Linear sums of certain analytic functions

Abstract:

Let $f$ belong to a certain subclass of the class of functions which are regular in the unit disc $E = \{ z:|z| < 1\}$. Suppose that $\phi = \phi (f,f’,f'')$ and $\psi = \psi (f,f’,f'')$ are regular in $E$ with $\operatorname {Re} \phi > 0$ in $E$ and $\operatorname {Re} \psi \ngtr 0$ in the whole of $E$. In this paper we consider the following two new types of problems: (i) To find the ranges of the real numbers $\lambda$ and $\mu$ such that $\operatorname {Re} (\lambda \phi + \mu \psi ) > 0$ in $E$. (ii) To determine the largest number $\rho ,0 < \rho < 1$, such that $\operatorname {Re} (\phi + \psi ) > 0$ in $|z| < \rho$.