Generalization of the Schwarz–Pick inequality and its application to the extreme problems of approximation of holomorphic functions - Savchuk, V.; Savchuk, M. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.5We propose a method for the pointwise estimation of the derivative \[\left(\frac{f(z)-S_n(f)(z)}{z^n}\right)',\quad n\in\mathbb Z_+,\] where $S_n(f)$ is a partial sum of the Taylor series of a bounded holomorphic function in the unit circle $\mathbb D$ in terms of the absolute value of a function or the quantities of the type of best approximations. Sharp inequalities are obtained and the corresponding extreme functions are described. As a result, we deduce the Schwartz–Pick inequality and obtain the solutions of several extreme problems for the pointwise approximations of bounded holomorphic functions by the Fejér  means of the Taylor series.