336 research outputs found
Geometric Properties of Partial Sums of Univalent Functions
The th partial sum of an analytic function is the polynomial . A survey of the
univalence and other geometric properties of the th partial sum of univalent
functions as well as other related functions including those of starlike,
convex and close-to-convex functions are presented
Certain subclasses of multivalent functions defined by new multiplier transformations
In the present paper the new multiplier transformations
\mathrm{{\mathcal{J}% }}_{p}^{\delta }(\lambda ,\mu ,l) (\delta ,l\geq
0,\;\lambda \geq \mu \geq 0;\;p\in \mathrm{% }%\mathbb{N} )} of multivalent
functions is defined. Making use of the operator two new subclasses and \textbf{\ }of multivalent analytic
functions are introduced and investigated in the open unit disk. Some
interesting relations and characteristics such as inclusion relationships,
neighborhoods, partial sums, some applications of fractional calculus and
quasi-convolution properties of functions belonging to each of these subclasses
and
are
investigated. Relevant connections of the definitions and results presented in
this paper with those obtained in several earlier works on the subject are also
pointed out
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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