Relations between the generalized Bessel functions and the Janowski class

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 $z{u}_{\lambda }^{\prime }(z)$ to be Janowski starlike in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ are obtained.