Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ := {z ∈ C : |z| < 1}, normalized by f(0) = f'(0) - 1 = 0 and such that Im z Im f(z) ≥ 0 for z ∈ Δ. Moreover, let us denote: T(2) := {f ∈ T : f(z) = -f(-z) for z ∈ Δ} and TM, g := {f ∈ T : f ≺ Mg in Δ}, where M > 1, g ∈ T ∩ 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 ∈ T : f < Mg in Δ}, where M > 1, g ∈ T, which we denote by TM, g. Furthermore, we broaden the class TM, g for the case M ∈ (0, 1) in the following way: TM, g = {f ∈ T : |f(z)| ≥ M|g(z)| for z ∈ Δ}, g ∈ T.
@article{bwmeta1.element.doi-10_2478_v10062-010-0006-x,
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 UMCS, Mathematica},
volume = {64},
year = {2010},
pages = {75-80},
zbl = {1213.30031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-010-0006-x}
}
Leopold Koczan; Katarzyna Trąbka-Więcław. Subclasses of typically real functions determined by some modular inequalities. Annales UMCS, Mathematica, Tome 64 (2010) pp. 75-80. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-010-0006-x/
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