The DCP radius of the exponential function

Abstract

We prove that e^rz is direction convexity preserving (DCP) for all 0<r≤ 1. This extends a previous result of G. Kurth (Kurth, G., 1991, Über variationsvermindernde Transformationen. Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilian-Universität Würzburg, Würzburg.).

Keywords:

• Direction convex
• Convex harmonic functions
• Convolution
• 2000 Mathematics Subject Classification: 30C45

Acknowledgements

The authors wish to thank Luis Salinas for useful discussions. Stephan Ruscheweyh received partial support from DGIP-UTFSM-Chile (grant 240222), from FONDECYT-CHILE (grants 1040366 (2004–2006) and 7040069 (2004)) and from the German-Israeli Foundation (grant G-643-117.6/1999).

Notes

Dedicated to Professor Heinrich Begehr on his 65th birthday.

Additional information

Notes on contributors

Gajendra B. Thapa

Dedicated to Professor Heinrich Begehr on his 65th birthday.