On semi-typically real functions which are generated by a fixed semi-typically real function

Abstract

Let $\mathcal{A}$ denote the family of all functions that are analytic in the unit disk $\Delta := \{ z \in \mathbb{C} : |z|<1 \}$ and normalized by $f(0)=f'(0)-1=0$. In this paper, we investigate the class $\mathcal{T}_G$ defined as follows \[\mathcal{T}_G:= \left\{ \sqrt{F(z)G(z)} : F \in \mathcal{T} \right\},\quad G \in \mathcal{T},\] where $\mathcal{T}$ denotes the class of all semi-typically real functions i.e. $\mathcal{T} := \{ F \in \mathcal{A} : F(z)>0 \iff z \in (0,1) \}$. We find the sets $\bigcup_{G \in \mathcal{T}} \mathcal{T}_G$ and $\bigcap_{G \in \mathcal{T}} \mathcal{T}_G$, the set of all extreme points of $\mathcal{T}_G$ and the set of all support points of $\mathcal{T}_G$. Moreover, for the fixed $G$, we determine the radii of local univalence, of starlikeness and of univalence of $\mathcal{T}_G$.