A continuous function f = u + iv is a complex valued harmonic function in a complex domain C if both u and v are real harmonic in C. In any simply connected domain D ⊂ C we can write f(z) = h + g, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f . A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h′(...

A continuous function f u iv is a complex-valued harmonic function in a complex domain Ω if both u and v are real and harmonic inΩ. In any simply connected domainD ⊂ Ω, we can write f h g, where h and g are analytic inD. We call h the analytic part and g the coanalytic part of f . A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that |h′ z | ...

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

The purpose of the present paper is to establish connections between various subclasses of harmonic functions by applying certain convolution operator involving hypergeometric functions. To be more precise, we investigate such connections with Goodman-Rønning-type harmonic univalent functions in the open unit disc U .

Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane. Introduction A well-known criterion due to Becker [5] states that if a locally univalent analytic function φ in the unit disk D satisfies (1) sup z∈D ∣∣∣∣φ′′(z) φ′(z) ∣∣∣∣ (1− ...

A 2p-times continuously differentiable complex-valued function f = u+ iv in a simply connected domainΩ ⊆ C is p-harmonic if f satisfies the p-harmonic equation ∆p f = 0. In this paper, we investigate the properties of p-harmonic mappings in the unit disk |z| < 1. First, we discuss the convexity, the starlikeness and the region of variability of some classes of p-harmonic mappings. Then we prove...

A recent result of Yalcin [9] appeared in Applied Mathematics Letters (2005), concerning the convolution of two harmonic univalent functions in class SH(m,n, α) is improved. AMS (MOS) Subject Classification Codes: 30C45