# Subclasses of typically real functions determined by some modular inequalities

• Volume: 64, Issue: 1, page 75-80
• ISSN: 2083-7402

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## Abstract

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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.

## How to cite

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Leopold Koczan, and Katarzyna Trąbka-Więcław. "Subclasses of typically real functions determined by some modular inequalities." Annales UMCS, Mathematica 64.1 (2010): 75-80. <http://eudml.org/doc/268267>.

@article{LeopoldKoczan2010,
abstract = {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.},
author = {Leopold Koczan, Katarzyna Trąbka-Więcław},
journal = {Annales UMCS, Mathematica},
keywords = {Typically real functions; majorization; subordination; typically real functions},
language = {eng},
number = {1},
pages = {75-80},
title = {Subclasses of typically real functions determined by some modular inequalities},
url = {http://eudml.org/doc/268267},
volume = {64},
year = {2010},
}

TY - JOUR
AU - Leopold Koczan
AU - Katarzyna Trąbka-Więcław
TI - Subclasses of typically real functions determined by some modular inequalities
JO - Annales UMCS, Mathematica
PY - 2010
VL - 64
IS - 1
SP - 75
EP - 80
AB - 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.
LA - eng
KW - Typically real functions; majorization; subordination; typically real functions
UR - http://eudml.org/doc/268267
ER -

## References

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1. Duren, P. L., Univalent Functions, Springer-Verlag, New York, 1983.
2. Goodman, A. W., Univalent Functions, Mariner Publ. Co., Tampa, 1983.
3. Koczan, L., On classes generated by bounded functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 52 (2) (1998), 95-101.
4. Koczan, L., Szapiel, W., Extremal problems in some classes of measures (IV). Typically real functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 43 (1989), 55-68. Zbl0743.30023
5. Koczan, L., Zaprawa, P., On typically real functions with n-fold symmetry, Ann. Univ. Mariae Curie-Skłodowska Sect. A 52 (2) (1998), 103-112. Zbl1010.30019
6. Rogosinski, W. W., Über positive harmonische Entwicklugen und tipisch-reelle Potenzreichen, Math. Z. 35 (1932), 93-121 (German). Zbl0003.39303

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