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

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 
 
 μ
 
 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.


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 U (see [7][8][9][10][11]). e objective of this manuscript is twofold. Firstly, Bessel functions of the first kind of order μ 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.

(6)
From (6), we have the identity relation where p � (b − 1/2). It is easy to observe from (7) that where Let ϕ(z) be a convex univalent function in U with ϕ(0) � 1 and Reϕ(z) > 0 in U. Ma and Minda and Kim examined the classes C(ϕ), S * (ϕ) (see [17]), and K(ϕ) (see [18]) using the subordination techniques. In particular, [20] and [21]. and We reduce to the classes S * s and C s , consisting of functions which are starlike and convex with respect to symmetrical points [12,13,[22][23][24]. e following lemmas are the key tools to prove our main results.

Suppose also that λ(z) is analytic in
which implies that Lemma 4 (see [21,27]). Let ψ(z) be convex in U with

Results and Discussion
Replacing z with − z in (18) and using the fact that which combined with (18) gives By subordination property,

Corollary 2. Every function f(z) in S *
s is a close-to-convex function.
where D 1 (A, B) is a constant that depends on A and B, while , then by Cauchy eorem, where A n is given by (6) and p(z)≺(1 + Az/1 + Bz), z ∈ U. From eorem 1, we knew that for ϕ(z) is an odd starlike function of Janowski type. us, By Cauchy-Schwarz inequality, subordination property, and Lemma 1 for the case B ≠ 0, (40) implies Observe that since − 1 ≤ B < 1, we have us, Let r � 1 − 1/n(n ⟶ ∞); we obtain from (6) and (44) that 4 Journal of Mathematics where D 1 (A, B) is given in eorem 4. For the case B � 0, we implement Lemma 1 and subordination property in (41) and follow the procedures for the case B ≠ 0 to obtain which completes the proof by using (6).
where D 2 (A, B) is a constant that depends on A and B, while D 2 (A) only depends on A.
Proof. Let f ∈ S s μ,b,c [A, B]. en, there exists an analytic function w(z) � ∞ n�1 w n z n with w(0) � 0 and |w(z)| < 1(z ∈ U) such that which is equivalent to us,
for some c m , n + 1 ≤ m < ∞. Squaring the moduli of both sides of (56), integrating around the circle |z| � r, and using Parseval's theorem, we note that (58) Taking limit as r ⟶ 1 − , we obtain the required result. In the following theorem, erf(x) for an arbitrary x 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.