Sharp inequalities for positive functions

Abstract

In this article we determine the extreme points for the families of positive and positive real functions. We prove a number of new sharp inequalities in addition to quantifying and generalizing some well-known results. We also investigate sharp bounds on |p'(z)| and |p′′(z)| for positive real functions, unknown since 1932.

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

Keywords:

• Positive functions
• Functions of positive real part
• Extreme points

Keywords:

• 2000 Mathematics Subject Classifications:
• Primary: 30D50
• Secondary 30D99

Notes

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