On the Janowski convexity and starlikeness of the confluent hypergeometric function

Abstract

For $-1 \leq B < A \leq 1$, conditions on $A$, $B$, $a$, $c$ are determined that ensure the confluent hypergeometric function $\Phi(a;c;z)$ satisfies the subordination $\Phi(a; c; z) \prec (1+Az)/ (1+Bz)$. This gives rise to conditions for $(c/a)( \Phi(a; c; z)-1)$ to be close-to-convex, $\Phi(a; c; z)$ to be Janowski convex, and $z\Phi(a; c; z)$ to be Janowski starlike.