Strong differential superordination.
In this work we consider the class of analytic functions G(α, γ), which is a subset of gamma-starlike functions introduced by Lewandowski, Miller and Złotkiewicz in Gamma starlike functions, Ann. Univ. Mariae Curie- Skłodowska, Sect. A 28 (1974), 53-58. We discuss the order of strongly starlikeness and the order of strongly convexity in this subclass.
The aim of this paper is to present a new method of proof of an analytic characterization of strongly starlike functions of order (α,β). The relation between strong starlikeness and spirallikeness of the same order is discussed in detail. Some well known results are reproved.
MSC2010: 30C45, 33C45
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...