Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

NORM ESTIMATES OF THE PRE-SCHWARZIAN DERIVATIVES FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS

Yong Chan Kima1 and Toshiyuki Sugawaa2

a1 Department of Mathematics Education, Yeungnam University, 214-1 Daedong, Gyongsan 712-749, Korea (kimyc@yu.ac.kr)

a2 Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan (sugawa@math.sci.hiroshima-u.ac.jp)

Abstract

A sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.

(Received April 28 2004)

Keywords

  • Primary 30C45;
  • Secondary 30A32;
  • pre-Schwarzian derivative;
  • uniformly locally univalent;
  • close-to-convex function