An Estimate of the First Derivative by the Laplacian

Abstract

In this note a particular case of the following general problem is considered: how to control lower order derivatives by higher ones, at least over a sequence of points. The following particular case is proved: if a C2 negative-valued function h=h(w) depends on one complex variable in the unit disc and h(1)=hw(1)=0, then the first derivative hw is controlled by the Laplacian of h over a sequence of points converging to w=1. Such kind of estimates have applications to delicate problems of convexity with respect to various families of functions.

Department(s)

Mathematics and Statistics

Keywords and Phrases

real functions; Laplacian

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2002 Real Analysis Exchange, All rights reserved.

Publication Date

01 Jan 2002

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