arXiv subjects

Saminathan Ponnusamy

Publications and source records attributed to Saminathan Ponnusamy.

At least 55 records · Page 3Linked to original sources

arXiv2021

Schwarz-Pick and Landau type theorems for solutions to the Dirichlet-Neumann problem in the unit disk

The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $(\partial_z\partial_{\overline{z}})^2 w=g$ in the unit disc $\ID$, $w=γ_0$ and $\partial_ν\partial_z\partial_{\overline{z}}w=γ$ on $\mathbb{T}$ (the unit circle), $\frac{1}{2πi}\int_{\mathbb{T}}w_{ζ\overlineζ}(ζ)\frac{dζ}ζ=c$, where $\partial_ν$ denotes differentiation in the outward normal direction. More precisely, we obtain Schwarz-Pick type inequalities and Landau type theorem for solutions to the Dirichlet-Neumann problem.

math.CV↗

arXiv2020

Complex symmetric weighted Composition Differentiation Operators

In this note, we completely characterize complex symmetric weighted composition differentiation operator on the Hardy space $H^2$ with respect to the conjugation operator $C_{λ,α}$. Meanwhile, the normal and self-adjoint of the weighted composition differentiation operators on the Hardy space $H^2$ are also studied. This note could be considered as a continuation of the work initiated by Fatehi and Hammond.

math.FA↗

arXiv2020

Improved Bohr's inequality for mappings defined on simply connected domains

In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore, we examine the case where a harmonic mapping is defined in a disk containing $\mathbb{D}$ and obtain a Bohr type inequality.

math.CV↗

arXiv2020

Refined Bohr-type inequalities with area measure for bounded analytic functions

In this paper, we establish five new sharp versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing Schwarz function in place of the initial coefficients in the power series representations of the functions involved and thereby, we generalize several related results of earlier authors.

math.CV↗

arXiv2020

Properties of Normal Harmonic Mappings

In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings.

math.CV↗

arXiv2020

Refined Bohr inequality for bounded analytic functions

In this article, by combining appropriate refined Bohr's inequalities with some techniques concerning bounded analytic functions defined in the unit disk, we generalize and improve several Bohr type inequalities for such functions.

math.CV↗

arXiv2020

Refinement of the Classical Bohr Inequality

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to $1$, then the sum of absolute values of its terms is less than or equal to $1$ for the subdisk $|z|<1/3$ and $1/3$ is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present a refined version of Bohr's inequality along with few other related improved versions of previously known results.

math.CV↗

arXiv2020

Bohr type inequalities for functions with a multiple zero at the origin

Recently, there has been a number of good deal of research on the Bohr's phenomenon in various setting including a refined formulation of his classical version of the inequality. Among them, in \cite{PaulPopeSingh-02-10} the authors considered the cases in which the above functions have a multiple zero at the origin. In this article, we present a refined version of Bohr's inequality for these cases and give a partial answer to a question from \cite{PaulPopeSingh-02-10} for the revised setting.

math.CV↗

arXiv2020

On certain quasiconformal and elliptic mappings

Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson's equation $Δf=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.

math.CV↗

arXiv2020

Bohr--Rogosinski inequalities for bounded analytic functions

In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)=\sum_{n=0}^\infty a_nz^n$ in the unit disk $|z| < 1$, in which we replace the coefficients $a_n$ $(n= 0,1,\ldots ,N)$ of the power series by the derivatives $f^{(n)}(z)/n!$ $(n= 0,1,\ldots ,N)$. Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form $f = h + \overline{g}$, where the analytic part $h$ is bounded by $1$ and that $|g'(z)| \le k|h'(z)|$ in $|z| < 1$ and for some $k \in [0,1]$.

math.CV↗

arXiv2020

Bohr's phenomenon for the classes of Quasi-subordination and $K$-quasiregular harmonic mappings

In this paper, we investigate the Bohr radius for $K$-quasiregular sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by Liu and Ponnusamy \cite{LP2019} and, in particular, we settle affirmatively the two conjectures proposed by them. Furthermore, we establish two refined versions of Bohr's inequalities and determine the Bohr radius for the derivatives of analytic functions associated with quasi-subordination.

math.CV↗

arXiv2020

Sharp Bohr type inequality

This article is devoted to the sharp improvement of the classical Bohr inequality for bounded analytic functions defined on the unit disk. We also prove two other sharp versions of the Bohr inequality by replacing the constant term by the absolute of the function and the square of the absolute of the function, respectively.

math.CV↗

arXiv2020

Composition operators on Hardy spaces of the homogenous rooted trees

In \cite{CO-Tp-spaces}, the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space $\mathbb{T}_{p}$ defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator $C_ϕ$ to be bounded on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces. Also, we discuss the compactness of $C_ϕ$ on $\mathbb{T}_{p}$ and finally prove there are no compact composition operators on $\mathbb{T}_{p,0}$ spaces.

math.CV↗

arXiv2019

Rotations and convolutions of harmonic convex mappings

In this paper, we consider the convolutions of slanted half-plane mappings and strip mappings and generalize related results in general settings. We also consider a class of harmonic mappings containing slanted half-plane mappings and strip mappings and, as a consequence, we prove that the convex combination of such mappings is convex.

math.CV↗

arXiv2019

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.

math.CV↗

arXiv2019

Bohr radius for subordination and $K$-quasiconformal harmonic mappings

The present article concerns the Bohr radius for $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic function $φ$, and the purpose is to look into two cases: when $φ$ is convex, or a general univalent function in $\ID$. The results state that if $h(z) =\sum_{n=0}^{\infty}a_n z^n$ and $g(z)=\sum_{n=1}^{\infty}b_n z^n$, then $$\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist (φ(0),\partialφ(\ID)) ~\mbox{ for $r\leq r^*$} $$ and give estimates for the largest possible $r^*$ depending only on the geometric property of $φ(\ID)$ and the parameter $K$. Improved versions of the theorems are given for the case when $b_1 = 0$ and corollaries are drawn for the case when $K\rightarrow \infty$.

math.CV↗

arXiv2019

On classes of meromorphic locally univalent functions defined by differential inequalities

n this article we consider functions meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions which also contains some known results. We include few open problems for further research.

math.CV↗

arXiv2019

Differential Inequalities and Univalent Functions

Let ${\mathcal M}$ be the class of analytic functions in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. $$ Functions in $\mathcal{M}$ are known to be univalent in $\ID$. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\mathcal{M}$ are shown to be starlike in $\ID$. However we conjecture that functions in $\mathcal{M}$ are not necessarily starlike, as apparently supported by other examples.

math.CV↗