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Start Over Coverage 733-742

2. On typically real functions which are generated by a fixed typically real function

Creator:
Sobczak-Kneć, Magdalena and Trabka-Wiȩcław, Katarzyna
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
typically real functions, superdomain of local univalence, radius of local univalence, radius of starlikeness, and radius of univalence
Language:
English
Description:
Let ${\rm T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{ z \in \mathbb {C} \colon |z|<1 \}$, normalized by $f(0)=f'(0)-1=0$ and such that $\mathop {\rm Im} z \mathop {\rm Im} f(z) \geq 0$ for $z \in \Delta $. In this paper we discuss the class ${\rm T}_g$ defined as \[{\rm T}_g:= \{ \sqrt {f(z)g(z)} \colon f \in {\rm T} \},\quad g \in {\rm T}.\] We determine the sets $\bigcup _{g \in {\rm T}} {\rm T}_g$ and $\bigcap _{g \in {\rm T}} {\rm T}_g$. Moreover, for a fixed $g$, we determine the superdomain of local univalence of ${\rm T}_g$, the radii of local univalence, of starlikeness and of univalence of ${\rm T}_g$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public