Relations between the generalized Bessel functions and the Janowski class

TL;DR

This paper establishes conditions under which generalized Bessel functions are subordinate to Janowski functions and explores their convexity and starlikeness properties within the unit disk.

Contribution

It provides new sufficient conditions linking generalized Bessel functions to Janowski class functions, including convexity and starlikeness criteria.

Findings

Conditions for subordination to Janowski functions.

Criteria for Janowski convexity of generalized Bessel functions.

Conditions for Janowski starlikeness of related functions.

Abstract

We are interested in finding the sufficient conditions on $A$, $B$, $\lambda$, $b$ and $c$ which ensure that the generalized Bessel functions ${u}_{\lambda}:={u}_{\lambda,b,c}$ satisfies the subordination ${u}_{\lambda}(z) \prec (1+Az)/ (1+Bz)$. Also, conditions for which ${u}_{\lambda}(z)$ to be Janowski convex, and $z{u}'_{\lambda}(z)$ to be Janowski starlike in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$ are obtained.