Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, Katarzyna Trąbka-Więcław


Let \(\mathrm{T}\) be the family of all typically real functions, i.e. functions that are analytic in the unit disk \(\Delta := \{ z \in \mathbb{C} : |z|<1 \}\), normalized by \(f(0)=f'(0)-1=0\) and such that Im \(z\) Im \(f(z)\) \(\geq 0\) for \(z \in \Delta\). Moreover, let us denote: \(\mathrm{T}^{(2)}:=  \{f \in \mathrm{T}: f(z)=-f(-z) \text{ for } z \in \Delta \}\) and \(\mathrm{T}^{M,g} :=  \{ f \in \mathrm{T}: f \prec Mg \text{ in } \Delta \}\), 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 \(\{ f \in \mathrm{T}: f \ll Mg \text{ in } \Delta \}\), 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:
\(\mathrm{T}_{M,g} = \left\{ f \in \mathrm{T} : |f(z)| \geq M |g(z)| \text{ for } z \in \Delta \right\}\), \(g \in \mathrm{T}\).


Typically real functions; majorization; subordination

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Duren, P. L., Univalent Functions, Springer-Verlag, New York, 1983.

Goodman, A. W., Univalent Functions, Mariner Publ. Co., Tampa, 1983.

Koczan, L., On classes generated by bounded functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 52 (2) (1998), 95-101.

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.

Koczan, L., Zaprawa, P., On typically real functions with n-fold symmetry, Ann. Univ. Mariae Curie-Skłodowska Sect. A 52 (2) (1998), 103-112.

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

Data publikacji: 2016-07-29 22:06:17
Data złożenia artykułu: 2016-07-29 21:32:18


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Copyright (c) 2010 Leopold Koczan, Katarzyna Trąbka-Więcław