4 research outputs found
Injectivity of sections of convex harmonic mappings and convolution theorems
In the article the authors consider the class of
sense-preserving harmonic functions defined in the unit disk
and normalized so that and , where
and are analytic in the unit disk. In the first part of the article we
present two classes and of
functions from and show that if
and , then the harmonic convolution is a univalent
and close-to-convex harmonic function in the unit disk provided certain
conditions for parameters and are satisfied. In the second
part we study the harmonic sections (partial sums) where , and denote the -th partial sums of
and , respectively. We prove, among others, that if
is a univalent harmonic convex mapping,
then is univalent and close-to-convex in the disk for
, and is also convex in the disk for
and . Moreover, we show that the section of is not convex in the disk but is shown to be convex
in a smaller disk.Comment: 16 pages, 3 figures; To appear in Czechoslovak Mathematical Journa
Remarks on the univalence criterion of Pascu and Pascu
We consider a recent work of Pascu and Pascu [‘Neighbourhoods of univalent functions’, Bull. Aust. Math. Soc. (2) (2011), 210–219] and rectify an error that appears in their work. In addition, we study certain analogue results for sense-preserving harmonic mappings in the unit disk . As a corollary to this result, we derive a coefficient condition for a sense-preserving harmonic mapping to be univalent in .
10.1017/S000497271300061