On Bessel Functions Related with Certain Classes of Analytic Functions with respect to Symmetrical Points

Abstract

In the present investigation, subclasses of analytic functions with respect to symmetrical points which are defined by the generalized Bessel functions of the first kind of order are introduced. Furthermore, some alluring geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results are studied. Moreover, some consequences of our results are also given.

1. Introduction

Geometric function theory (GFT) is the area of complex analysis which deals with the geometric characterization of analytic functions, established around the turn of the twentieth century[1]. It is a known fact that the study of special functions plays a significant role in GFT. One reason is that solutions of extremal problems can be frequently written in terms of special function. Another reason is that some important conformal mappings are given by special function. For example, the conformal mapping of an annulus onto the complement of two closed segments on the real axis and the conformal mapping of a square onto a rectangle are expressed by elliptic functions (see [2]). In recent times, the solution of Bieberbach conjecture by de Branges is obtained with the help of special functions [3].

Bessel function is one of the most significant special functions. It is therefore important for solving many problems in engineering, physics, and mathematics (see [4, 5]). For instance, it is used for velocity and stress derivation in the rotational flow of Burge’s fluid flowing through an unbounded round channel [6].

In recent times, many researchers paid their attention on establishing various conditions under which a Bessel function has some certain geometric properties such as close-to-convexity (univalency), starlikeness, and convexity in frame of a unit disc (see [7–11]).

The objective of this manuscript is twofold. Firstly, Bessel functions of the first kind of order is used to introduce new generalized starlike and convex functions with respect to symmetrical points, which was first initiated and studied by Sakaguchi [12] and Das and Sign [13]. Moreover, we examine some interesting geometric properties of these classes, which include inclusion property, integral-preserving properties, coefficients, and distortion results.

2. Materials and Methods

Now, we give some basic preliminaries and definitions that play the integral part in obtaining our main results.

Consider  (the set of complex numbers) and the second-order linear homogenous differential equationwhich is a natural extension of Bessel’s equation. The solution (see [14]) of (1) has a series representation:

Differential equation (2) allows the investigation of Bessel function of the first kind of order [15, 16] (the case b = c = 1), modified Bessel function [15, 16] (the case b = 1, c = −1), and the spherical Bessel Function [16] (the case b = 2, c = 1). Using the well-known Pochhammer symbol with , we consider the function defined by the transformation

Let denote the class of normalized analytic functions in given by the representation

Then, the convolution of and denoted by is defined byand we say is subordinate to (written as ) if there exists a Schwarz function such that .

Let be an operator defined by

From (6), we have the identity relationwhere . It is easy to observe from (7) thatwhere

Let be a convex univalent function in with and in . Ma and Minda and Kim examined the classes (see [17]), and (see [18]) using the subordination techniques. In particular, for , [19] and [20] and and [21].

Definition 1. Let . Then, if and only ifand if and only ifWe note that . If , we set and . It is worthy of note that if in (6) and , then the classes and reduce to the classes and , consisting of functions which are starlike and convex with respect to symmetrical points [12, 13, 22–24].
The following lemmas are the key tools to prove our main results.

Lemma 1. (see [19, 25]). If , then

Lemma 2. (see [26]). Let be an univalent function in . Then, there exists with such that for all ,

Lemma 3. (see [21]). Let be convex in with . Suppose also that is analytic in with . If is analytic in with p (0) = 1, thenwhich implies that

Lemma 4. (see [21, 27]). Let be convex in with . If is analytic in with , then

3. Results and Discussion

Theorem 1. If , then .

Proof. Let . Then,Replacing with in (18) and using the fact that is an odd function, we havewhich combined with (18) givesBy subordination property, we have that .

Corollary 1. The function belongs to and hence is univalent in .

Setting and choosing in Theorem 1, we are led to the result of Sakaguchi [12] contained in the following corollary.

Corollary 2. Every function in is a close-to-convex function.

Theorem 2. Suppose and . Then,

Proof. ConsiderFrom relation (7), (22) can be written asDifferentiating (23) and applying (8), we obtaini.e.,It follows from (8) thatwhereIn view of (25) and (26), we obtainSince and , then . Hence, by Lemma 3, , i.e., . This completes the proof.

Corollary 3. Let and . Then,

Corollary 4. Suppose that all the conditions of Theorem 2 are satisfied. Then,

Proof By Theorem 2. we have that

Theorem 3. Let be defined by the integral transformationand suppose . Then, .

Proof. Letwhere is analytic in with . From (32) and applying the operator , we obtaini.e.,Differentiating (35) logarithmically, we obtainand in view of Theorem 1, it follows thatwhich by Lemma 4 implies . Thus, .

Corollary 5. Let . Then, .

Proof.

Corollary 6. Let and . Then, . Similarly, if , then .

Setting and choosing in Theorem 3, we are led to the results of Das and Sign [13] contained in the following corollaries.

Corollary 7. Let . Then, .

Corollary 8. Let . Then, .

Theorem 4. If , then for ,where is a constant that depends on and , while only depends on .

Proof. Since , then by Cauchy Theorem,where is given by (6) and . From Theorem 1, we knew that for , the function is an odd starlike function of Janowski type. Thus,By Cauchy–Schwarz inequality, subordination property, and Lemma 1 for the case , (40) impliesObserve that since , we haveThus,Let ; we obtain from (6) and (44) thatwhere is given in Theorem 4.
For the case , we implement Lemma 1 and subordination property in (41) and follow the procedures for the case to obtainwhich completes the proof by using (6).

Remark 1. If we allow and choose in Theorem 3, it follows thatSince is a subclass of the class of close-to-convex function (see [12]), it shows that our index of is a nice one.

Theorem 5. Let . Then,where is a constant that depends on and , while only depends on .

Proof. Let be a complex number with . Then, by Cauchy Theorem, we obtainwhere . Following the techniques used in Theorem 4 and using Lemma 2, we obtainTaking and choosing , we have the result. The case also follows the same procedures.

Theorem 6. If , then

Proof. Let . Then, there exists an analytic function with and such thatwhich is equivalent toThus,where we have used (6), (9), and (53). Comparing the coefficients of in (54), we obtainThe coefficients combination on the right side of (55) depends only upon the coefficients combination of the left side. Therefore, we can write (54) asfor some . Squaring the moduli of both sides of (56), integrating around the circle , and using Parseval’s theorem, we note thatTherefore,Taking limit as , we obtain the required result.
In the following theorem, for an arbitrary denotes the error function and we need an Euler integral representation for the special class of hypergeometric functions given in [28] and defined as follows.
For ,

Theorem 7. Let . Then, for ,whereEquality is obtained for the functionif and .

Proof. Since , thenwhere by Theorem 1 and . Using (41) and subordination property, we have thatBy adopting (13) and (64), we obtainApplying hypergeometric function (59), we obtain the upper bound for the case . In case , applying (13) and using (64) in (66), we obtainThis establishes the upper bound. To prove the lower bound, we consider a point such that . Let be an arc in which is mapped by the function onto a line segment connecting origin to the point and lying completely in the image of under . Thus, by (13) and (65), we obtainAdopting similar procedures as used in finding the upper bounds from (65), we obtain the desired result.
For , we obtain a more reduced form of Theorem 7 which is contained in the following corollary.

Corollary 9. If , thenThis bound cannot be improved.

Corollary 10. If , thenThis bound is sharp.

Remark 2. In respect of the lower bounds of for the classes and , given by (69) and (70), respectively, we note that the disc of the maximum radius is contained in the image domain if and , respectively, where .
In view of Remark 2, we note that as in the lower bound of both Corollaries 9 and 10, we have the following results giving the omission values for the classes and .

Corollary 11. Let and be such that . Then, .

Corollary 12. Let and be such that . Then, .

4. Conclusion

Bessel functions are essential in many branches of mathematics and applied mathematics. Recently, there has been a clear interest on Bessel and hypergeometric functions from the point of view of geometric function theory. As a result, we presented some subclasses of analytic functions with respect to symmetrical points, which were associated with Bessel function. The geometric properties of these aforementioned classes which include integral-preserving properties, coefficients, and distortion results were studied. As a consequence of our investigation, some relevant special cases were pointed out. In addition, to capture more new results under the current examination, new idea and applications can be investigated with some positive and novel outcomes in various fields of science, especially in mathematics. These new investigations will be presented in future research work being processed by the authors of the present article.

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.

Acknowledgments

The authors would like to thank the Rector of COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environment. The fourth author is supported by UKM grant (GUP-2019-032).