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 $C^2$ negative-valued function $h=h(w)$ depends on one complex variable in the unit disc and $h(1)=h_w(1)=0$, then the first derivative $h_w$ 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.
Publié le : 2002-05-14
Classification:
Real functions,
Laplacian,
26B25,
31A05,
32F05,
52A41
@article{1150118731,
author = {Dwilewicz, Roman},
title = {An estimate of the first derivative by the Laplacian.},
journal = {Real Anal. Exchange},
volume = {28},
number = {1},
year = {2002},
pages = { 145-152},
language = {en},
url = {http://dml.mathdoc.fr/item/1150118731}
}
Dwilewicz, Roman. An estimate of the first derivative by the Laplacian.. Real Anal. Exchange, Tome 28 (2002) no. 1, pp. 145-152. http://gdmltest.u-ga.fr/item/1150118731/