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S. Ponnusamy

Publications and source records attributed to S. Ponnusamy.

At least 37 records · Page 2Linked to original sources
arXiv2014

Uniform close-to-convexity radius of sections of functions in the close-to-convex family

The authors consider the class $\F$ of normalized functions $f$ analytic in the unit disk $\ID$ and satisfying the condition $${\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2},\quad z\in\D. $$ Recently, Ponnusamy et al. \cite{samy-hiroshi-swadesh} have shown that $1/6$ is the uniform sharp bound for the radius of convexity of every section of each function in the class $\F$. They conjectured that $1/3$ is the uniform univalence radius of every section of $f\in \F$. In this paper, we solve this conjecture affirmatively.

math.CV↗
arXiv2013

On quasisymmetry of quasiconformal mappings and its applications

Suppose that $f: D\to D'$ is a quasiconformal mapping, where $D$ and $D'$ are domains in ${\mathbb R}^n$, and that $D$ is a broad domain. Then for every arcwise connected subset $A$ in $D$, the weak quasisymmetry of the restriction $f|_A: A\to f(A)$ implies its quasisymmetry, and as a consequence, we see that the answer to one of the open problems raised by Heinonen from 1989 is affirmative under the additional condition that $A$ is arcwise connected. As an application, we establish nine equivalent conditions for a bounded domain, which is quasiconformally equivalent to a bounded and simply connected uniform domain, to be John. This result is a generalization of the main result of Heinonen from [Quasiconformal mappings onto John domains, \textit{Rev. Math. Iber.,} {\bf 5} (1989), 97--123].

math.CV↗
arXiv2012

Lipschitz spaces and bounded mean oscillation of harmonic mappings

In this paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz number of planar harmonic mappings in terms of bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BMO_{2}$ as a Banach space..

math.CV↗
arXiv2012

Harmonic Close-to-convex Functions and Minimal Surfaces

In this paper, we study the family ${\mathcal C}_{H}^0$ of sense-preserving complex-valued harmonic functions $f$ that are normalized close-to-convex functions on the open unit disk $\mathbb{D}$ with $f_{\bar{z}}(0)=0$. We derive a sufficient condition for $f$ to belong to the class $\CC_{H}^0$. We take the analytic part of $f$ to be $zF(a,b;c;z)$ or $zF(a,b;c;z^2)$ and for a suitable choice of co-analytic part of $f$, the second complex dilatation $w(z)=\bar{f_{\bar{z}}}/f_z$ turns out to be a square of an analytic function. Hence $f$ is lifted to a minimal surface expressed by an isothermal parameter. Explicit representation for classes of minimal surfaces are given. Graphs generated by using Mathematica are used for illustration.

math.CV↗
arXiv2012

Univalent harmonic mappings with integer or half-integer coefficients

Let ${\mathcal S}$ denote the set of all univalent analytic functions $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ on the unit disk $|z|<1$. In 1946 B. Friedman found that the set $\mathcal S$ 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 $f$ on the unit disk with integer or half-integer coefficients for the analytic and co-analytic parts of $f$. 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.

math.CV↗
arXiv2012

Convolutions of slanted half-plane harmonic mappings

Let ${\mathcal S^0}(H_γ)$ denote the class of all univalent, harmonic, sense-preserving and normalized mappings $f$ of the unit disk $\ID$ onto the slanted half-plane $H_γ:=\{w:\,{\rm Re\,}(e^{iγ}w) >-1/2\}$ with an additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in $\ID$ is not necessarily even univalent in $\ID$. In this paper, we fix $f_0\in{\mathcal S^0}(H_{0})$ and show that the convolutions of $f_0$ and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solves an open problem of Dorff, et. al. \cite{DN} in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.

math.CV↗
arXiv2012

Coefficient Conditions for Harmonic Univalent Mappings and Hypergeometric Mappings

In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic close-to-convex (resp. fully starlike) functions involving Gaussian hypergeometric functions. In addition, we present a convolution characterization for a class of univalent harmonic functions discussed recently by Mocanu, and later by Bshouty and Lyzzaik in 2010. Our approach provide examples of harmonic polynomials that are close-to-convex and starlike, respectively.

math.CV↗
arXiv2012

Weighted Lipschitz continuity, Schwarz-Pick's Lemma and Landau-Bloch's theorem for hyperbolic-harmonic mappings in $\mathbb{C}^{n}$

In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz-Pick type theorem for hyperbolic-harmonic mappings and apply it to prove the existence of Landau-Bloch constant for mappings in $α$-Bloch spaces.

math.CV↗
arXiv2012

The minimal surfaces over the slanted half-planes, vertical strips and single slit

In this paper, we discuss the minimal surfaces over the slanted half-planes, vertical strips, and single slit whose slit lies on the negative real axis. The representation of these minimal surfaces and the corresponding harmonic mappings are obtained explicitly. Finally, we illustrate the harmonic mappings of each of these cases together with their minimal surfaces pictorially with the help of mathematica.

math.CV↗
arXiv2012

On some properties of solutions of the $p$-harmonic equation

A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a simply connected domain $Ω\subseteq\mathbb{C}$ is \textit{p-harmonic} if $f$ satisfies the $p$-harmonic equation $Δ^pf=0.$ 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 the existence of Landau constant for the class of functions of the form $Df=zf_{z}-\barzf_{\barz}$, where $f$ is $p$-harmonic in $|z|<1$. Also, we discuss the region of variability for certain $p$-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for bi- and tri-harmonic mappings.

math.CV↗
arXiv2012

Radius problems associated with pre-Schwarzian and Schwarzian derivatives

Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible $r\in (0,1)$ for which the pre-Schwarzian norm of the dilation $r^{-1}f(rz)$ is not greater than a prescribed number for normalized univalent functions $f(z)$ in the unit disk. Similar results concerning the Schwarzian derivative are also obtained

math.CV↗
arXiv2012

Area integral means, Hardy and weighted Bergman spaces of planar harmonic mappings

In this paper, we investigate some properties of planar harmonic mappings. First, we generalize the main results in \cite{CPW3} and \cite{HT}, and then discuss the relationship between area integral means and harmonic Hardy spaces or harmonic weighted Bergman spaces. At the end, coefficient estimates of mappings in weighted Bergman spaces are obtained.

math.CV↗
arXiv2012

Univalence of the average of two analytic functions

Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\ID=\{z:\,|z|<1\}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_nz^n.$ Let $\mathcal{U}$ denote the set of all $f\in \mathcal{A}$, $f(z)/z\neq 0$ and satisfying the condition $$ | f'(z) (\frac{z}{f(z)})^{2}-1 | < 1 {for $z\in \ID$}. $$ Functions in ${\mathcal U}$ are known to be univalent in $\ID$. For $α\in [0,1]$, let $$ \mathcal{N}(α)= \{f_α:\, f_α(z)=(1-α)f(z)+α\int_0^z\frac{f(t)}{t}\,dt, {$f\in\mathcal{A}$ with $|a_n|\leq n$ for $n\geq 2$}\}. $$ In this paper, we first show that the condition $\sum_{n=2}^{\infty}n|a_n|\leq 1$ is sufficient for $f$ to be in ${\mathcal U}$ and the same condition is necessary for $f\in {\mathcal U}$ in case all $a_n$'s are negative. Next, we obtain the radius of univalence of functions in the class $\mathcal{N}(α)$. Also, for $f,g\in \mathcal{U}$ with $\frac{f(z)+g(z)}{z}\neq 0$ in $\ID$, $F(z)=(f(z)+g(z))/2$, and $G(z)=r^{-1}F(rz)$, we determine a range of $r$ such that $G\in {\mathcal U}$. As a consequence of these results, several special cases are presented.

math.CV↗
arXiv2011

Landau-Bloch constants for functions in $α$-Bloch spaces and Hardy spaces

In this paper, we obtain a sharp distortion theorem for a class of functions in $α$-Bloch spaces, and as an application of it, we establish the corresponding Landau's theorem. These results generalize the corresponding results of Bonk, Minda and Yanagihara, and Liu, respectively. We also prove the existence of Landau-Bloch constant for a class of functions in Hardy spaces and the obtained result is a generalization of the corresponding result of Chen and Gauthier.

math.CV↗
arXiv2011

On harmonic combination of univalent functions

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (λ)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1| <λ~for $z\in \ID$, $$ for some $λ\in (0,1]$. In this paper, among other things, we study a "harmonic mean" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\frac{z}{F(z)}=1/2(\frac{z}{f(z)}+\frac{z}{g(z)}), $$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.

math.CV↗