ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$. We establish sufficient conditions under which a transformed analytic function is subordinate (or superordinate) to a univalent target function. The results include sharp bounds obtained through identification of the best dominant and best subordinant, leading to the development of a comprehensive sandwich-type theorem. These findings extend classical results into the setting of symmetric q-calculus, highlighting the unifying potential of this operator in the theory of analytic and univalent functions.

The next two lemmas are derived using a method similar to the approach presented in [48].

Lemma 6. 

Assume that υ is a one-to-one analytic function on U and that ϑ and φ are holomorphic in a region, D, that includes the image $\upsilon \left(U\right)$, such that $\phi \left(w\right)$ remains nonzero for all $w\in \upsilon \left(U\right)$. Define $\Theta \left(z\right)=z{\tilde{D}}_q\upsilon \left(z\right)\phi \left(\upsilon \left(z\right)\right)$ and $h\left(z\right)=\vartheta \left(\upsilon \left(z\right)\right)+\Theta \left(z\right)$. Assume that the following conditions hold:

• 1. Θ is a starlike and injective function within U;
• 2. $Re\left({\displaystyle \frac{z{\tilde{D}}_{q}h\left(z\right)}{\Theta \left(z\right)}}\right)>0$, given any $z\in U$.

Provided that s is holomorphic in U and satisfies $s\left(0\right)=\upsilon \left(0\right)$, $s\left(U\right)\subseteq D$, and

$$\vartheta \left(s\left(z\right)\right)+z{\tilde{D}}_{q}s\left(z\right)\phi \left(s\left(z\right)\right)\prec \vartheta \left(\upsilon \left(z\right)\right)+z{\tilde{D}}_q\upsilon \left(z\right)\phi \left(\upsilon \left(z\right)\right)=h\left(z\right),$$

(22)

then $s\left(z\right)\prec \upsilon \left(z\right)$ and $\upsilon \left(z\right)$ is the best dominant.

Proof. 

Suppose that $\upsilon$ is a one-to-one mapping in U, and set $\Theta \left(z\right)=z{\tilde{D}}_q\upsilon \left(z\right)\phi \left(\upsilon \left(z\right)\right)$ and $h\left(z\right)=\vartheta \left(\upsilon \left(z\right)\right)+\Theta \left(z\right)$, where $\vartheta$ and $\phi$ are complex differentiable functions in a region, D, including $\upsilon \left(U\right)$, and $\phi \left(w\right)$ remains nonvanishing for every w in $\upsilon \left(U\right).$ Assume that conditions 1 and 2 hold. Define $m\left(z\right)=\vartheta \left(s\left(z\right)\right)+z{\tilde{D}}_{q}s\left(z\right)\phi \left(s\left(z\right)\right)$, with s being holomorphic in U, $s\left(0\right)=\upsilon \left(0\right)$, and $s\left(U\right)\subseteq D$. According to our hypothesis, $m\left(z\right)$ is subordinate to $h\left(z\right).$ Now, letting q approach 1 from below, the symmetric q-difference operator ${\tilde{D}}_{q}f\left(z\right)$ tends to the conventional differential operator $f^{\prime }\left(z\right)$. Therefore, (22) becomes

$$\vartheta \left(s\left(z\right)\right)+z{s}^{\prime }\left(z\right)\phi \left(s\left(z\right)\right)\prec \vartheta \left(\upsilon \left(z\right)\right)+z{\upsilon }^{\prime }\left(z\right)\phi \left(\upsilon \left(z\right)\right).$$

By applying the known lemma for the classical derivative 1, it follows that $s\left(z\right)\prec \upsilon \left(z\right)$ and $\upsilon$ constitutes the best majorizing function. □

Lemma 7. 

Assume that υ is a convex injective mapping on U and let ν and φ be holomorphic functions defined over a region, D, that encompasses the image $\upsilon \left(U\right)$. Assume that the following conditions hold:

1. For every $z\in U$, the real part of $\left({\displaystyle \frac{{\tilde{D}}_q\nu \left(\upsilon \left(z\right)\right)}{\phi \left(\upsilon \left(z\right)\right)}}\right)$ is positive;

2.The function $\psi \left(z\right)=z{\tilde{D}}_q\upsilon \left(z\right)\phi \left(\upsilon \left(z\right)\right)$ is both starlike and injective in U. Assuming that $s\left(z\right)\in \mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$, with $s\left(U\right)\subseteq D$ and $\nu \left(s\left(z\right)\right)+z{\tilde{D}}_{q}s\left(z\right)\phi \left(s\left(z\right)\right)$ being univalent in U and

$$\nu \left(\upsilon \left(z\right)\right)+z{\tilde{D}}_q\upsilon \left(z\right)\phi \left(\upsilon \left(z\right)\right)\prec \nu \left(s\left(z\right)\right)+z{\tilde{D}}_{q}s\left(z\right)\phi \left(s\left(z\right)\right),$$

it follows that $\upsilon \left(z\right)\prec s\left(z\right)$ and υ is the best subordinant.

Proof. 

This proof mirrors the method used in the previous lemma. □

The following theorem provides a sharp subordination result, where the symmetric q-differential operator acts on the function and the best dominant is explicitly identified.

Theorem 4. 

Assume that $f\in {\mathcal{A}}_p$ and that the function ${\displaystyle \frac{{\tilde{V}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ belongs to the class $\mathcal{H}\left(U\right)$, where $z\in U\setminus \left\{0\right\}$. Suppose that $\upsilon \left(z\right)$ is a convex univalent function in U, normalized by the condition $\upsilon \left(0\right)=1$. Let $m,n\in \mathbb{C}$, with $n\ne 0$, and suppose that the following condition is satisfied:

$$Re\left\{{\displaystyle \frac{m}{n}}+n+{\displaystyle \frac{z{\tilde{D}}_q^{\left(2\right)}\upsilon \left(z\right)}{{\tilde{D}}_q\upsilon \left(z\right)}}\right\}>0,z\in U.$$

(23)

The function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$ is given by

$${\psi }_{\eta ,q,p}^i\left(m,n;z\right)={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\left[m+{\displaystyle \frac{n}{\eta }}\left({\displaystyle \frac{{\tilde{V}}_{\eta ,q,p}^{i+1}f\left(z\right)}{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}}-1\right)\right].$$

(24)

If the subordination

$${\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec m\upsilon \left(z\right)+nz{\tilde{D}}_q\upsilon \left(z\right),$$

(25)

is satisfied in U, then the subordination relation

$${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec \upsilon \left(z\right),z\in U\setminus \left\{0\right\},$$

(26)

holds and υ represents the finest majorant.

Proof. 

Let the following auxiliary expression be given:

$$s\left(z\right):={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}},$$

(27)

where $z\in U\setminus \left\{0\right\},$$f\in {\mathcal{A}}_p$. It is evident that z is analytic in U and satisfies the normalization criterion $s\left(0\right)=1$.

By applying logarithmic symmetric q-differentiation, $s\left(z\right)$, on both sides of (27), we obtain

$${\displaystyle \frac{z{\tilde{D}}_{q}s\left(z\right)}{s\left(z\right)}}={\displaystyle \frac{z{\tilde{D}}_q{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{{\tilde{V}}_{\eta ,q,p}^{i}f\left(z\right)}}-{\widetilde{\left[p\right]}}_q.$$

Using identity (11), this expression reduces to

$${\displaystyle \frac{z{\tilde{D}}_{q}s\left(z\right)}{s\left(z\right)}}={\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i+1}f\left(z\right)}{{\tilde{V}}_{\eta ,q,p}^{i}f\left(z\right)}}-1\right).$$

(28)

We now introduce two auxiliary functions defined by $\vartheta \left(w\right):=aw$ and $\phi \left(w\right):=b$, where $m,n\in \mathbb{C}$ and n$\ne 0$. The function $\vartheta$ is clearly analytic in $\mathbb{C}\setminus \left\{0\right\}$, and $\phi$, being constant and nonzero, is also analytic in $\mathbb{C}\setminus \left\{0\right\}$ and non-vanishing in $\mathbb{C}\setminus \left\{0\right\}$.

Next, we define the function $\Theta \left(z\right)=z{\tilde{D}}_q\upsilon \left(z\right)\phi \left(\upsilon \left(z\right)\right)=nz{\tilde{D}}_q\upsilon \left(z\right)$, and consequently, we define $h\left(z\right)=\vartheta \left(\upsilon \left(z\right)\right)+\Theta \left(z\right)=m\upsilon \left(z\right)+nz{\tilde{D}}_q\upsilon \left(z\right)$.

Through straightforward calculation, we arrive at the following result:

$$\begin{array}{ccc}\hfill Re\left({\displaystyle \frac{z{\tilde{D}}_{q}h\left(z\right)}{\Theta \left(z\right)}}\right)& =& \\ \hfill Re\left\{{\displaystyle \frac{m}{n}}+b+{\displaystyle \frac{z{\tilde{D}}_q^{\left(2\right)}\upsilon \left(z\right)}{{\tilde{D}}_q\upsilon \left(z\right)}}\right\}& >& 0.\hfill \end{array}$$

By substituting from identity (28), we obtain the following relation:

$$\begin{array}{ccc}& & ms\left(z\right)+nz{\tilde{D}}_{q}s\left(z\right)=m{\displaystyle \frac{{\tilde{V}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}+{\displaystyle \frac{n}{\eta }}{\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\left({\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i+1}f\left(z\right)}{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}}-1\right)\hfill \\ & =& {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\left[m+{\displaystyle \frac{n}{\eta }}\left({\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i+1}f\left(z\right)}{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}}-1\right)\right].\hfill \end{array}$$

Based on (25), we can conclude that

$$ms\left(z\right)+nz{\tilde{D}}_{q}s\left(z\right)\prec m\upsilon \left(z\right)+nz{\tilde{D}}_q\upsilon \left(z\right).$$

Therefore, the requirements of the subordination in Lemma 6 are fully met, which implies that $s\left(z\right)\prec \upsilon \left(z\right)$ for all $z\in U\setminus \left\{0\right\}$. In other words, ${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec \upsilon \left(z\right)$, $z\in U\setminus \left\{0\right\},$ where $\upsilon$ serves as the best dominant for this subordination. □

Corollary 2. 

Let $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}}$ and $-1\le R<S\le 1$ and suppose that condition (23) is satisfied. If $f\in {\mathcal{A}}_p$ and

$${\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec m{\displaystyle \frac{1+Sz}{1+Rz}}+nz{\displaystyle \frac{S-R}{\left(1+Bqz\right)\left(1+B{q}^{-1}z\right)}},$$

for the complex constants $m,n\in \mathbb{C}$ and $n\ne 0,$$-1\le B<A\le 1$, where the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$ is defined in (24), then the following subordination holds:

$${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec {\displaystyle \frac{1+Sz}{1+Rz}},z\in U,$$

and the function ${\displaystyle \frac{1+Sz}{1+Rz}}$ is the best dominant.

Proof. 

By choosing $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}}$, with parameters in the range of $-1\le R<S\le 1,$ and applying Theorem 4, the desired conclusion follows immediately as a particular case. □

Remark 7. 

The function $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}},$ $z\in U$, where $-1\le R<S\le 1$, was originally introduced by Janowski [49]. It defines a conformal mapping of U onto a circular area in $\mathbb{C}$, exhibiting reflectional symmetry across the real line. Specifically, the image of U under $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}}$ is a disk centered at ${\displaystyle \frac{1-SR}{1-R^2}}$ with a radius of ${\displaystyle \frac{S-R}{1-R^2}}$. In the limiting case when $R=-1$, the image degenerates into a right half-plane. The real axis intercepts of the circular image— i.e., the endpoints of the diameter—are located at $M=\left({\displaystyle \frac{1-S}{1-R}},0\right)$ and $N=\left({\displaystyle \frac{1+S}{1+R}},0\right)$. These mappings, commonly referred to as Janowski functions, play a fundamental role in geometric function theory. They are extensively used in the characterization and analysis of various subclasses of univalent, starlike, and convex functions.

Corollary 3. 

Suppose that $\upsilon \left(z\right)={\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}},$ $0\le \xi <1$ and that condition (23) is satisfied. If $f\in {\mathcal{A}}_p$ and the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$, as given in (24), satisfies the subordination relation

$${\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec m{\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}}+nz{\displaystyle \frac{2\left(1-\xi \right)}{\left(1-qz\right)\left(1-q^{-1}z\right)}},$$

for some complex constants, $m,n\in \mathbb{C}$, and $n\ne 0,$ then the subordination

$${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec {\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}},$$

holds, and the function ${\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}}$ is the best dominant.

Proof. 

When choosing $\upsilon \left(z\right)={\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}},$ with $0\le \xi <1$ in Theorem 4, the stated result follows immediately as a particular instance of the general theorem. □

Corollary 4. 

Let $\upsilon \left(z\right)={\displaystyle \frac{1+z}{1-z}}$ and assume that condition (23) is satisfied. Suppose that $f\in {\mathcal{A}}_p$ and the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$, defined in (24), satisfies the subordination

$${\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec m{\displaystyle \frac{1+z}{1-z}}+nz{\displaystyle \frac{2}{\left(1-qz\right)\left(1-q^{-1}z\right)}},$$

for some complex constants, $m,n\in \mathbb{C}$, with $n\ne 0$. Then it follows that

$${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec {\displaystyle \frac{1+z}{1-z}},$$

and ${\displaystyle \frac{1+z}{1-z}}$ represents the best dominant.

Proof. 

This conclusion is an immediate consequence of Theorem 4 when selecting the specific function $\upsilon \left(z\right)={\displaystyle \frac{1+z}{1-z}}$, which satisfies the hypotheses of the theorem and corresponds to a particular case within its framework. □

Corollary 5. 

Let $\upsilon \left(z\right)={\displaystyle \frac{1+z}{{\left(1-z\right)}^2}},$ $z\in U$ and suppose that condition (23) is satisfied. If $f\in {\mathcal{A}}_p$ and the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$, as defined in (23), satisfies the subordination

$${\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec m{\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}+nz{\displaystyle \frac{\left(q^{-1}-q\right)z^2+\left(q^{-2}-q^2\right)z-3\left(q^{-1}-q\right)}{\left(q-q^{-1}\right){\left(1-qz\right)}^2{\left(1-q^{-1}z\right)}^2}},$$

for $m,n,c\in \mathbb{C}$, with $c\ne 0,$ then the subordination relation

$${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec {\displaystyle \frac{1+z}{{\left(1-z\right)}^2}},$$

holds, and the function ${\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}$ serves as the best dominant.

Proof. 

The result is obtained as a direct consequence of Theorem 4 by choosing $\upsilon \left(z\right)={\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}$, which satisfies the theorem’s assumptions and yields the stated conclusion as a particular case. □

The following theorem establishes a second-order differential superordination involving the symmetric q-differential operator ${\tilde{\mathcal{V}}}_{\eta ,q,p}^i$. It offers sufficient conditions under which a given univalent function dominates, in the sense of superordination, a transformed analytic function. A major contribution of this result is the explicit identification of the best subordinant—that is, the maximal function (with respect to subordination) among all the admissible candidates that satisfy the prescribed differential inequality. This theorem not only complements the earlier subordination result but also emphasizes the intrinsic duality of subordination and superordination in geometric function theory. Moreover, the framework of symmetric q-calculus enables a more general and nuanced formulation, providing enhanced analytical flexibility in capturing the behavior of the associated solutions.

Theorem 5. 

Consider υ to be an injective and convex mapping within U, normalized such that $\upsilon \left(0\right)=1$. Suppose that there exist constants, $m,n\in \mathbb{C},$ with $n\ne 0$, for which the following condition is satisfied:

$$Re\left({\displaystyle \frac{m{\tilde{D}}_q\upsilon \left(z\right)}{n}}\right)>0.$$

(29)

Assume that $f\in {\mathcal{A}}_p$ and that the function ${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ belongs to the class $\mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$. Further, let ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$, defined as in (24), be univalent in U. If the subordination condition

$$m\upsilon \left(z\right)+nz{\tilde{D}}_q\upsilon \left(z\right)\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right),$$

(30)

holds in U, then the subordination

$$\upsilon \left(z\right)\prec {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}},$$

(31)

follows, and the function υ is the best subordinant.

Proof. 

Let $s\left(z\right):={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ and $z\in U\setminus \left\{0\right\}$, $f\in {\mathcal{A}}_p$.

We introduce the auxiliary functions:

$$\nu \left(w\right):=mw\mathrm{and}\phi \left(w\right):=n.$$

Obviously, $\nu$ is a holomorphic function on $\mathbb{C}$, and $\phi$ is holomorphic on $\mathbb{C}$ and nonzero and non-vanishing in $\mathbb{C}\setminus \left\{0\right\}.$

It can be readily verified that

$${\displaystyle \frac{{\tilde{D}}_q\nu \left(\upsilon \left(z\right)\right)}{\phi \left(\upsilon \left(z\right)\right)}}={\displaystyle \frac{m{\tilde{D}}_q\upsilon \left(z\right)}{n}}.$$

Based on assumption (15), we have

$$Re\left({\displaystyle \frac{{\tilde{D}}_q\nu \left(\upsilon \left(z\right)\right)}{\phi \left(\upsilon \left(z\right)\right)}}\right)=Re\left({\displaystyle \frac{m{\tilde{D}}_q\upsilon \left(z\right)}{n}}\right)>0$$

Now, employing the subordination condition in (30), we obtain

$$m\upsilon \left(z\right)+nz{\tilde{D}}_q\upsilon \left(z\right)\prec ms\left(z\right)+nz{\tilde{D}}_{q}s\left(z\right).$$

An application of (7) then guarantees the subordination

$$\upsilon \left(z\right)\prec s\left(z\right)={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}},z\in U,$$

and further establishes that $\upsilon$ is the best subordinant. □

Corollary 6. 

Let $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}}$, where $-1\le R<S\le 1$. Assume that condition (29) holds. If $f\in {\mathcal{A}}_p$ and the function $s\left(z\right)={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ belongs to the class $\mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$, and if the subordination

$$m{\displaystyle \frac{1+Sz}{1+Rz}}+nz{\displaystyle \frac{S-R}{\left(1+Rqz\right)\left(1+R{q}^{-1}z\right)}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right),$$

holds in $U,$ for constants $m,n\in \mathbb{C}$, with $n\ne 0$, where ${\psi }_{\beta }^{m,n}\left(m,n;z\right)$ is defined in (24), then the subordination relation

$${\displaystyle \frac{1+Sz}{1+Rz}}\prec {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}},z\in U,$$

is satisfied, and the function $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}}$ represents the best subordinant.

Proof. 

The result follows immediately by applying Theorem 5 with the specific choice $\upsilon \left(z\right)={\displaystyle \frac{1+Sz}{1+Rz}}$ and $-1\le R<S\le 1$, provided that the conditions are satisfied. □

Corollary 7. 

Assume that $\upsilon \left(z\right)={\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}},0\le \xi <1,z\in U$ and assume that condition (29) is satisfied. Suppose that the function $f\in {\mathcal{A}}_p$ and ${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$$\in \mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$. Assume that the subordination

$$m{\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}}+nz{\displaystyle \frac{2\left(1-\xi \right)}{\left(1-qz\right)\left(1-q^{-1}z\right)}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right),$$

holds in U for the complex constants m and n, with $n\ne 0$, and where ξ is within the range $\left(0,1\right]$, where ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$ corresponds to the definition in (24). In that case, the subordination result

$${\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}}\prec {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$$

holds, and the function ${\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}}$ is the best possible subordinant.

Proof. 

The corollary emerges as a special case of Theorem 5 when choosing $\upsilon \left(z\right)={\displaystyle \frac{1+\left(1-2\xi \right)z}{1-z}}$ for $0<\xi \le 1.$ □

Corollary 8. 

Let $\upsilon \left(z\right)={\displaystyle \frac{1+z}{1-z}},$ $z\in U$ and assume that condition (29) is satisfied. Let $f\in {\mathcal{A}}_p$, and define the function $s\left(z\right)={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$. Consider also that $s\left(z\right)\in \mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$ and that the subordination

$$m{\displaystyle \frac{1+z}{1-z}}+nz{\displaystyle \frac{2}{\left(1-qz\right)\left(1-q^{-1}z\right)}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right),$$

holds for some constants, $m,n\in \mathbb{C}$, with $n\ne 0,$ where the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$ is defined in (24). Then the subordination

$${\displaystyle \frac{1+z}{1-z}}\prec {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}},$$

holds, and the function ${\displaystyle \frac{1+z}{1-z}}$ is the best subordinant.

Proof. 

The conclusion follows directly by applying Theorem 5 with the specific choice $\upsilon \left(z\right)={\displaystyle \frac{1+z}{1-z}}$, which is a convex univalent function in U, satisfying the requirement $\upsilon \left(0\right)=1$. The stated assumptions ensure that all the hypotheses of the theorem are satisfied, and thus the desired result follows. □

Corollary 9. 

Let $\upsilon \left(z\right)={\displaystyle \frac{1+z}{{\left(1-z\right)}^2}},$ $z\in U$ and suppose that condition (29) holds. Let $f\in {\mathcal{A}}_p$ and define the function $s\left(z\right)={\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$. Assume additionally that $s\left(z\right)\in \mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$ and that the subordination

$$m{\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}+nz{\displaystyle \frac{\left(q^{-1}-q\right)z^2+\left(q^{-2}-q^2\right)z-3\left(q^{-1}-q\right)}{\left(q-q^{-1}\right){\left(1-qz\right)}^2{\left(1-q^{-1}z\right)}^2}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right),$$

holds for constants $m,n\in \mathbb{C}$, with $n\ne 0,$ where ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$ is defined in (24). Then the subordination

$${\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}\prec {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}},$$

holds, and the function ${\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}$ is the best subordinant.

Proof. 

The result follows as a direct application of Theorem 5 when taking $\upsilon \left(z\right)={\displaystyle \frac{1+z}{{\left(1-z\right)}^2}}$, a convex univalent function in the unit disk U, normalized by $\upsilon \left(0\right)=1$. Given the stated assumptions, all the hypotheses of the theorem are satisfied, thereby ensuring the validity of the conclusion. □

By jointly utilizing the statements of Theorems 4 and 5, a unified conclusion is formulated in the form of a double subordination theorem. This finding combines the structures of both lower and upper function dominance, producing a bidirectional inclusion for the analytic function under consideration. Specifically, it demonstrates that the function transformed using the symmetric q-differential operator is bounded above and below by two extremal functions—identified as the best subordinant and best dominant, respectively—each satisfying specific geometric or analytic criteria. This formulation offers a sharp and symmetric enclosure of the transformed function, highlighting the effectiveness of combining subordination and superordination techniques within the structure of symmetric q-calculus. Sandwich-type theorems of this nature are particularly significant in geometric function theory, as they deliver precise information about functional behavior and constraints within complex domains.

Theorem 6. 

Let ${\upsilon }_1$ and ${\upsilon }_2$ be holomorphic and one-to-one functions in the unit disk U, where for every $z\in U$, both ${\upsilon }_1\left(z\right)$ and ${\upsilon }_2\left(z\right)$ do not vanish. Furthermore, assume that the functions $z{\tilde{D}}_q{\upsilon }_1\left(z\right)$ and $z{\tilde{D}}_q{\upsilon }_1\left(z\right)$ are both starlike and univalent in U. Suppose that ${\upsilon }_1$ satisfies condition (23) and ${\upsilon }_2$ satisfies condition (29), and let $f\in {\mathcal{A}}_p$ be such that ${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ belongs to the class $\mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$. Assume also that the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$ as defined in (24) is univalent in U, and that for constants $m,n\in \mathbb{C}$, with $n\ne 0$, the following double subordination holds:

$$m{\upsilon }_1\left(z\right)+nz{\tilde{D}}_q{\upsilon }_1\left(z\right)\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec m{\upsilon }_2\left(z\right)+nz{\tilde{D}}_q{\upsilon }_2\left(z\right),$$

Then it follows that

$${\upsilon }_1\left(z\right)\prec {\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}\prec {\upsilon }_2\left(z\right),$$

and the functions ${\upsilon }_1$ and ${\upsilon }_2$ represent, respectively, the best subordinant and the best dominant in this sandwich-type subordination.

Let ${\upsilon }_1\left(z\right)={\displaystyle \frac{1+S_{1}z}{1+R_{1}z}}$, ${\upsilon }_2\left(z\right)={\displaystyle \frac{1+S_{2}z}{1+R_{2}z}}$, assuming an ordered relation among the parameters, $-1\le R_2<R_1<S_1<S_2\le 1$. Under these constraints, the following corollary is valid.

Corollary 10. 

Assume that the conditions specified in (23) and (29) are satisfied for the functions ${\upsilon }_1\left(z\right)={\displaystyle \frac{1+S_{1}z}{1+R_{1}z}}$ and ${\upsilon }_2\left(z\right)={\displaystyle \frac{1+S_{2}z}{1+R_{2}z}}$, respectively, where the parameters are subject to the ordering $-1\le R_2<R_1<S_1<S_2\le 1$. Let $f\in {\mathcal{A}}_p$, and suppose that the function ${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ belongs to the class $\mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$. Assume further that the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$, defined in (24), is univalent in U, and that for constants $m,n\in \mathbb{C}$, with $n\ne 0$, the following double subordination holds:

$$m{\displaystyle \frac{1+S_{1}z}{1+R_{1}z}}+nz{\displaystyle \frac{S_1-R_1}{\left(1+R_{1}qz\right)\left(1+R_1{q}^{-1}z\right)}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right)$$

$$\prec m{\displaystyle \frac{1+S_{2}z}{1+R_{2}z}}+nz{\displaystyle \frac{S_2-R_2}{\left(1+R_{2}qz\right)\left(1+R_2{q}^{-1}z\right)}},$$

It can therefore be concluded that

$${\displaystyle \frac{1+S_{1}z}{1+R_{1}z}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec {\displaystyle \frac{1+S_{2}z}{1+R_{2}z}},$$

and consequently, the functions ${\displaystyle \frac{1+S_{1}z}{1+R_{1}z}}$ and ${\displaystyle \frac{1+S_{2}z}{1+R_{2}z}}$ serve, respectively, as the best subordinant and best dominant within this double subordination framework.

Corollary 11. 

Assume that the conditions specified in (23) and (29) are satisfied for the functions ${\upsilon }_1\left(z\right)={\displaystyle \frac{1+\left(1-2{\xi }_1\right)z}{1-z}}$ and ${\upsilon }_2\left(z\right)={\displaystyle \frac{1+\left(1-2{\xi }_2\right)z}{1-z}}$, respectively, where the parameters fulfill the inequality $0\le {\xi }_2\le {\xi }_1\le 1$. Let $f\in {\mathcal{A}}_p$, and suppose that the function ${\displaystyle \frac{{\tilde{\mathcal{V}}}_{\eta ,q,p}^{i}f\left(z\right)}{z^p}}$ belongs to the class $\mathcal{H}\left[\upsilon \left(0\right),1\right]\cap Q$. Furthermore, assume that the function ${\psi }_{\eta ,q,p}^i\left(m,n;z\right)$, defined in (24), is univalent in U, and that for constants $m,n\in \mathbb{C}$, with $n\ne 0$, the following double subordination holds:

$$m{\displaystyle \frac{1+\left(1-2{\xi }_1\right)z}{1-z}}+nz{\displaystyle \frac{2\left(1-{\xi }_1\right)}{\left(1-qz\right)\left(1-q^{-1}z\right)}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right)$$

$$\prec m{\displaystyle \frac{1+\left(1-2{\xi }_2\right)z}{1-z}}+nz{\displaystyle \frac{2\left(1-{\xi }_2\right)}{\left(1-qz\right)\left(1-q^{-1}z\right)}},$$

Accordingly, we conclude that

$${\displaystyle \frac{1+\left(1-2{\xi }_1\right)z}{1-z}}\prec {\psi }_{\eta ,q,p}^i\left(m,n;z\right)\prec {\displaystyle \frac{1+\left(1-2{\xi }_2\right)z}{1-z}},$$

and therefore, the functions ${\displaystyle \frac{1+\left(1-2{\xi }_1\right)z}{1-z}}$ and ${\displaystyle \frac{1+\left(1-2{\xi }_2\right)z}{1-z}}$ serve as the optimal subordinant and dominant, respectively, within this subordination framework.

4. Comments and Reflections

Within this work, we investigated a family of holomorphic functions characterized by a symmetric generalized derivative in the sense of q-calculus, framed within the context of geometric function theory. In the Section 1, we established sufficient conditions based on coefficient inequalities for functions to belong to a specific subclass associated with the operator. Additionally, we analyzed how this class behaves as certain parameters vary, leading to several inclusion results that generalize known properties from a classical setting to the context of symmetric q-calculus.

In the Section 2, we developed a general theory of differential subordination and superordination involving functions transformed by the operator. We provided sufficient conditions for a function to be subordinate or superordinate to a given univalent function, and we identified both the best dominant and the best subordinant for certain cases. As a result, an integrated analytic condition was established, encompassing both the extremal behaviors of the function.

In summary, the findings broaden traditional frameworks and highlight the adaptability and usefulness of the symmetric quantum difference method in exploring the analytic behavior and structural characteristics of complex functions. The approach outlined in this work paves the way for future exploration of multivalent, harmonic, and bi-univalent function classes subjected to symmetric q-transformations, and also suggests promising applications in areas such as quantum calculus and mathematical physics.