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 $C^2$ 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.