39,916 research outputs found
COEFFICIENT CONDITIONS FOR HARMONIC CLOSE-TO-CONVEX FUNCTIONS
New sufficient conditions, concerned with the coefficients of harmonic
functions in the open unit disk normalized
by , for to be harmonic close-to-convex functions
are discussed. Furthermore, several illustrative examples and the image domains
of harmonic close-to-convex functions satisfying the obtained conditions are
enumerated.Comment: 11 pages, 6 figure
Radial oscillation of harmonic functions in the Korenblum class
We study radial behavior of harmonic functions in the unit disk belonging to
the Korenblum class. We prove that functions which admit two-sided Korenblum
estimate either oscillate or have slow growth along almost all radii
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
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