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International Journal of Mathematics and Mathematical Sciences
Volume 13, Issue 2, Pages 287-293
http://dx.doi.org/10.1155/S0161171290000424

On α-convex functions of order β with m-fold symmetry

Department of Mathematics, Northwest University, Xian, China

Received 23 July 1987; Revised 22 September 1987

Copyright © 1990 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This note is a continuation of the previous work [1,2,3]. First we get a new subordination for α-convex functions of order β when α=12β, which implies the rotation theorem for (12β)/m-convex functions of order β with m-fold symmetry. Then we extend the known results on α-convex functions of order β to the functions with m-fold symmetry. In particular, we give the sharp order of convexity of α-convex functions of order β with m-fold symmetry for α1, which is analogous in sharpness to a result given by Miller, Mocanu and Reade [1].