Certain Class of Analytic Functions Connected with -Analogue of the Bessel Function

Abstract

The focus of this article is the introduction of a new subclass of analytic functions involving q-analogue of the Bessel function and obtained coefficient inequities, growth and distortion properties, radii of close-to-convexity, and starlikeness, as well as convex linear combination. Furthermore, we discussed partial sums, convolution, and neighborhood properties for this defined class.

1. Introduction

Let specify the category of analytic functions and represent on the unit disc with normalization and such that a function has the extension of the Taylor series on the origin in the form

Indicated by , the subclass of is composed of functions that are univalent in .

Then, a function of is known as starlike and convex of order if it delights the pursingfor specific , respectively, and we express by and the subclass of , which is expressed by the aforesaid functions, respectively. Also, indicated by , the subclass of is made up of functions of this formand let . There are interesting properties in the and classes which were thoroughly studied by Silverman [1] and Alessa et al. [2].

The intense devotion of scientists has recently fascinated the study of the -calculus. The great focus in many fields of mathematics and physics is due to its benefits. In the analysis of many subclasses of analytic functions, the importance of the -derivative operator is very evident from its applications.

The concept of -star functions was originally proposed by Ismail et al. [3] in the year 1990. However, in the sense of Geometric Function Theory, a firm basis of the use of the -calculus was effectively developed. For example, in the rotational flow of Burge’s fluid flowing through an unbounded round channel [3], it is used to derive velocity and stress.

After that, numerous mathematicians have carried out remarkable studies, which play a significant role in the development of geometric function theory. Furthermore, Srivastava [4] recently published a survey-cum-expository analysis article that could be useful for researchers and scholars working on these topics. The mathematical description and applications of the fractional q-calculus and fractional q-derivative operators in geometric function theory were systematically investigated in this survey-cum-expository analysis article [4]. In particular, Srivastava et al. [5] also considered some function groups of conical region related q-star like functions. For other recent investigations involving the -calculus, one may refer to [6–13].

One of the most significant special functions is the Bessel function. As a result, it is important for solving a wide range of problems in engineering, physics, and mathematics (see [14]). In recent years, several researchers have focused their efforts on forming different types of relationships. Many researchers have recently focused on determining the various conditions under which a Bessel function has geometric properties such as close-to-convexity, starlikeness, and convexity in the frame of a unit disc .

The first-order Bessel function is defined by the infinite series [15]:where stands for a function of Gamma. Szasz and Kupan [16] and Thirupathi Reddy and Venkateswarlu [17] have recently explored the univalence of the first-kind normalization Bessel function defined by

For , El-Deeb and Bulboaca [18] defined the -derivative operator as follows:where

Using (7), we are going to define two products in the text:(1)The -shifted factorial is given for any nonnegative integer :(2)The -generalized Pochhammer symbol for any positive number is defined by

For , and , El-Deeb and Bulboaca [18] defined by (see [19])

A simple computation shows thatwhere the function is supplied with the function

El-Deeb and Bulboaca [18] introduced the linear operator using the definition of -derivative along with the idea of convolutions defined by

Remark 1 (see [18]). We can easily verify from the definition relation (13) that the next relation holds for all :Now, we propose a new subclass of concerning - analogue of the Bessel function as follows.

Definition 1. For and , we say is in if it fulfils the requirementAlso, we indicate by .

2. Coefficient Inequalities

This section gives us an adequate requirement for a function given by (1) to be in .

Theorem 1. A function is assigned to the class if

Proof. Since and , now if we putrhen its just a matter of proving it .
Indeed, if , then we have .
This implies (16) holds.
If , then there exist a coefficient for some . The consequence is that . Further note thatBy (16), we obtainHence, we obtainThen, .

Theorem 2. Let be given by (3). Then, the function

Proof. In view of Theorem 1, to examine it, fulfils the coefficient inequality (16). If , then the functionsatisfies . This implies thatNoting that in the open interval , this is the real continuous function with , and we haveNow,and consequently, by (24), we obtainLetting , we get .
This proves the converse part.

Remark 2. If a function of form (3) belongs to the class , thenThe equality holds for the functions

3. Distortion Theorem

In the section, the distortion limits of the functions are owned by the class .

Theorem 3. Let and . Then,

The approximation is sharp, with the extreme function indicated by (28).

Proof. Since , we apply Theorem 2 to attainEquation (29) follows. In a similar way, for , the inequalitiesare satisfied, which leads to (30).

4. Radii of Close-to-Convexity and Starlikeness

A close-to-convex and star-like radius of this class is obtained in this section.

Theorem 4. Let be specified by (3) in . Then, is the order of close-to-convex in the disc where

The estimate is sharp with the extremal function is indicated by (28).

Proof. If and is order of close-to-convex , then we obtainFor the L.H.S of (34), we obtainWe know thatThus, (34) holds true ifand hence proved.

Theorem 5. Let . Then, is order of starlike in the disc , where

The estimate is sharp with the extremal function indicated by (28).

Proof. We have and is order of starlike , and we haveFor the L.H.S of (39), we have is bigger than the R.H.S of the left relation ifWe know thatThus, (39) is true ifIt yields starlikeness of the family.

5. Convex Linear Combinations

Theorem 6. Let and

Then, in the way it can be expressed:and .

Proof. If a function is of the form and , thenwhich provides (21), and hence , by Theorem 2.
On the contrary, if is in the class , then we may setand .
Then, the function is of form (45).

6. Partial Sums

Silverman [20] examined partial sums for the function given by (1) and established through

In this paragraph, in the class , partial function sums can be considered and sharp lower limits can be reached for the function. True component ratios are to and to .

Theorem 7. Let fulfil (16). Then,where

Proof. Clearly, .
Thus, by Theorem 1, we obtainand it is good enough to show . Applying (51), we think thatwhich givesand hence proved.

Theorem 8. Let fulfil (16). Then,where

Proof. Clearly, .
Thus, by Theorem 1, we obtainto show . Implementing (57), we attainwhich givesand hence proved.

Theorem 9. Let fulfil (16). Then,where

Proof. By settingthe evidence is close to that of (51) and (52) theorems, so the specifics are omitted.

7. Convolution Properties

We will prove in this section that the class is closed by convolution.

Theorem 10. Let of the form,be regular in . If , then the function is in the class . Here, the symbol denotes to the Hadamard product.

Proof. Since , we haveEmploying the last inequality and the fact thatwe obtainand hence, in view of Theorem 1, the result follows.

8. Neighborhood Property

Following [21, 22], we defined the -neighbourhood of the function by

Definition 2. The function is defined in the class if the function occurs in such a way that the function is :

Theorem 11. If andthen .

Proof. Let . We then find from which easily implies the coefficient inequalitySince , we have from equation (16) thatand hence proved.

9. Conclusions

This research has introduced q-analogue of the Bessel function and studied some basic properties of geometric function theory. Accordingly, some results related to coefficient estimates, growth and distortion properties, convex linear combination, partial sums, radii of close-to-convexity and starlikeness, convolution, and neighborhood properties have also been considered, inviting future research for this field of study.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. And, all the authors have read and approved the final version manuscript.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.