Inclusion of generalized Bessel functions in the Janowski class

TL;DR

This paper establishes conditions under which generalized Bessel functions belong to the Janowski class, ensuring properties like subordination, close-to-convexity, convexity, and starlikeness in the unit disk.

Contribution

It provides new sufficient conditions for generalized Bessel functions to satisfy various geometric function properties within the Janowski class.

Findings

Conditions for subordination of generalized Bessel functions.

Criteria for close-to-convexity of scaled Bessel functions.

Parameters ensuring Janowski convexity and starlikeness.

Abstract

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.