A Study of New Class of Star-Like Functions Associated by Symmetric -Calculus

Abstract

As of late quantum calculus is broadly utilized in different parts of mathematics. Uniquely, the hypothesis of univalent functions can be newly portrayed by utilizing -calculus. In this paper, we utilize our recently presented symmetric -number to characterize new symmetric -derivative of analytic function in the open unit disk . Utilizing , we introduce new class of analytic star-like functions and examine some fascinating results.

1. Introduction

The mathematical study of -calculus has been a topic of great interest for researchers due to its wide applications in different fields. Some of the earlier work on the applications of -calculus was introduced by Jackson [1]. Later on, q-calculus attained much popularity among the researchers. Recently, -calculus has gained the attention of researchers because of its huge applications in mathematics and physics. The in-depth analysis of -calculus was firstly discussed by Jackson [1, 2], where he defined -integral and -derivative in a very systematic way. Recently, authors are using these -integral and -derivative to define new subclasses of the class of univalent functions and obtained variety of new results. Extending the idea of -number, which contains only one variable , the -number which contains two independent parameters and was independently considered by Chakrabarti and Jagannathan [3]. As -calculus or quantum calculus originated by using -number, similarly by using -number, the -calculus or postquantum calculus has been studied and discussed by several researchers (see, for example, Duran et al. [4] and references therein). Let . Furthermore, is normalized analytic, if is single valued and differentiable for along and is represented as

The symbol is used for this type of functions. Let be given by (1). Then,

Assume that is analytic. Furthermore,and is presented as

Let If is a constant in D and for all z₀ in D, we have

At that point, we state that is a star-formed region. Geometrically, is star-like if is a star-shaped region. We meant by the family of these functions. Analytically, we attain

Assume that If , then

Geometrically,

The family containing all the convex function is represented as . Analytically ,

Note

Let be presented as in (1), and the convolution is characterized aswhere

Let be two functions. Then, , if there exist a Schwarz function analytic along , such that . It can be found in [5] that if , thenand for details, see [6].

Kanas and Wisniowska [7] presented the conic regions bywith . Assume

Therefore,

Identified with the region , the accompanying functions are extremal and :

The families and are fascinating subfamilies of and are characterized in [8] for all , , as

Note from [8] that

Geometrically,

2. Postquantum Calculus

Extending the idea of -number, the -number with two variables and was independently considered in [3].

For the twin basic number or -number is defined for aswhich is the normal speculation of -number with the end goal that for , we have

Furthermore, note that .

In 1991, Chakrabarti and Jagannathan [3] characterized -derivative of as

Furthermore, if , then , and for (1), we have

We remark here as

We now extend the idea in [9] and define symmetric -number as follows:

Analogous to -derivative defined by (22), the symmetric -derivative is characterized as

If

In this manner for (1), we acquire

Equivalently, by using the same technique as in [9], it can be seen that

Assume that denotes the subfamily of with negative coefficients such as

For further developments about class and its subclasses, one can refer to a wide range of extraordinary articles written by famous mathematicians (see [10–13] and references therein).

Presently, utilizing the symmetric -derivative characterized by (27) of , we present another subclass of as follows.

Definition 1. Letand. Assume thatcharacterized by (1). Furthermore,Geometrically,where is given byand is defined as in (13).
Utilizing the functions defined in [14], we have

Remark 1. Likewise, we set .
Note that on the off chance that , at that point

3. Main Results

Now in this segment, we will demonstrate our principle results. It is worthy to mention here that our main results are extension of results studied by Kanas et al. [9]. We utilize symmetric -derivative operator to obtain these results.

The accompanying lemmas are useful to prove main result.

Lemma 1. Assume thatand. Furthermore, are analytic inalong. Then,and for details, see [15].

Lemma 2. For the sequenceand,is a subordinating factor sequence (see [15]).

Now we will extend the existing results of [13] with the help of symmetric -derivative.

Theorem 1. Assume thatand letbe defined by (1). If the inequalityholds true, then

Proof. To prove (39), it is adequate to prove thatConsiderNow using (1) and (29), we obtainThis can be composed asFrom (43), we obtain (39) and the proof is completed.

Remark 2. In Theorem 1, by putting , we have the result presented in [9].

Corollary 1. Assume thatand letbe presented as in (1). Ifholds true for , , then

Remark 3. It can be seen that the quantityfor all , unless otherwise mentioned.
Now using (39), we can find some special members of . One of them is the following.

Corollary 2. Let forIf, forthe following inequality:holds, then Especially

For , we obtain the following known results (see [9]).

Corollary 3. Let, and . If, for, the following inequality:holds, then Especially

Also, for we obtain the following.

Corollary 4. LetAn essential and adequate condition ofbelongs tosuch as

The accompanying function yields quality.

Proof. Proof immediately follows by using Theorem 1.
On the off chance that we take in Corollary 4, we have the accompanying corollary (see [9]).

Corollary 5. Let A necessary and sufficient condition forof the form, to be in the classis that

The result is sharp and accompanying function yields quality.

Theorem 2. Let, and . Assumebelongs toFurthermore, yields

Accompanying function yields quality for (54).

Proof. Let By using Theorem 1, we haveThis means thatFurthermore if ,andCombining (58) and (59), we obtain (54).

Theorem 3. Suppose that. Moreover,

If

Proof. Note that to prove (61) for , it is equivalent to proveBy Lemma 1, it is sufficient to show thatSetThis means thatNow using (39), we get and this finishes the proof.

Theorem 4. Let. Then,that is,

The constant cannot be increased.

Proof. Let .
Then, consider

Therefore, by using the definition of subordinating factor sequence, (66) holds true, ifis a subordinating factor sequence with Equivalently, by Lemma 2, we have

Now for , we can write

It can be seen that the function defined byis increasing for ; furthermore, by utilizing affirmation (39) of Theorem 1, we have

This equivalently proves the subordination condition given by (66). The inequality given by (67) can be obtained from (66) by taking . Now from (47), we obtained the following function:

Function (66) becomes

It is easily verified that

This implies that the constant is possible.

Theorem 5. LetIf we setthen

Proof. Suppose thatUtilizing (77), we getSince , this means that , and using the assertion (51) of Corollary 4, we obtain .
Conversely, suppose that . Making use of (39), we may setThen,which is required.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.