Harmonic univalent mappings have attracted the serious attention of complex analysts only after the appearance of a basic paper by Clunie and Sheil-Small [4] in 1984. These researchers laid the foundation for the study of harmonic univalent mappings over the unit disk as a generalization of analytic univalent functions. Interestingly, almost at the same time, the famous Bieberbach conjecture wh...

Abstract In this paper we introduce a new family Ā(α, β, γ) of harmonic univalent functions with negative coefficients using fractional calculus operator Ωλwhich contains various well known classes of harmonic univalent functions as well as many new ones. We obtain coefficient estimate, distortion theorem and various other properties for functions belonging to the class Ā(α, β, γ) in the unit d...

Let q1 and q2 belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions p to satisfy the double subordination chain q1(z)≺ p(z)≺ q2(z). The differential sandwich-type result obtained is applied to normalized univalent functions and to Φ-like functions.

Let be the class of functions which are analytic in open unit disk and having form . Denote to for all that univalent Then, let denote bi-univalent In this paper, we obtain second Hankel determinant certain classes function defined by subordinations particular, determine initial coefficients obtained upper bound functional

A continuous function f with domain X and range f(X) in R n is weakly univalent if there is a sequence of continuous one-to-one functions on X converging to f uniformly on bounded subsets of X. In this article, we establish, under certain conditions, the connectedness of an inverse image f ?1 (q). The univalence results of Radulescu-Radulescu, Mor e-Rheinboldt, and Gale-Nikaido follow from our ...

Let R D be the algebra generated in Sobolev space W22 D by the rational functions with poles outside the unit disk D. In this paper, we study the similarity invariant of the multiplication operators Mg in L R D , when g is univalent analytic on D or Mg is strongly irreducible. And the commutants of multiplication operators whose symbols are composite functions, univalent analytic functions, or ...