2,990 research outputs found
Univalent harmonic mappings with integer or half-integer coefficients
Let denote the set of all univalent analytic functions
on the unit disk . In 1946 B.
Friedman found that the set of those functions which have integer
coefficients consists of only nine functions. In a recent paper Hiranuma and
Sugawa proved that the similar set obtained for the functions with half-integer
coefficients consists of twelve functions in addition to the nine. In this
paper, the main aim is to discuss the class of all sense-preserving univalent
harmonic mappings on the unit disk with integer or half-integer
coefficients for the analytic and co-analytic parts of . Secondly, we
consider the class of univalent harmonic mappings with integer coefficients,
and consider the convexity in real direction and convexity in imaginary
direction of these mappings. Thirdly, we determine the set of univalent
harmonic mappings with half-integer coefficients which are convex in real
direction or convex in imaginary direction.Comment: 26 pages, 12 figures, The paper is submitted to a journa
Schwarz's Lemmas for mappings satisfying Poisson's equation
For , and a given continuous function
, we establish some Schwarz type lemmas for
mappings of into satisfying the PDE: , where is a subset of . Then we apply these
results to obtain a Landau type theorem.Comment: 14 page
John disk and -quasiconformal harmonic mappings
The main aim of this article is to establish certain relationships between
-quasiconformal harmonic mappings and John disks. The results of this
article are the generalizations of the corresponding results of Ch.~Pommerenke
\cite{Po}.Comment: 18 pages, 1 figur
On univalent log-harmonic mappings
We consider the class univalent log-harmonic mappings on the unit disk.
Firstly, we obtain necessary and sufficient conditions for a complex-valued
continuous function to be starlike or convex in the unit disk. Then we present
a general idea, for example, to construct log-harmonic Koebe mapping,
log-harmonic right half-plane mapping and log-harmonic two-slits mapping and
then we show precise ranges of these mappings. Moreover, coefficient estimates
for univalent log-harmonic starlike mappings are obtained. Growth and
distortion theorems for certain special subclass of log-harmonic mappings are
studied. Finally, we propose two conjectures, namely, log-harmonic coefficient
and log-harmonic covering conjectures.Comment: 16 pages; This paper was with Studia Scientiarum Mathematicarum
Hungarica since May 2017; Finally returned by saying that they could not find
a suitable referee
Region of variability for functions with positive real part
For \gamma\in\IC such that and , let
denote the class of all analytic functions
in the unit disk with and {\rm Re\,} \left
(e^{i\gamma}P(z)\right)>\beta\cos\gamma \quad \mbox{ in ${\mathbb D}$}. For
any fixed and , we shall
determine the region of variability for
when ranges over the class As a consequence,
we present the region of variability for some subclasses of univalent
functions. We also graphically illustrate the region of variability for several
sets of parameters.Comment: 21 pages with 10 figures, to appear in 201
On the generalized Zalcman functional in the close-to-convex family
Let denote the class of all functions
analytic and univalent in the unit disk
\ID. For , Zalcman conjectured that for . This conjecture has been verified only certain values
of for and for all for the class
of close-to-convex functions (and also for a couple of other classes). In this
paper we provide bounds of the generalized Zalcman coefficient functional
for functions in and for all ,
where is a positive constant. In particular, our special case settles
the open problem on the Zalcman inequality for (i.e. for the
case and ).Comment: 14 pages. The article has been with a journa
Sharp inequalities for logarithmic coefficients and their applications
I. M. Milin proposed, in his 1971 paper, a system of inequalities for the
logarithmic coefficients of normalized univalent functions on the unit disk of
the complex plane. This is known as the Lebedev-Milin conjecture and implies
the Robertson conjecture which in turn implies the Bieberbach conjecture. In
1984, Louis de Branges settled the long-standing Bieberbach conjecture by
showing the Lebedev-Milin conjecture. Recently, O.~Roth proved an interesting
sharp inequality for the logarithmic coefficients based on the proof by de
Branges. In this paper, following Roth's ideas, we will show more general sharp
inequalities with convex sequences as weight functions and then establish
several consequences of them. We also consider the inequality with the help of
de Branges system of linear ODE for non-convex sequences where the proof is
partly assisted by computer. Also, we apply some of those inequalities to
improve previously known results.Comment: 20 pages, 4 figure
Bohr radius for subordination and -quasiconformal harmonic mappings
The present article concerns the Bohr radius for -quasiconformal
sense-preserving harmonic mappings in the unit disk
for which the analytic part is subordinated to some analytic
function , and the purpose is to look into two cases: when
is convex, or a general univalent function in \ID. The results state that if
and , then
\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist
(\varphi(0),\partial\varphi(\ID)) ~\mbox{ for $r\leq r^*$} and give
estimates for the largest possible depending only on the geometric
property of \varphi (\ID) and the parameter . Improved versions of the
theorems are given for the case when and corollaries are drawn for
the case when .Comment: 15 pages; To appear in Bulletin of the Malaysian Mathematical
Sciences Societ
On Harmonic -Bloch and -Bloch-type mappings
The aim of this paper is twofold. One is to introduce the class of harmonic
-Bloch-type mappings as a generalization of harmonic -Bloch mappings
and thereby we generalize some recent results of harmonic -Bloch-type
mappings investigated recently by Efraimidis et al. \cite{EGHV}. The other is
to investigate some subordination principles for harmonic Bloch mappings and
then establish Bohr's theorem for these mappings and in a general setting, in
some cases.Comment: 17 pages; Comments are welcom
Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm
In this paper, we consider the class of uniformly locally univalent harmonic
mappings in the unit disk and build a relationship between its pre-Schwarzian
norm and uniformly hyperbolic radius. Also, we establish eight ways of
characterizing uniformly locally univalent sense-preserving harmonic mappings.
We also present some sharp distortions and growth estimates and investigate
their connections with Hardy spaces. Finally, we study subordination principles
of norm estimates.Comment: 28 pages; The article is to appear in Indagationes Mathematica
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