Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, and Katarzyna Trąbka-Więcław. "Subclasses of typically real functions determined by some modular inequalities." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 54.1 (2010): null. <http://eudml.org/doc/289740>.

@article{LeopoldKoczan2010,
abstract = {Let $\mathrm \{T\}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta := \lbrace z \in \mathbb \{C\} : |z|<1 \rbrace $, normalized by $f(0)=f^\{\prime \}(0)-1=0$ and such that Im $z$ Im $f(z)$$\ge 0$ for $z \in \Delta $. Moreover, let us denote: $\mathrm \{T\}^\{(2)\}:=~ \lbrace f \in \mathrm \{T\}: f(z)=-f(-z) \text\{ for \} z \in \Delta \rbrace $ and $\mathrm \{T\}^\{M,g\} :=~ \lbrace f \in \mathrm \{T\}: f \prec Mg \text\{ in \} \Delta \rbrace $, where $M>1$, $g \in \mathrm \{T\} \cap \mathrm \{S\}$ and $\mathrm \{S\}$ consists of all analytic functions, normalized and univalent in $\Delta $.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\lbrace f \in \mathrm \{T\}: f \ll Mg \text\{ in \} \Delta \rbrace $, where $M>1$, $g \in \mathrm \{T\}$, which we denote by $\mathrm \{T\}_\{M,g\}$. Furthermore, we broaden the class $\mathrm \{T\}_\{M,g\}$ for the case $M \in (0,1)$ in the following  way:$\mathrm \{T\}_\{M,g\} = \left\lbrace f \in \mathrm \{T\} : |f(z)| \ge M |g(z)| \text\{ for \} z \in \Delta \right\rbrace $, $g \in \mathrm \{T\}$.},
author = {Leopold Koczan, Katarzyna Trąbka-Więcław},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Typically real functions; majorization; subordination},
language = {eng},
number = {1},
pages = {null},
title = {Subclasses of typically real functions determined by some modular inequalities},
url = {http://eudml.org/doc/289740},
volume = {54},
year = {2010},
}

TY - JOUR
AU - Leopold Koczan
AU - Katarzyna Trąbka-Więcław
TI - Subclasses of typically real functions determined by some modular inequalities
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2010
VL - 54
IS - 1
SP - null
AB - Let $\mathrm {T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta := \lbrace z \in \mathbb {C} : |z|<1 \rbrace $, normalized by $f(0)=f^{\prime }(0)-1=0$ and such that Im $z$ Im $f(z)$$\ge 0$ for $z \in \Delta $. Moreover, let us denote: $\mathrm {T}^{(2)}:=~ \lbrace f \in \mathrm {T}: f(z)=-f(-z) \text{ for } z \in \Delta \rbrace $ and $\mathrm {T}^{M,g} :=~ \lbrace f \in \mathrm {T}: f \prec Mg \text{ in } \Delta \rbrace $, where $M>1$, $g \in \mathrm {T} \cap \mathrm {S}$ and $\mathrm {S}$ consists of all analytic functions, normalized and univalent in $\Delta $.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\lbrace f \in \mathrm {T}: f \ll Mg \text{ in } \Delta \rbrace $, where $M>1$, $g \in \mathrm {T}$, which we denote by $\mathrm {T}_{M,g}$. Furthermore, we broaden the class $\mathrm {T}_{M,g}$ for the case $M \in (0,1)$ in the following  way:$\mathrm {T}_{M,g} = \left\lbrace f \in \mathrm {T} : |f(z)| \ge M |g(z)| \text{ for } z \in \Delta \right\rbrace $, $g \in \mathrm {T}$.
LA - eng
KW - Typically real functions; majorization; subordination
UR - http://eudml.org/doc/289740
ER -