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## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2010 | 54 | 1 |
Tytuł artykułu

### Subclasses of typically real functions determined by some modular inequalities

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EN
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EN
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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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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