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

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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

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1331153410

Digital Object Identifier
doi:10.36045/bbms/1331153410

Mathematical Reviews number (MathSciNet)
MR2952797

Zentralblatt MATH identifier
1242.30014

Subjects
Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C55: General theory of univalent and multivalent functions

Keywords
Typically real functions set of local univalence radius of local univalence radius of starlikeness radius of univalence extreme points support points

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


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