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 517.5 We propose a method for the pointwise estimation of the derivative \[(f(z)-S_n(f)(z)z^n)’,nZ_+,\] where S_n(f) is a partial sum of the Taylor series of a bounded holomorphic function in the unit circle 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.