On integral functions with derivatives univalent in a circle

Abstract

It is proved that if the increasing sequence $n_p$  of natural numbers satisfies the condition $n_{p+1}/n_p→1 (p→\infty)$ and all derivatives $f^{(n_p)}$  of the analytic function $f$ in $D=\{z : |z | < 1\}$ are univalent in $D$, then $f$ is an entire function. At the same time, for each increasing sequence $(n_p)$ natural numbers such that $n_{p+1}/n_p→1 (p→\infty)$ there exists an analytic function $f$  in $D$  all of whose derivatives $f^{(n_p)}$ are univalent in $D$  and $\partial D$  is the boundary for $f$. The growth of entire functions with derivatives univalent in the disc $D$ is also studied.