On the Existence of Functions being Univalent in Half-plane together with their Derivatives

Abstract

In articles [1]-[3] a class of functions being univalent in unit disc with all their derivatives was investigated. It was proved such functions exist and must be the entire functions of exponential type. The example of such a function is an exponential function z f (z) = e . In this work the question on existence of functions, beeing univalent in half-plane with all their derivatives was raised and the negative answer to such question was given.

Keywords

• class of univalent functions
• unit disc
• half-plane
• existence