On the Mutual Change of Values of the Derivative and Third Coefficient in a Class of Regular Functions

Abstract

Let T be the class of functions \( f(z)=z+\sum_{n=2}^{\infty }{c}_n{z}^n \) regular and typically real in the disk |z| < 1. Sharp estimates for the derivative f^′(r)(0 < r < 1) in terms of the value c₃ and sharp estimates for the coefficient c₃ in terms of f′(r) are obtained.