Differential subordination and superordination studies involving symmetric functions using a $q$-analogue multiplier operator

1.   Introduction

The original results obtained in this work are connected to the geometric function theory, and they were obtained using methods based on subordination and with the help of a $q$-calculus operator. The main notions that define the context of the research are first presented.

Let $H$ be the class of analytic functions in the open unit disc $\mathbb{D} : = \{\varsigma \in \mathbb{C}:\left\vert \varsigma \right\vert < 1\}$.

A notable subclass of $H$ is denoted by $H[a, n]$ and contains functions $f\in H$ of the form

Another remarkable subclass of $H$ is denoted by $A(n)$ and consists of functions $f\in H$ of the form

with $n\in \mathbb{N} = \{1, 2, ...\}$ and written as $A = A(1).$ The subclass of $A$ represented by

denotes the class of convex functions in the unit disk $\mathbb{D}$.

The notion of subordination ^[1,2,3] is characterized by the following:

If $f$ and $\hslash$ are analytic in $\mathbb{D}$, $f$ is said to be subordinate to $\hslash$, denoted by $f(\varsigma)\prec \hslash (\varsigma)$, if there exists an analytic function $\varpi$; with $\varpi (0) = 0$ and $\left\vert \varpi (\varsigma)\right\vert < 1$ for all $\varsigma \in \mathbb{D}$, such that $f(\varsigma) = \hslash (\varpi (\varsigma))$. Moreover, if the function $\hslash$ is univalent in $\mathbb{D}$, then the following equivalence holds:

For a function $f\in A(n)$ and a function $\hslash$ of the form

the well-known convolution product is defined as:

Many studies involving the $q$-derivative and the $q$-integral operators described by Jackson ^[4,5] have emerged in recent years due to the multiple applications of those operators in various branches of mathematics and other related fields. A comprehensive review regarding the quantum calculus apects applied in the geometric function theory was done ^[6], and Kanas and Răducanu ^[7] presented the $q$-analogue of the Ruscheweyh differential operator and looked into some of its features by utilizing the concept of convolution. Aldweby and Darus ^[8], Mahmood and Sokol ^[9] and others analyzed many types of analytical functions defined by the $q$-analogue of the Ruscheweyh differential operator. Multivalent analytic functions were investigated involving the $q$ -difference operator ^[10], and bi-univalent analytic functions are investigated under a similar operator in ^[11]. Analytic functions were investigated in a conic domain using $q$-calculus ^[12] and applications of the subordination concept and $q$-calculus were given ^[13,14,15]. The Faber polynomial expansion method was applied on bi-univalent functions using a $q$-integral operator ^[16], and the $q$-derivative linked Gegenbauer polynomials for certain bi-univalent functions ^[17]. Close-to-convex functions were investigated in association with $q$ -Srivastava-Attiya in operator ^[18], an extended $q$-analogue of multiplier transformation was used for subordination and superordination studies ^[19] and $q$-analogue of the Choi-Saigo-Srivastava operator was associated for the study presented ^[20].

The pleasant results recently obtained by combining the aforementioned quantum calculus components into the geometric function theory are what inspired the introduction of the new findings in this study. We were motivated to further explore the $q$-analogue of the multiplier transformation after reading about its applications in the definition of new subclasses of univalent functions, as well as after taking into account recent findings involving another quantum calculus operator and the classical theories of differential subordination and superordination ^[21,22,23].

The studies presented above motivated the use of the linear extended multiplier operator, which was recently defined using certain quantum calculus means ^[24] for the investigations applying the theories of differential subordination and superordiantion presented in this paper connected to new classes of analytic functions.

The fundamental concepts of the $q$-calculus, created by Jackson ^[4] and relevant to our research, will now be discussed. This method can also be applied to higher dimensional domains.

Jackson ^[4,5] defined the $q$-derivative operator $D_{q}$ of a function $f$:

As a remark, for a function $f$ written as (1.1), it implies

where $[\vartheta]_{q}$ is the $q$-bracket of $\vartheta$, that is

and

The definition of the $q$-number shift factorial for every nonnegative integer $\vartheta$ is

Wang et al. ^[20] the notion of the $q$-derivative and the concept of the convolution, the $q$-analogue Choi-Saigo-Srivastava operator $I_{\alpha, \beta }^{q}:A\rightarrow A$,

where

where $[\beta, q]_{\vartheta }$ is the $q$-generalized Pochhammer symbol for $\beta > 0$ defined by

Thus,

while

In ^[24], an extended multiplier operator was defined applying the operator $I_{\alpha, \beta }^{q}$ as follows:

Definition 1. ^[24] For $\mu \geq 0$ and $\tau > -1$, with the aid of the operator $I_{\alpha, \beta }^{q},$ we will define the new linear extended multiplier $q$-Choi-Saigo-Srivastava operator $D_{\alpha, \beta }^{m, q}(\mu, \tau):A\rightarrow A$ as follows:

where $\mu \geq 0$, $\tau > -1$, $m\in \mathbb{N}_{0}$, $\alpha > -1$, $\beta > 0$ and $0 < q < 1$.

If $f\in A$ has the form (1.1) from (1.4) and the above definition, it follows that

From (1.3) and (1.5), we find that

where

We note that

Assuming that $\lambda, \hbar \in H$, suppose

If $\lambda$ satisfies the first order differential subordination

then $\lambda$ is called to be a solution of the differential subordination in $(1.7).\ $The function $\varkappa$ is called a dominant of the solutions of the differential subordination in (1.7) if $\lambda (\varsigma)\prec \varkappa (\varsigma)$ for all the functions $\lambda$ satisfying (1.7). A dominant $\widetilde{\varkappa }$ is said to be the best dominant of (1.7) if $\widetilde{\varkappa }(\varsigma)\prec \varkappa (\varsigma)$ for all the dominants $\varkappa$.

If the following first order differential superordination is met by $\lambda$,

then $\lambda$ is called to be a solution of the differential superordination in (1.8). The function $\varkappa$ is called a subordinant of the solutions of the differential superordination in (1.8) if $\varkappa (\varsigma)\prec$ $\lambda (\varsigma)$ for all the functions $\lambda$ satisfying (1.8). A subordinant $\widetilde{ \varkappa }$ is said to be the best subordinant of (1.8) if $\varkappa (\varsigma)\prec \widetilde{\varkappa }(\varsigma)$ for all the subordinants $\varkappa$.

Miller and Mocanu ^[25] obtained sufficient conditions on the functions $\hbar$, $\varkappa$ and $\Phi$ for which the following implication holds:

Using the results presented ^[25], Bulboacă ^[26] investigated several classes of first-order differential superordinations and also considered superordination preserving integral operators ^[27]. Ali et al. ^[28] developed on Bulboacă's results and obtained sufficient conditions for specific normalized analytic functions $f(\varsigma)$ to satisfy

where $\varkappa _{1}$ and $\varkappa _{2}$ are univalent functions in $\mathbb{D}$ normalized with $\varkappa _{1}(0) = \varkappa _{2}(\varsigma) = 1.$

The function $f(\varsigma)$ defined by (1.1) is said to be a member of the class denoted by $S_{s}^{\ast }$ of starlike functions with respect to symmetric points if it satisfies the following condition:

The class $S_{s}^{\ast }$ was introduced by Sakaguchi ^[29] as a subclass of close-to-convex functions, and, hence, univalent in $\mathbb{D}$. It is also known that the class of convex functions and the class of odd starlike functions, with respect to the origin, are also included in $S_{s}^{\ast }$ ^[29,30].

Using this class as inspiration, Aouf et al. ^[31] developed and investigated the class $S_{s, n}^{\ast }T(1, 1)$ of functions $n$-starlike with respect to symmetric points, consisting of functions $f\in A$ with $a_{\vartheta }\leq 0$ for $\vartheta \geq 2$, and satisfying the inequality

where $D^{n}$ is the Sălăgean operator ^[32].

The classes defined in ^[30,31] are generalized by the following new class of functions, defined in this paper by applying the $D_{\alpha, \beta }^{m, q}(\mu, \tau)f(\varsigma)$ operator seen in Definition 1. The new class is introduced here as follows:

Definition 2. The function $f\in A(n)$ complying

is said to belong to the class $\aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D)$ if the following subordination condition is satisfied:

Using specific values for the parameters $\mu, \tau, \alpha, \beta$ and $q$ the following subclasses appear:

$(i)$ For $q\rightarrow 1^{-}$, the class $\Im _{\alpha, \beta, \mu }^{m, \tau }(\eta, \delta, C, D)$ is obtained as follows:

with the operator $\mathcal{L}_{\alpha, \beta }^{m}(\mu, \tau)f(\varsigma)$ given by (1.7);

$(ii)$ For $q\rightarrow 1^{-}$and $m = 0,$ the class $N^{\eta, \delta }(n, C, D)$ is obtained and rectifies the class introduced by Muhammad and Marwan ^[33] as follows:

The study exposed in this research tries to connect the special class of analytic functions with coefficients defined by the $q$-analogue operator with the differential subordination and superordination theory. As a result, certain sharp differential subordination and superordination results are investigated in the following theorems and corollaries for the functions belonging to the class $\aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D)$.

2.   Materials and methods

In order to prove the new differential subordination and superordination findings, the following known results will be used.

Definition 3. ^[3] (Definition 2.2b., p. 21). Denote by $\wp$ the set of all functions $f\ (\varsigma)\ $that are analytic and injective on $\overline{\mathbb{D}}\setminus E(f)$, where

and are such that $f^{\prime }(\zeta)\neq 0$ for $\zeta \in \partial \mathbb{D}\setminus E(f)$.

Lemma 1. ^[3] (Theorem 1b., p. 71). Let $h$ be a convex function in $\mathbb{D}$ with $h(0) = a$ and let $\gamma \in \mathbb{C}$ with $\mathrm{{Re}}(\gamma)\geq 0$. If $p\in H[a, n]$ and

then

The function $q$ is convex and is the best dominant of (2.1).

Lemma 2. ^[34] (Lemma 2.2., p. 3). Let $q$ be univalent in $\mathbb{D}$ with $q(0) = 1.$ Let $\xi, \varphi \in \mathbb{C}$ with $\varphi \neq 0,$ and suppose that

If $\lambda$ is analytic in $\mathbb{D}$ and

then $\lambda (\varsigma)\prec q(\varsigma)$, and $q$ is the best dominant of (2.2).

From ^[25] (Theorem 6, p. 820), we could easily obtain the following lemma:

Lemma 3. Let $q$ be convex in $\mathbb{D}$ and $\lambda \neq 0$, with $\mathrm{{Re}}\left(\lambda \right) \geq 0$. If $\breve{g}\in H[q(0), 1]\cap \wp$ such that $\breve{g}(\varsigma)+\lambda \varsigma \breve{g}^{\prime }(\varsigma)$ is univalent in $\mathbb{D}$, then

implies $q(\varsigma)\prec \breve{g}(\varsigma)$ and $q$ is the best subordinant of (2.3).

Lemma 4. ^[35]. Let $\mathcal{F}$ be analytic and convex in $\mathbb{D}$ and $0\leq \lambda \leq 1$. If $f, g\in A$ such that $f(\varsigma)\prec \mathcal{F}(\varsigma)$ and $g(\varsigma)\prec \mathcal{F}(\varsigma),$ then

3.   Results

The remainder of this paper assumes, unless otherwise stated, $\eta \in \mathbb{C},$ $0 < \delta < 1, -1\leq D < C\leq 1, \mu \geq 0$, $\tau > -1, m\in \mathbb{N} _{0}, \alpha > -1$, $\beta > 0$, $0 < q < 1$ and all the powers are understood as principle values.

Theorem 1. Consider $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D)$ and $\eta \in \mathbb{C} ^{\ast } = \mathbb{C} \backslash \{0\}, \ \delta > 0$ satisfying $\mathrm{{Re}}\eta \geq 0.$ Then,

and $q$ is convex $q\in$ $H[1, n]$ and is the best dominant.

Proof. Define the function $\omega (\varsigma)$ by

This function $\omega (\varsigma) \in H$ complies $\omega (0) = 1$. Differentiating (3.1) with respect to $\varsigma$ logarithmically, we have

Since

where,

we have

with

Moreover,

with unknowns $\eta _{j}, j\geq 1,$ we have

and equating the corresponding coefficients it follows that

Hence

Applying (3.1), the following can be written as

By employing the well-known binomial power expansion formula, we obtain

Now, from the subordination in (3.2) and using Lemma 1 for $\gamma = \frac{\delta }{\eta },$ the desired result is obtained. □

Using in Theorem 1 the assumption $q\rightarrow 1^{-}$, the following corollary emerges:

Corollary 1. Consider $f\in \Im _{\alpha, \beta, \mu }^{m, \tau }(\eta, \delta, C, D)$ and $\eta \in \mathbb{C} ^{\ast } = \mathbb{C} \backslash \{0\}, \ \delta > 0$ with $\mathrm{{Re}}\eta \geq 0.$ Then,

and $q$ is convex $q\in$ $H[1, n]$ and is the best dominant.

Remark 1. From Theorem 1 the following inclusion relation can be written:

Furthermore, the following inclusion relation holds for the class $\aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D)$:

Theorem 2. If $\eta _{1}, \eta _{2}\in \mathbb{R}$ such that $0\leq \eta _{1}\leq \eta _{2}\ $and $-1\leq D_{1}\leq D_{2} < C_{2}\leq C_{1}\leq 1,$ then

Proof. If $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta _{2}, \delta, C_{2}, D_{2})$, since $-1\leq D_{1}\leq D_{2} < C_{2}\leq C_{1}\leq 1,$ it is easy to show that

that is $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta _{1}, \delta, C_{1}, D_{1}),$ hence, the assertion in (3.3) holds for $\eta _{1} = \eta _{2}.$

If $0\leq \eta _{1} < \eta _{2}$ and considering Remark 1 and (3.4), it follows $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(0, \delta, C_{1}, D_{1})$, that is

A simple computation shows that

Moreover,

and the function $\frac{1+C_{1}\varsigma }{1+D_{1}\varsigma }$ with $-1\leq D_{1} < C_{1}\leq 1$ is analytic and convex in $\mathbb{D}$. Considering relation (3.6), using the subordination results given by (3.4) and (3.5) and using Lemma 4, we deduce that

that is $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta _{1}, \delta, C_{1}, D_{1}).$ □

Using in Theorem 2 the assumption $q\rightarrow 1^{-}$, the following corollary emerges:

Corollary 2. If $\eta _{1}, \eta _{2}\in \mathbb{R}$ such that $0\leq \eta _{1}\leq \eta _{2}\ $and $-1\leq D_{1}\leq D_{2} < C_{2}\leq C_{1}\leq 1,$ then

Example 1. Use $C_{1} = 1$ and $D_{1} = -1$ in Theorem 2 and Corollary 2. Let $\eta _{1}, \ \eta _{2}\in \mathbb{R}$ such that $0\leq \eta _{1}\leq \eta _{2}\ $and $-1\leq D_{2} < C_{2}\leq 1.$ We obtain

(i) If $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta _{2}, \delta, C_{2}, D_{2}),$ then

(ii) $f\in \Im _{\alpha, \beta, \mu }^{m, \tau }(\eta _{2}, \delta, C_{2}, D_{2}),$ then

Theorem 3. Consider $q\in K$, with $q(0) = 1$, $\ \eta \in \mathbb{C} ^{\ast }$ such that

If $f\in A(n)$ complies (1.9) and the following subordination is satisfied:

then

and $q(\varsigma)$ is the best dominant of (3.8).

Proof. The function $f\in A(n)$ is assumed to comply (1.9), hence, the function defined by (3.1) satisfies $\omega \in H$, with $\omega (0) = 1$. As done for proving Theorem 1, differentiating (3.1) with respect to $\varsigma$ gives that (3.8) is equivalent to

Thus, by Lemma 2, for $\xi = 1$ and $\varphi = \frac{\eta }{ \delta }$ we get $\omega (\varsigma)\prec q(\varsigma),$ and $q(\varsigma)$ is the best dominant of (3.8). □

Taking $q(\varsigma) = \frac{1+C\varsigma }{1+D\varsigma }$ with $-1\leq D < C\leq 1$ in Theorem 3, the following corollary holds:

Corollary 3. Let $\eta \in \mathbb{C} ^{\ast }$ such that

If $f\in A(n)$ complies (1.9) and the following subordination is satisfied:

then

and $\frac{1+C\varsigma }{1+D\varsigma }$ is the best dominant of (3.10).

Proof. For $q(\varsigma) = \frac{1+C\varsigma }{1+D\varsigma },$ the condition in (3.7) reduces to

Since

it is easy to check that (3.11) holds, if and only if, the assumption in (3.9) is satisfied whenever $-1\leq D < 1.$ □

Using in Theorem 3 the assumption $q\rightarrow 1^{-}$, the next corollary is obtained:

Corollary 4. Let $q\in K$, with $q(0) = 1$, $\ \eta \in \mathbb{C} ^{\ast }.$ Suppose that $q$ complies (3.7). If $f\in A(n)$ satisfies the subordination

then

and $q(\varsigma)$ is the best dominant of (3.12).

Theorem 4. Let $\eta \in \mathbb{C} ^{\ast }, \mathrm{{Re}}\eta \geq 0\mathit{\ }$and $q\in K,$ with $q(0) = 1$. Let $f\in A(n)$ such that

and, consider the function

that is univalent in $\mathbb{D}$. If

then

and $q(\varsigma)$ is the best dominant of (3.14).

Proof. Considering that the function $\omega$ is defined by (3.1), we know that $\omega \in H[q(0), m]$, and using (3.14) we have that $\omega \in H[q(0), 1]\cap \wp$. As in the proof of Theorem 1, differentiating (3.1) with respect to $\varsigma$ we get

Then, by applying Lemma 3 for $\lambda = \frac{\eta }{\delta },$ the desired result is obtained. □

Using in Theorem 4 the assumption $q(\varsigma) = \frac{1+C\varsigma }{1+D\varsigma }$ with $-1\leq D < C\leq 1$ the next corollary can be written:

Corollary 5. Let $\eta \in \mathbb{C} ^{\ast }, \mathrm{{Re}}\eta \geq 0.$ If $f\in A(n)$ such that the assumptions in (3.13) and (3.14) hold and satisfy the subordination

then

and $\frac{1+C\varsigma }{1+D\varsigma }$ is the best dominant of (3.16).

Using in Theorem 4 the assumption $q\rightarrow 1^{-}$, the next result can be derived:

Corollary 6. Let $\eta \in \mathbb{C} ^{\ast },$ $\mathrm{{Re}}\eta \geq 0$ and $q\in K$ with $q(0) = 1$. Let $f\in A(n)$ such that

and consider the function

that is univalent in $\mathbb{D}$. If

then

and $q(\varsigma)$ is the best dominant of (3.17).

Combining Theorems 3 and 4, the following sandwich-type theorem can be stated.

Theorem 5. Let $q_{1}, q_{2}\in K$, with $q_{1}(0) = q_{2}(0) = 1$, and let $\ \eta \in \mathbb{C} ^{\ast },$ $\mathrm{{Re}}\eta \geq 0.$ If $f\in A(n)$ such that the assumptions in (3.13) and (3.14) hold, then

implies that

and $q_{1}$ and $q_{2}$ are respectively the best subordinant and best dominant of (3.20).

Combining Corollaries 4 and 6, the following sandwich-type result is stated.

Corollary 7. Let $q_{1}, q_{2}\in K$ with $q_{1}(0) = q_{2}(0) = 1$, and let $\ \eta \in \mathbb{C} ^{\ast },$ $\mathrm{{Re}}\eta \geq 0.$ If $f\in A(n)$ such that the assumptions in (3.17) and (3.18) hold, then

implies that

and $q_{1}$ and $q_{2}$ are respectively the best subordinant and best dominant of (3.21).

Using $q_{i} = e^{r_{i}\varsigma }$ with $0 < r_{1} < r_{2}\leq 1, \ i = 1, 2\ $in Theorem 5 and Corollary 7, the following examples are constructed:

Example 2. (i) Let $\ \eta \in \mathbb{C} ^{\ast }$ with $\mathrm{{Re}}\eta \geq 0.$ If $f\in A(n)$ such that the assumptions in (3.13) and (3.14) hold, then

where $\Upsilon$ and $\Phi$ are given in Theorem 5, and $e^{r_{1}\varsigma }$ and $e^{r_{2}\varsigma }$are, respectively, the best subordinant and the best dominant.

(ii) If $f\in A(n)$ such that the assumptions in (3.17) and (3.18) hold, then

where $\widetilde{\Upsilon }$ and $\widetilde{\Phi }$ are given in Corollary 7, and $e^{r_{1}\varsigma }$ and $e^{r_{2}\varsigma }$are, respectively, the best subordinant and the best dominant.

Theorem 6. If $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(0, \delta, 1-2\sigma, -1)$ with $0\leq \sigma < 1,$ then $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, 1-2\sigma, -1)$ for $\left\vert \varsigma \right\vert < R,$ where

Proof. For $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(0, \delta, 1-2\sigma, -1)$ with $0\leq \sigma < 1,$ let the function $\omega$ be defined by

Then the function $\omega$ is analytic in $\mathbb{D}$ with $\omega (0) = 1$, and since $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(0, \delta, 1-2\sigma, -1)$ is equivalent to

it follows that $\mathrm{{Re}}\omega (\varsigma) > 0.$ As in the proof of Theorem 1, since $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(0, \delta, 1-2\sigma, -1)$ with $0\leq \sigma < 1$, we deduce that

and from the relation (3.23), we get $\omega \in H[1, n]$. Therefore, the following estimate holds

that represents the result of Shah ^[36] (the inequality (6), p. 240, for $\alpha = 0$), which generalizes Lemma 2 ^[37].

Simple calculations demonstrate that

hence, we have

and the righthand side of (3.24) is positive provided that $r < R$, where $R$ is given by (3.22). □

Theorem 7. Suppose that $f\in \aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D)$, let $\eta \in \mathbb{C} ^{\ast }$ with $\mathrm{{Re}}\eta \geq 0$ and $-1\leq D < C\leq 1.$ Then, the following inequalities hold:

$(i)$

$(ii)$ For $\left\vert \varsigma \right\vert = r < 1,$ we have

All these inequalities are the best possible.

Proof. Applying the results given by Theorem 1 for the hypothesis of this theorem, we obtain that

and the convex function $q\in H[1, n]$ is the best dominant.

Therefore,

and

In addition, since

we get

while

implies that

The inequalities of (3.25) and (3.26) are the best possible because the subordination in (3.27) is sharp. □

Using in Theorem 7 the assumption $q\rightarrow 1^{-}$, we state the corollary:

Corollary 8. Suppose that $f\in \Im _{\alpha, \beta, \mu }^{m, \tau }(\eta, \delta, C, D)$, let $\eta \in \mathbb{C} ^{\ast }$ with $\mathrm{{Re}}\eta \geq 0$ and $-1\leq D < C\leq 1.$ Then,

$(i)$

$(ii)$ For $\left\vert \varsigma \right\vert = r < 1,$ we have

All these inequalities are the best possible.

Using in Theorem 7 the assumptions $q\rightarrow 1^{-}$ and $m = 0$, we obtain the corollary:

Corollary 9. Suppose that $f\in N^{\eta, \delta }(n, C, D)$, let $\eta \in \mathbb{C} ^{\ast }$ with $\mathrm{{Re}}\eta \geq 0$ and $-1\leq D < C\leq 1.$ Then,

$(i)$

$(ii)$ For $\left\vert \varsigma \right\vert = r < 1,$ we have

All these inequalities are the best possible.

4.   Conclusions

There has been a resurgence of interest in the study of $q$-series and $q$ -polynomials and related topics, which has a history dating back to the 19th century as a result of the creation of quantum groups and their applications in mathematics and physics beginning in 1980. This study introduces the class $\aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D))$ of normalized analytic functions by using the linear extended multiplier $q$ -Choi-Saigo-Srivastava operator in the open unit disk $\mathbb{D}$ given by Definition 1. Some applications of the theory of differential subordination differential superordination, and sandwich-type results were obtained here for the class $\aleph _{\alpha, \beta, \mu }^{m, q, \tau }(\eta, \delta, C, D))$ with interesting corollaries obtained when particularizing the parameters of the defined class.

Future investigations can be done on the newly defined class considering coefficient estimates ^[38,39]. The classes obtained in this paper can be investigated using the newer theories of strong and fuzzy differential subordination and superordination ^[19,40]. Also, new classes of other types of functions could be investigated using the same linear extended multiplier $q$-Choi-Saigo-Srivastava operator like it is done for bi-univalent functions ^[41] or for meromorphic functions ^[42,43].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The article processing charges for this paper was supported by the University of Oradea, Romania.

Conflict of interest

The authors declare no conflict of interest.