Some Convolution Properties of Multivalent Analytic Functions

1.   Introduction

Quantum calculus, or q-calculus, is of great interest to academics for its diverse applications in scientific domains, especially in geometric function theory. In 1996, Ismail ^[1] was the first to define and study the class of q-starlike functions and established many properties associated with the class. The Russell operator, which is a generalization of the differential operator, plays a key role in characterizing subclasses of analytic functions by examining their geometric properties, such as starlikeness and convexity ^[2]. The Bernardi operator is similarly used to define classes of functions with geometric constraints that are important in the study of conformal mappings and their applications in complex analysis ^[3]. Later, in 2013, Mohammed and Darus ^[4] introduced the q-derivative operator, which uses the convolution structure of normalized analytic functions and q-hypergeometric functions. In 2014, Aldweby and Darus ^[5] introduced the q-analogue of the Ruscheweyh differential operator. Over the years, there has been a growing exploration of the connection between q-calculus and geometric function theory. The q-Salagean differential operator was introduced in ^[6]. Many authors successfully used these operators to investigate the properties of both known and new classes of analytic functions ^[7,8].

In ^[9], Selvakumaran et al. developed q-integral operators for analytic functions using fractional q-calculus and examined the convex characteristics of these operators on specific classes of analytic functions. In ^[10], the authors introduced the q-Bernardi integral operator and studied its integral-preserving features. The q-analogue of the Noor integral operator was presented in ^[11]. In ^[12], the q-Srivastava Attiya operator and q-multiplier transformation were presented in relation to a specific q-Hurwitz-Lerch zeta function. By linking these q-operators with the idea of subordinations, several subclasses of analytic functions have been identified and examined.

Many researchers contributed to the theory by obtaining coefficient estimates that contain the initial coefficients of q-classes of biunivalent functions. The Fekete-Szego functional and Henkel determinants were studied for these classes. Some recent studies have also focused on new families of meromorphic functions ^[13].

The versatility and potential of q-calculus, as demonstrated by its ability to enhance our theoretical understanding of analytic functions and its applicability in various scientific domains, have made it an active area of research. The continued exploration and advancement of q-calculus in the field of geometric function theory are expected to yield valuable contributions to both the theoretical and practical aspects of this field.

This work explores new classes of analytic functions using the q-difference operator in the open unit disk, inspired by recent developments in q-calculus and its applications to analytic functions. Previous research has examined classical operator properties, but there is a gap in understanding q-operators, particularly in terms of inclusion relations, integral preservation, and convolution identities. To fill this gap, we examine the key characteristics of the new q-classes. In particular, we define inclusion relations across q-classes, investigate integral-preserving characteristics of the q-integral operator, and deduce convolution identities. These findings enhance the framework of q-calculus in geometric function theory, offering new insights and tools for future research.

2.   Materials and methods

This section provides some mathematical preliminaries that are utilized in this paper.

Let $\mathcal{A}$ denote the class of the functions expressed as follows:

which are analytic in the open unit disk

Definition 2.1. (^[14]) The convolution (Hadamard product) for two analytic functions $f(z), g(z)\in$ $\mathcal{A}$ is defined as

In the following definition, we will refer to a well-known function with two key conditions, called a Schwarz function. These requirements are essential for applying various mathematical results and theorems related to analytic functions, ensuring predictable and consistent behavior within the unit disk.

Definition 2.2. (^[14]) We say that two functions $f(z)$ and $g(z)$ are subordinate to one another and we write $f(z)\prec g(z)$ if there is a Schwarz function $w(z)$ with the conditions that

and

Additionally, in the case where the function $g(z)$ is univalent in $U$, the subsequent equivalent relationship is valid:

Definition 2.3. (^[15]) For each non-negative integer $k$, the $q$-number, denoted by $[k]_q$, is defined by

where

and

Example 2.4.

and

Example 2.5. For non-negative integers $r$ and $s$:

In ^[16], Jackson defined the $q$-derivative and $q$-integral $D_q$: $\mathcal{A} \rightarrow \mathcal{A}$ as follows:

Definition 2.6. (^[15]) The $q$-derivative operator of $f(z)$ is defined by the formula

and the $q$-integral is defined by the formula

provided that the series converges.

Therefore, for $f \in$ $\mathcal{A}$, we conclude that:

and

Let $\mathcal{P}$ be the class of functions $\phi(z)$, which map the unit disk $U$ analytically onto the right-half plane. These functions play a major role in the field of geometric function theory. Many fundamental results have been established in regard to this class of functions. Any function $\phi$ belonging to the class $\mathcal{P}$ has the following representation form

such that

and

These functions are usually called the Caratheodory functions or functions with a positive real part. We shall utilize these kind of functions with the $q$-derivative using the subordination concept in the following two definitions:

Definition 2.7. Let $\phi$ $\in$ $\mathcal{P}$, $0\leq \gamma < 1$. A function f $\in$ $\mathcal{A}$ is said to be in the class $\mathcal{S}_{ q}(\gamma, \phi)$, if and only if

where $D_q$ is the $q$-derivative operator.

Definition 2.8. Let $\phi$ $\in$ $\mathcal{P}$, $0\leq \gamma < 1$. A function f $\in$ $\mathcal{A}$ is said to be in the class $\mathcal{C}_{q}(\gamma, \phi)$, if and only if

where $D_q$ is the $q$-derivative operator.

We note that for special values of the parameter $\gamma$ and the function $\phi$, with $(q \rightarrow -1)$, we obtain the famous classes as follows:

(ⅰ)

and

then,

(ⅱ)

and

then,

(ⅲ)

and

then,

We recall the q-differential operator $\mathcal{R}_{ q}^{\lambda}$, which was introduced in ^[5] and is also referred to as the $q$-analogue of the Rusheweyh operator, defined as follows:

where $f\in A$, $\lambda > -1$, and $q\in (0, 1)$.

As $q\rightarrow 1^{-1}$, we observe

where $\mathcal{R}^{\lambda}$ is the most familiar Ruscheweyh differential operator.

It can also be shown that this $q$-operator is q-hypergeometric in nature as

where $_{2}\Phi_{1}$ is the Gauss $q$-hypergeometric function (see ^[17]).

The identity relation can be derived from Eq (2.3)

To discuss our main results, we state the following lemma:

Lemma 2.9. ^[18] Let $h(z)$ be convex in $U$ with $h(0) = 1$, and let $P$: $U \rightarrow \mathbb{C}$ with $Re (P(z)) > 0$ in $U$. If

is analytic in $U$, then

implies that $p(z)\prec h(z)$.

3.   Results

This section presents the main results, which include introducing two new classes of analytic functions $\mathcal{S}_{q}^{\lambda}(\gamma; \phi)$ and $\mathcal{C}_{q}^{\lambda}(\gamma; \phi)$ along with their inclusion relation, integral preserving properties, and convolution properties.

First, utilizing the $q$-operator $\mathcal{R}_{q}^{\lambda}$, we define distinct classes of analytic functions for $\phi \in P$ and $0 \leq \gamma < 1$,

and

Moreover, we observe that

Next, we prove the following lemma with the help of Lemma 2.9.

Lemma 3.1. Let $\beta$ and $\eta$ be complex numbers with $\beta \neq 0$ and let $\psi(z)$ be convex in $U$ with

and

If

is analytic in $U$, then

implies that

Proof. By setting

and we have

then

By Lemma 2.9, we get

This completes the proof. □

3.1.   Inclusion relations

Theorem 3.2. Let $0\leq \gamma < 1$ and $\phi \in \mathfrak{P}$ with

Then

Proof. Let $f\in \mathcal{S}_{q}^{\lambda+1}(\gamma; \phi)$ and suppose that

Then $\psi$ is analytic in $U$ with

and

for all $z\in U$. Combining Eqs (2.4) and (3.6), we get

By employing logarithmic $q$-differentiation on both sides of Eqs (3.7) and (3.6), we obtain

Given the validity of Eq (3.4), applying Lemma 3.1 to Eq (3.8) results in

that is $f\in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$, which implies that the assertion (3.5) of Theorem 3.2 holds. □

Theorem 3.3. Let $0\leq \gamma < 1$ and $\phi$ $\in$ $\mathcal{P}$. Then

Proof. Applying the relation (3.3) and Theorem 3.2, we have

which evidently proves Theorem 3.3. □

In this place, if we set

in Theorems 3.2 and 3.3, we have the following consequence:

Corollary 3.4. Let $\lambda > -1, 0\leq \gamma < 1$. Then

and

3.2.   Integral preserving properties

In this section, we discuss some integral preserving properties for the $q$-integral operator defined in ^[10].

Theorem 3.5. Let $f \in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$ with

Then $F(f)\in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$, where $F_{z}$ is the $q$-Bernardi integral operator defined by

Proof. Let $f \in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$. Then from Eq (3.10), we find that

By setting

we observe that $\mathfrak{p}$ is analytic in $\mathbb{U}$ with

and

for all $z\in U$. It follows from Eqs (3.11) and (3.12) that

Applying logarithmic $q$-differentiation on both sides of Eq (3.13), and using Eq (3.12), we obtain

Since

an application of Lemma 3.1 to Eq (3.14) yields

and we readily deduce that the assertion of Theorem 3.5 holds true, which means that $F_{z}\in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$. This completes the proof. □

In the same manner of Theorem 3.3, one can get the next result:

Corollary 3.6. Let $f \in \mathcal{C}_{q}^{\lambda}(\gamma; \phi)$. Then $F_{z}(f)\in \mathcal{C}_{q}^{\lambda}(\gamma; \phi)$.

Theorem 3.7. Let $F$ be defined by (3.10). If $f\in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$, $\alpha > 0$, and

then

Proof. By employing the operator $\mathcal{R}_{q}^{\lambda}$, we get

By taking the $q$-derivative, we have

This relation is equivalent to

which implies that

and this is equivalent to

We obtain that

and

If

for all $z\in U$, then

We employ $D_q^\mathfrak{n}$ and $D_q^{\mathfrak{n}+1}$ in the last inequality, we obtain

Hence, the proof is complete. □

3.3.   Convolution properties

In this section, we derive certain convolution properties for the class $\mathcal{S}_{q}^{\lambda}(\gamma; \phi)$.

Theorem 3.8. Let $f \in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$. Then

where $w$ is analytic in $U$ with

and

Proof. Suppose that $f \in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$ and Eq (3.1) can be written as follows:

where $w$ is analytic in $U$ with

and

We now find from Eq (3.16) that

which, upon $q$-integration, yields

It follows from Eq (3.20) that

Now if we convolute both sides of Eq (3.21) by the expression $\Big(z+ \sum_{k = 2}^{\infty} \frac{[\lambda]_{q}![k-1]_{q}!}{[k+\lambda-1]_{q}!} z^k\Big)$, the assertion (3.15) of Theorem 3.8 is obtained. □

In the following, we derive a result related to functions in the class $\mathcal{S}_{q}^{\lambda}(\gamma; \phi)$ by examining their convolution with a special analytic function.

Theorem 3.9. Let $f$ $\in$ $\mathcal{A}$ and $\phi$ $\in$ $\mathcal{P}$. Then $f \in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$ if and only if

Proof. Suppose that $f\in \mathcal{S}_{q}^{\lambda}(\gamma; \phi)$. Then Eq (3.1) is equivalent to

The condition (3.23) can be written as follows:

On the other hand, we have

Substituting Eqs (3.26) and (3.27) in Eq (3.25), we readily get the convolution property (3.22) asserted by Theorem 3.9.

The proof is complete. □

4.   Conclusions

In this study, we discuss subclasses of starlike and convex functions associated with the $q$-Ruschewehy differential operator and the $q$-Bernardi integral operator. To define these q-classes of analytic functions, we use the concept of $q$-derivatives. For the newly defined classes, we investigate inclusion relations and integral preservation properties. We highlight several intriguing properties of convolution. In the future, this work will inspire other authors to make contributions in this area for numerous generalized subclasses of the $q$-classes of starlike and convex functions. Furthermore, extensions of the current work can be to consider higher-order q-derivatives or generalizations of the operators discussed. Lastly, they can apply these findings to multivalent and meromorphic functions.

Author contributions

K. R. Alhindi: conceptualization, validation, formal analysis, investigation, resources, writing–review and editing, visualization; K. M. K. Alshammari: resources, writing–review and editing; H. A. Aldweby: conceptualization, validation, formal analysis, investigation, resources, writing–original draft preparation, writing–review and editing. All authors have read and agreed to the published version of the manuscript.

Conflict of interest

The authors declare no conflicts of interest.