Sharp results on linear combination of simple expressions of analytic functions

Abstract

In this paper we use methods from the theory of differential subordinations to study the linear combination $azf^{"}(z) + bf^{'}(z) + c\frac{f(z)}{z}$ and give sharp bounds over the module, the argument and the real part of $\alpha f^{'}(z) + \beta\frac{f(z)}{z}$.