## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Katarzyna Trąbka-Więcław

#### 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 $\mathcal{T}_G$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 1 (2012), 81-90.

Dates
First available in Project Euclid: 7 March 2012

https://projecteuclid.org/euclid.bbms/1331153410

Digital Object Identifier
doi:10.36045/bbms/1331153410

Mathematical Reviews number (MathSciNet)
MR2952797

Zentralblatt MATH identifier
1242.30014

#### Citation

Trąbka-Więcław, Katarzyna. On semi-typically real functions which are generated by a fixed semi-typically real function. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 1, 81--90. doi:10.36045/bbms/1331153410. https://projecteuclid.org/euclid.bbms/1331153410