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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:\(\mathrm{T}_{M,g} = \left\{ f \in \mathrm{T} : |f(z)| \geq M |g(z)| \text{ for } z \in \Delta \right\}\), \(g \in \mathrm{T}\).
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2010
online
2016-07-29
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autor
Bibliografia
- 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).
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Bibliografia
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bwmeta1.element.ojs-doi-10_17951_a_2010_54_1_75-80