Saminathan Ponnusamy

• Ph.D
• Professor at Indian Institute of Technology Madras

Consider the family of locally univalent analytic functions h in the unit disk \(|z|<1\) with the normalization \(h(0)=0\), \(h'(0)=1\) and satisfying the condition where \(0<\alpha \le 1\). The aim of this article is to show that this family has several elegant properties such as involving Blaschke products, Schwarzian derivative and univalent har...

Recently, the Wang et al. \cite{wwrq} proposed a coefficient conjecture for the family ${\mathcal S}_H^0(K)$ of $K$-quasiconformal harmonic mappings $f = h + \overline{g}$ that are sense-preserving and univalent, where $h(z)=z+\sum_{k=2}^{\infty}a_kz^k$ and $g(z)=\sum_{k=1}^{\infty}b_kz^k$ are analytic in the unit disk $|z|<1$, and the dilatation $...

In this paper we give a Banach algebra structure by the Duhamel product and an invertibility criterion for Besov spaces \(B_{p}\). We also describe the extended eigenvalues of the Volterra integral operator V. In the last section of the paper, motivated by the work of Karapetrović and Mashreghi: namely, \(\Vert f*g\Vert _{A_{q}}\le \Vert D^{1}f\Ver...

Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new multidimensional analogues of the refined Bohr inequalities for bounded holomorphic mappings on the unit ball...

Consider the family of locally univalent analytic functions $h$ in the unit disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the condition $${\real} \left( \frac{z h''(z)}{\alpha h'(z)}\right) <\frac{1}{2} ~\mbox{ for $z\in \ID$,} $$ where $0<\alpha\leq1$. The aim of this article is to show that this family has several elegant...

The main aim of this paper is to investigate properties of certain class of logharmonic mappings. Initially, we establish the argument principle of sense-preserving mappings \(F=h\overline{g}+H\overline{G}\), where h, g, H and G are analytic functions. As applications, we obtain a direct extension of Rouché’s theorem, open mapping theorem and minim...

In this paper, we establish several new versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing φ={φn(r)}n=0∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargi...

In this paper, we investigate properties of harmonic entire mappings. First, we study lower order of harmonic entire mappings. For a harmonic entire mappings f of order \(\rho \), we also discuss the case \(\rho =\infty \) by introducing the quantities \(\rho (k)\), \(\tau (k)\), \(\lambda (k)\), \(\omega (k)\), and also the case \(\rho =0\) by stu...

Let $\mathcal{H}_0$ denote the set of all sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\ID$, normalized with $h(0)=g(0)=g'(0)=0$ and $h'(0)=1$. In this paper, we investigate some properties of certain subclasses of $\mathcal{H}_0$, including inclusion relations and stability analysis by precise examples, coefficient bound...

The purpose of this paper is to study the Schwarz-Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: $\Delta(\Delta f)=g$, where $g:$ $\overline{\ID}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\ID}$ denotes the closure of the unit disk $\ID$ in the complex plane $\mathbb...

In this paper, we mainly use the Fréchet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space, namely, mappings from Un to U (resp. Un to Un). In addition, we discuss whether or not there is a constant term in f, and we obtain two redefined Bohr inequalities in Un. Finally, we redefine the Bohr inequality...

The purpose of the note is to explore the invariance properties of sphericalization and flattening in quasi-metric spaces. We show that the Ahlfors regular and doubling property of quasi-metric spaces are preserved under sphericalization and flattening transformations. As an application, we give an improvement of a recent result in [21].

In this article, we first establish an asymptotically sharp Koebe type covering theorem for harmonic K-quasiconformal mappings. Then we use it to obtain an asymptotically Koebe type distortion theorem, a coefficients estimate, a Lipschitz characteristic and a linear measure distortion theorem of harmonic K-quasiconformal mappings. At last, we give...

In this note, we mainly study operator-theoretic properties on the Besov space $B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates fo...

In this paper, we establish two new versions of Landau-type theorems for pluriharmonic mappings with a bounded distortion. Then using these results, we derive three Bloch-type theorems of pluriharmonic mappings, which improve the corresponding results of Chen and Gauthier.

In this article, we mainly study the weighted composition-differentiation operator on the weighted Bergman space Aα2 and the derivative Hardy space S12, which characterize complex symmetric weighted composition-differentiation operator Du,φ. Moreover, the normality and self-adjointness of Du,φ are also discussed.

We introduce definitions of pre-Schwarzian and Schwarzian derivatives for logharmonic mappings, and basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the...

Assume that \(f\) lies in the class of starlike functions of order \(\alpha\in[0,1)\), that is, which are regular and univalent for \(|z|<1\) and such that Re\(\left(\frac{zf'(z)}{f(z)}\right)>\alpha\) for \(|z|<1.\) In this paper we show that for each \(\alpha\in[0,1)\), the following sharp inequality holds: \(|f(re^{i\theta})|^{-1}\int_{0}^{r}|f'...

In this paper, we investigate properties of harmonic entire mappings. Firstly, we give the characterizations of the order and the type for a harmonic entire mapping \(f=h+\overline{g}\), respectively, and also consider the relationship between the order and the type of f, h, and g. Secondly, we investigate the harmonic mappings \(f=h+\overline{g}\)...

We consider the class of univalent log-harmonic mappings on the unit disk. Firstly, we present general idea of constructing log-harmonic Koebe mappings, log-harmonic right half-plane mappings and log-harmonic two-slits mappings and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike map...

In this note, we mainly study operator-theoretic properties on Besov space $B_{1}$ on the unit disc. This space is the minimal Mobius invariant space. Firstly, we consider the boundedness of Volterra type operators. Secondly, we prove that Volterra type operators belong to the Deddens algebra of composition operator. Thirdly, we obtain estimates fo...

In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the un...

In this paper, we introduce a concept of quasihyperbolic John spaces (with center) and provide a criteria to determine spaces to be quasihyperbolic John. As an application, we provide a simple proof to show that a John space with a Gromov hyperbolic quasihyperbolization is quasihyperbolic John, quantitatively. This gives an affirmative answer to an...

In this paper, we introduce a concept of quasihyperbolic John spaces and provide a necessary and sufficient condition for a space to be quasihyperbolic John. Using this criteria, we exhibit a simple proof to show that a John space with a Gromov hyperbolic quasihyperbolization is quasihyperbolic John, quantitatively. This answers affirmatively to an...

Let \(f = P[F]\) denote the Poisson integral of F in the unit disk \({\mathbb {D}}\) with F being absolutely continuous in the unit circle \({\mathbb {T}}\) and \({\dot{F}}\in L^{p}({\mathbb {T}})\), where \({\dot{F}}(e^{it})=\frac{d}{dt} F(e^{it})\) and \(p\ge 1\). Recently, the author in Zhu (J Geom Anal, 2020) proved that (1) if f is a harmonic...

In this paper, we investigate properties of harmonic entire mappings. Firstly, we give the characterizations of the order and the type for a harmonic entire mapping $f=h+\overline{g}$, respectively, and also consider the relationship between the order and the type of $f$, $h$, and $g$. Secondly, we investigate the harmonic mappings $f=h+\overline{g...

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 $\ID$, $w=\gamma_0$ and $\partial_{\nu}\partial_z\partial_{\overline{z}}w=\gamma$ on $\mathbb{T}$ (the unit circle), $\frac{1}{2\pi i}\int_{\mathbb{T}}w_{\zeta\overline{\zeta}}(\zeta)\frac...

This book features selected papers from the 5th International Conference on Mathematics and Computing (ICMC 2019), organized by the School of Computer Engineering, Kalinga Institute of Industrial Technology Bhubaneswar, India, on February 6 – 9, 2019. Covering recent advances in the field of mathematics, statistics and scientific computing, the boo...

This book features selected papers from the 6th International Conference on Mathematics and Computing (ICMC 2020), organized by Sikkim University, Gangtok, Sikkim, India, during September 2020. It covers recent advances in the field of mathematics, statistics, and scientific computing. The book presents innovative work by leading academics, researc...

The paper is a review of the latest results of I. R. Kayumov and S. Ponnusamy on the Bohr inequality. An exact estimate in the strong Bohr inequality is obtained and the Bohr–Rogosinski radius for a certain class of subordinations is examined. All results are exact.

In this note, we completely characterize complex symmetric weighted composition differentiation operator on the Hardy space $H^2$ with respect to the conjugation operator $C_{\lambda,\alpha}$. 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 co...

Recently, there has been a number of good deal of research on the Bohr’s phenomenon in various settings including a refined formulation of his classical version of the inequality. Among them, in Paulsen et al. (Proc Lond Math Soc 85(2):493–512, 2002) the authors considered cases in which the above functions have a multiple zero at the origin. In th...

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 limi...

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 limi...

Let $f = P[F]$ denote the Poisson integral of $F$ in the unit disk $\mathbb{D}$ with $F$ being absolutely continuous in the unit circle $\mathbb{T}$ and $\dot{F}\in L_p(0, 2\pi)$, where $\dot{F}(e^{it})=\frac{d}{dt} F(e^{it})$ and $p\geq 1$. Recently, the author in \cite{Zhu} proved that $(1)$ if $f$ is a harmonic mapping and $1\leq p< 2$, then $f_...

The main purpose of this paper is twofold. First, a class of polyharmonic mappings with positive Jacobian and being univalent in the unit disk \({\mathbb {D}}\) are constructed. Second, we state and prove several theorems about the radius of univalence of a class of polyharmonic mappings, which gives a partial generalized answer to the radius probl...

In [16], the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space 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 T p and on T p,0 spaces and compute their operator norm. We also characterize invertible...

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 $\Delta f=g$ in $\mathbb{D}$, and then establish some sharp distortion the...

Let D‾ be the closure of the unit disk D in the complex plane C and g be a continuous function in D‾. In this paper, we discuss some characterizations of elliptic mappings f satisfying the Poisson's equation Δf=g in D, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, resp...

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 $C_\phi$ to be bounded on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces and co...

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.

Assume that $f$ lies in the class of starlike functions of order $\alpha \in [0,1)$, that is, which are regular and univalent for $|z|<1$ and such that $${\rm Re} \left (\frac{zf'(z)}{f(z)} \right ) > \alpha ~\mbox{ for } |z|<1. $$ In this paper we show that for each $\alpha \in [0,1)$, the following sharp inequality holds: $$ |f(re^{i\theta})|^{-1...

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

In this article, we prove the Riesz - Fejér inequality for complex-valued harmonic functions in the harmonic Hardy space hp for all p > 1. The result is sharp for p ∈ (1,2]. Moreover, we prove two variant forms of Riesz-Fejér inequality for harmonic functions, for the special case p = 2.

The aim of this paper is to reveal some complex dynamical properties of König’s and Steffensen’s methods for entire functions. Two procedures are presented for constructing infinite entire functions simultaneously so that any given finite pairs of prescribed cycles occur when the two methods are applied, respectively. Furthermore, these functions c...

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.

In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the un...

Let f=h+g‾ be a normalized and sense-preserving convex harmonic mapping in the unit disk D. In a recent paper, Ponnusamy and Sairam Kaliraj conjectured that there is a θ∈[0,2π)such that the function h+e iθ g is convex in D. In this article, we first disprove a more flexible conjecture: “Let f=h+g‾ be a convex harmonic mapping in the disk D. Then th...

The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings f=h+g¯ in the unit disk 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 D. The results state that if h(z)=∑n=0∞anzn and g(z)...

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-harmoni...

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 $\varphi$, and the purpose is to look into two cases: when $\varphi$ is convex, or a general univalent function in $\ID$. Th...

In this paper, we introduce the study of the Bohr phenomenon for a quasi-subordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $K$-quasiconfo...

Let \(f=h+\overline{g}\) be a harmonic univalent map in the unit disk \(\mathbb {D}\), where h and g are analytic. This paper finds an improved estimate for the second coefficient of h. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25, 1984), an...

In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of K-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investi...

In this paper, we introduce the study of the Bohr phenomenon for a quasi-subordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $K$-quasiconfo...

We consider the family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. Our objective in this paper is to estimate the difference of the moduli of successive coefficients, that is $\big | |a_{n+1}|-|a_n|\big |$, for $f$ belonging to the family of $\gamma$-spirallike functions of order $\alpha$. Our...

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-sta...

In this article, we first discuss the Lipschitz characteristic and the linear measure distortion of $K$-quasiconformal harmonic mappings. Then we give some characterizations of the radial John disks with the help of Pre-Schwarzian of harmonic mappings.

Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57 (5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a mor...

The object of this paper is to study the powered Bohr radius ρ p , p ∈ (1, 2), of analytic functions f(z) =Σ k=0∞ a k z k defined on the unit disk |z| < 1 and such that |f(z)| < 1 for |z| < 1. More precisely, if M pf (r) =Σ k=0∞ |a k | p r k , then we show that M pf (r) ≤ 1 for r ≤ r p where r ρ is the powered Bohr radius for conformal automorphism...

Let $\es$ be the family of analytic and univalent functions $f$ in the unit disk $\D$ with the normalization $f(0)=f'(0)-1=0$, and let $\gamma_n(f)=\gamma_n$ denote the logarithmic coefficients of $f\in {\es}$. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the famil...

Let $\es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$. Let $F$ be the inverse of the function $f\in\es$ with the series expansion %in a disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ \sum_{n=2}^{\infty}A_nw^n$ for $|w|<1/4$. The logarithmic inverse c...

The main aim of this paper is to study the Lipschitz continuity of certain (K,K′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(K, K^{\prime })$$\end{document}-quasic...

We prove several improved versions of Bohr's inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $|g'(z)|\le|h'(z)|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. \cite{KayPon2}, for example a term related to the area of the image of the disk $D(0,r)$...

In this article, we prove the Riesz-Fejér inequality for complex-valued harmonic functions in the harmonic Hardy space h^p for all p > 1. The result is sharp for p ∈ (1, 2]. Moreover, we prove two variant forms of Riesz-Fejér inequality for harmonic functions, for the special case p = 2.

In this article, we prove the Riesz - Fej\'er inequality for complex-valued harmonic functions in the harmonic Hardy space ${\bf h}^p$ for all $p > 1$. The result is sharp for $p \in (1,2]$. Moreover, we prove two variant forms of Riesz-Fej\'er inequality for harmonic functions, for the special case $p=2$.

The aim of this paper is twofold. One of them is to introduce the class of harmonic \(\nu \)-Bloch-type mappings as a generalization of harmonic \(\nu \)-Bloch mappings and thereby we generalize some recent results of harmonic 1-Bloch-type mappings investigated recently by Efraimidis et al. (Complex Var Elliptic Equ 62:1081–1092, 2017). The other i...

The object of this paper is to study the powered Bohr radius $\rho_p$, $p \in (1,2)$, of analytic functions $f(z)=\sum_{k=0}^{\infty} a_kz^k$ and such that $|f(z)|<1$ defined on the unit disk $|z|<1$. More precisely, if $M_p^f (r)=\sum_{k=0}^\infty |a_k|^p r^k$, then we show that $M_p^f (r)\leq 1$ for $r \leq r_p$ where $r_\rho$ is the powered Bohr...

We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $\alpha$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Fi...

In this paper, we shall discuss the family of biharmonic mappings for which maximum principle holds. As a consequence of our study, we present Schwarz Lemma for the family of biharmonic mappings. Also we discuss the univalency of certain class of biharmonic mappings.

Let $f=h+\overline{g}$ be a harmonic univalent map in the unit disk $\mathbb{D}$, where $h $ and $g$ are analytic. We obtain an improved estimate for the second coefficient of $h$. This indeed is the first qualitative improvement after the appearance of the papers by Clunie and Sheil-Small in 1984, and by Sheil-Small in 1990. Also, when the sup-nor...

Let $\mathcal{S}_H^0$ denote the class of all functions $f(z)=h(z)+\overline{g(z)}=z+\sum^\infty_{n=2} a_nz^n +\overline{\sum^\infty_{n=2} b_nz^n}$ that are sense-preserving, harmonic and univalent in the open unit disk $|z|<1$. The coefficient conjecture for $\mathcal{S}_H^0$ is still \emph{open} even for $|a_2|$. The aim of this paper is to show...

In this article we obtain two sharp results concerning the analytic part of harmonic mappings \(f=h+\overline{g}\) from the class \(\mathcal {S}^0_H(\mathcal {S})\) which was recently introduced by Ponnusamy and Sairam Kaliraj. For example, we get the sharp estimate for \(|\arg h'(z)|\) in the case when \(|z| \le 1/\sqrt{2}\) and obtain the sharp r...

We determine the Bohr radius for the class of all functions f of the form f(z)=zm∑k=0∞akpzkp analytic in the unit disk |z|<1 and satisfy the condition |f(z)|≤1 for all |z|<1. In particular, our result also contains a solution to a recent conjecture of Ali et al. [9] for the Bohr radius for odd analytic functions, solved by the authors in [17]. We c...

Dorff et al. \cite{DN} formulated an open problem concerning the convolution of two right half-plane mappings, where the normalization of the harmonic mappings has been considered incorrectly. Without realizing the error, the present authors considered the open problem (see \cite[Theorem 2.2]{LiPo1} and \cite[Theorem 1.3]{LiPo2}). In this paper, we...

There has been a number of problems closely connected with the classical Bohr inequality for bounded analytic functions defined on the unit disk centered at the origin. Several extensions, generalizations and modifications of it are established by many researchers and they can be found in the literature, for example, in the multidimensional setting...

Let ${\mathcal U}(\lambda)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some $0<\lambda \leq 1$. The logarithmic coefficients $\gamma_n$ of $f$ are defined by the formula $\log(f(z)/z)=2\sum_{n=1}^\infty \gamma_nz^n$. I...

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^...

In this article, we study the weighted composition operators preserving the class $\mathcal{P}_{\alpha}$ of analytic functions subordinate to $\frac{1+\alpha z}{1-z}$ for $|\alpha|\leq 1, \alpha \neq -1$. Some of its consequences and examples for some special cases are presented. Furthermore, we discuss about the fixed points of weighted compositio...

By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $\mathcal{H}^2$, the space of Dirichlet series with square summable coefficients, for the inducing symbol $\varphi(s)=c_1+c_{q}q^{-s}$ where $q\geq 2$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator.

In this article, we study the weighted composition operators preserving the class $\mathcal{P}_{\alpha}$ of analytic functions subordinate to $\frac{1+\alpha z}{1-z}$ for $|\alpha|\leq 1, \alpha \neq -1$. Some of its consequences and examples for some special cases are presented. Furthermore, we discuss about the fixed points of weighted compositio...

We consider König’s method for finding roots of polynomials. Let Kf,n be the function defined in König’s method for polynomial f of order n (n≥2). For any given complex number λ∈C and any given set Ω≔{z1,z2,…,zk} of k (k≥2) distinct complex numbers, we present a procedure of constructing a polynomial fn,λ for which Ω is a k-cycle of Kfn,λ,n with mu...

We determine the representation theorem, distortion theorem, coefficients estimate and Bohr’s radius for log-harmonic starlike mappings of order α, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of 1-slit log-harmonic mappings. Finally, we...

We determine the Bohr radius for the class of odd functions f satisfying |f(z)|≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|f(z)|\le 1$$\end{document} for all |z|...

The main aim of this article is to establish analogues of Landau’s theorem for solutions to the $\overline{\unicode[STIX]{x2202}}$ -equation in Dirichlet-type spaces.

In this paper, we shall discuss the family of biharmonic mappings for which maximum principle holds. As a consequence of our study, we present Schwarz Lemma for the family of biharmonic mappings. Also we discuss the univalency of certain class of biharmonic mappings.

For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda $ in $\ID$. Although functions in this family are known to be univalent in $\ID$, the coefficient conjecture a...

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds: h is bounded in . h satisfies the condition in with . both h and g are bounded in . h is bounded and . We also consider the problem of determining the Bohr r...

In this article, we define discrete analogue of generalized Hardy spaces and its separable subspace on a homogenous rooted tree and study some of its properties such as completeness, inclusion relations with other spaces, separability, growth estimate for functions in these spaces and their consequences. Equivalent conditions for multiplication ope...

We determine the Bohr radius for the class of all functions $f$ of the form $f(z)=\sum_{k=1}^\infty a_{kp+m} z^{kp+m}$ analytic in the unit disk $|z|<1$ and satisfy the condition $|f(z)|\le 1$ for all $|z|<1$. In particular, our result also contains a solution to a recent conjecture of Ali, Barnard and Solynin \cite{AliBarSoly} for the Bohr radius...