Geodesic
Subclasses of typically real functions determined by some modular inequalities
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 54 (2010) no. 1.

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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| lt;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 T^M,g :=  { f ∈T: f ≺ Mg in Δ}, where M gt;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 gt;1, g ∈T, which we denote by T_M,g. Furthermore, we broaden the class T_M,g for the case M ∈ (0,1) in the following  way:T_M,g = { f ∈T : |f(z)| ≥ M |g(z)| for z ∈Δ}, g ∈T.
Keywords: Typically real functions, majorization, subordination 
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Koczan, Leopold; Trąbka-Więcław, Katarzyna. Subclasses of typically real functions determined by some modular inequalities. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 54 (2010) no. 1. http://geodesic.mathdoc.fr/item/AUM_2010_54_1_a6/

[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.

[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.

[6] Rogosinski, W. W., Uber positive harmonische Entwicklugen und tipisch-reelle Potenzreichen, Math. Z. 35 (1932), 93–121 (German).