A generalization of Nehari's p-criterion for univalence

Abstract

Nehari's most general univalence criterion | Sf (z)| ≤2p(| z|) requires the even solution of u″+pu=0 to have no zeros, and to be nonincreasing on [0,1). In this article, we show univalence under a weaker form of the second assumption, namely, that pu ⁴ be nonincreasing on [0,1).

§Dedicated to Professor Peter L. Duren on the occasion of his 70 th birthday.

Keywords:

• Schwarzian derivative
• Univalent functions

Keywords:

• AMS 2000 Mathematics Subject Classifications: Primary 30C99
• Secondary 31A05
• 53A10

Acknowledgement

The authors were supported in part by FONDECYT Grant # 1030589.

Notes

§Dedicated to Professor Peter L. Duren on the occasion of his 70 th birthday.