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On semi-typically real functions

Leopold KoczanKatarzyna Trąbka-Więcław — 2009

Annales UMCS, Mathematica

Suppose that A is the family of all functions that are analytic in the unit disk Δ and normalized by the condition [...] For a given A ⊂ A let us consider the following classes (subclasses of A): [...] and [...] where [...] and S consists of all univalent members of A.In this paper we investigate the case A = τ, where τ denotes the class of all semi-typically real functions, i.e. [...] We study relations between these classes. Furthermore, we find for them sets of variability of initial coeffcients,...

Subclasses of typically real functions determined by some modular inequalities

Leopold KoczanKatarzyna Trąbka-Więcław — 2010

Annales UMCS, Mathematica

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...

Subclasses of typically real functions determined by some modular inequalities

Leopold KoczanKatarzyna Trąbka-Więcław — 2010

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ : = { z : | 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 T M , g : = { f T : f M g 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 M g in Δ } , where M > 1 , g T , which we denote by T M , g . Furthermore,...

On typically real functions which are generated by a fixed typically real function

Magdalena Sobczak-KnećKatarzyna Trąbka-Więcław — 2011

Czechoslovak Mathematical Journal

Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ : = { z : | z | < 1 } , normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 and such that Im z Im f ( z ) 0 for z Δ . In this paper we discuss the class T g defined as T g : = { f ( z ) g ( z ) : f T } , g T . We determine the sets g T T g and g T T g . Moreover, for a fixed g , we determine the superdomain of local univalence of T g , the radii of local univalence, of starlikeness and of univalence of T g .

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