Inclusion of generalized Bessel functions in the Janowski class

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 $(-4\kappa /c)(u_{p,b,c}(z)-1)$, $c\ne 0$ 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.

13 pages