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

Sobczak-Kneć, Magdalena, and Trąbka-Więcław, Katarzyna. "On typically real functions which are generated by a fixed typically real function." Czechoslovak Mathematical Journal 61.3 (2011): 733-742. <http://eudml.org/doc/196418>.

@article{Sobczak2011,
abstract = {Let $\{\rm T\}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\lbrace z \in \mathbb \{C\} \colon |z|<1 \rbrace $, normalized by $f(0)=f^\{\prime \}(0)-1=0$ and such that $\mathop \{\rm Im\} z \mathop \{\rm Im\} f(z) \ge 0$ for $z \in \Delta $. In this paper we discuss the class $\{\rm T\}_g$ defined as \[\{\rm T\}\_g:= \lbrace \sqrt\{f(z)g(z)\} \colon f \in \{\rm T\} \rbrace ,\quad g \in \{\rm T\}.\] 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 $\{\rm T\}_g$.},
author = {Sobczak-Kneć, Magdalena, Trąbka-Więcław, Katarzyna},
journal = {Czechoslovak Mathematical Journal},
keywords = {typically real functions; superdomain of local univalence; radius of local univalence; radius of starlikeness; radius of univalence; typically real functions; superdomain of local univalence; radius of local univalence; radius of starlikeness; radius of univalence},
language = {eng},
number = {3},
pages = {733-742},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On typically real functions which are generated by a fixed typically real function},
url = {http://eudml.org/doc/196418},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Sobczak-Kneć, Magdalena
AU - Trąbka-Więcław, Katarzyna
TI - On typically real functions which are generated by a fixed typically real function
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 733
EP - 742
AB - Let ${\rm T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\lbrace z \in \mathbb {C} \colon |z|<1 \rbrace $, normalized by $f(0)=f^{\prime }(0)-1=0$ and such that $\mathop {\rm Im} z \mathop {\rm Im} f(z) \ge 0$ for $z \in \Delta $. In this paper we discuss the class ${\rm T}_g$ defined as \[{\rm T}_g:= \lbrace \sqrt{f(z)g(z)} \colon f \in {\rm T} \rbrace ,\quad g \in {\rm T}.\] 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 ${\rm T}_g$.
LA - eng
KW - typically real functions; superdomain of local univalence; radius of local univalence; radius of starlikeness; radius of univalence; typically real functions; superdomain of local univalence; radius of local univalence; radius of starlikeness; radius of univalence
UR - http://eudml.org/doc/196418
ER -