Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan; Katarzyna Trąbka-Więcław

Leopold Koczan ; Katarzyna Trąbka-Więcław

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 54 (2010), / Harvested from The Polish Digital Mathematics Library

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Résumé

Let $T$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $Δ : = {z \in ℂ : | z | < 1}$ , normalized by $f (0) = f^{'} (0) - 1 = 0$ and such that Im $z$ Im $f (z)$ $\geq 0$ for $z \in Δ$ . Moreover, let us denote: $T^{(2)} : = {f \in T : f (z) = - f (- z) for z \in Δ}$ and $T^{M, g} : = {f \in T : f ≺ M g in Δ}$ , where $M > 1$ , $g \in T \cap S$ and $S$ consists of all analytic functions, normalized and univalent in $Δ$ .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 ${f \in T : f ≪ M g in Δ}$ , where $M > 1$ , $g \in T$ , which we denote by $T_{M, g}$ . Furthermore, we broaden the class $T_{M, g}$ for the case $M \in (0, 1)$ in the following way: $T_{M, g} = \{f \in T : | f (z) | \geq M | g (z) | for z \in Δ\}$ , $g \in T$ .

Publié le : 2010-01-01
EUDML-ID : urn:eudml:doc:289740

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     author = {Leopold Koczan and Katarzyna Tr\k abka-Wi\k ec\l aw},
     title = {Subclasses of typically real functions determined by some modular inequalities},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {54},
     year = {2010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2010_54_1_75-80}
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Leopold Koczan; Katarzyna Trąbka-Więcław. Subclasses of typically real functions determined by some modular inequalities. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 54 (2010) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2010_54_1_75-80/

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