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.
"An estimate of the first derivative by the Laplacian.." Real Anal. Exchange 28 (1) 145 - 152, 2002-2003.