Entire functions with univalent derivatives in a disc

It is proved that if the increasing sequence (n_p) of natural numbers satisfies the condition n_p+1/n_p 1 (p ) and all derivatives f^(np) of the analytic function f in D=¦¦< 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 (p ) there exists an analytic function f in D all of whose derivatives f^(np) are univalent in D and D is the boundary for f. The growth of entire functions with derivatives univalent in the disc D is also studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 400–406, March, 1991.