Inclusion of generalized Bessel functions in the Janowski class

Sufficient conditions on $A$, $B$, $p$, $b$ and $c$ are determined that will ensure the generalized Bessel functions ${u}_{p,b,c}$ satisfies the subordination ${u}_{p,b,c}(z) \prec (1+Az)/ (1+Bz)$. In particular this gives conditions for $(-4\kappa/c)({u}_{p,b,c}(z)-1)$, $c \neq 0$ to be close-to-convex. Also, conditions for which ${u}_{p,b,c}(z)$ to be Janowski convex, and $z{u}_{p,b,c}(z)$ to be Janowski starlike in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$ are obtained.


Introduction
Let A denote the class of analytic functions f defined in the open unit disk D = {z : |z| < 1} normalized by the conditions f (0) = 0 = f ′ (0) − 1. If f and g are analytic in D, then f is subordinate to g, written f (z) ≺ g(z), if there is an analytic self-map w of D satisfying w(0) = 0 and f = g • w. For −1 ≤ B < A ≤ 1, let P[A, B] be the class consisting of normalized analytic functions p(z) = 1 + c 1 z + · · · in D satisfying p(z) ≺ 1 + Az 1 + Bz .
The class S * [A, B] of Janowski starlike functions [8] consists of f ∈ A satisfying zf ′ (z) f (z) ∈ P[A, B]. These classes have been studied, for example, in [1,2]. A function f ∈ A is said to be close-to-convex of order β [7,12] if Re (zf ′ (z)/g(z)) > β for some g ∈ S * := S * (0). This article studies the generalized Beesel function u p (z) = u p,b,c (z) given by the power series where κ = p + (b + 1)/2 = 0, −1, −2, −3 · · · . The function u p (z) is analytic in D and solution of the differential equation if b,p,c in C ,such that κ = p + (b + 1)/2 = 0, −1, −2, −3 · · · and z ∈ D. This normalized and generalized Bessel function of the first kind of order p, also satisfy the following recurrence relation which is an useful tool to study several geometric properties of u p . There has been several works [3,4,15,16,5,6] studying geometric properties of the function u p (z), such as on its close-to-convexity, starlikeness, and convexity, radius of starlikeness and convexity.
In Section 2 of this paper, sufficient conditions on A, B, c, κ are determined that will ensure u p satisfies the subordination u p (z) ≺ (1 + Az)/(1 + Bz). It is to be understood that a computationally-intensive methodology with shrewd manipulations is required to obtain the results in this general framework. The benefits of such general results are that by judicious choices of the parameters A and B, they give rise to several interesting applications, which include extending the results of previous works. Using this subordination result, sufficient conditions are obtained for (−4κ/c)u ′ (z) ∈ P[A, B], which next readily gives conditions for (−4κ/c)(u p (z) − 1) to be close-toconvex. Section 3 gives emphasis to the investigation of u p (z) to be Janowski convex as well as of zu p (z) to be Janowski starlike.
The following lemma is needed in the sequel.

Close-to-convexity of the Bessel function
In this section, one main result on the close-to-convexity of the generalized Bessel function with several consequences are discussed in details.
|c| . (4) Further let A, B, κ and c satisfy either the inequality Proof. Define the analytic function p : D → C by Then, a computation yields and Thus, using the identities (9)-(11), the Bessel differential equation (2) can be rewrite as Assume Ω = {0}, and define Ψ(r, s, t; z) by cz.
Proof. Put A = 0 and B = −1 in Theorem 2.1. The condition (4) reduces to κ ≥ 1, which holds in all cases. It is sufficient to establish conditions (6) and (5), or equivalently, and For the case when c ≤ 0, both the inequality (17) and (18) hold as κ ≥ 1.
Next theorem gives the sufficient condition for close-to-convexity when |c| .
Further let A, B, κ and c satisfy either