arXiv subjects
Publications and source records attributed to Saminathan Ponnusamy.
This paper focuses on properties of \partial-biLipschitz mappings which were recently introduced by Bulter. We establish several characterizations for the class of \partial-biLipschitz mappings between domains in quasiconvex metric spaces. As an application, we show that a locally quasisymmetric equivalence between uniform metric spaces is quasimöbius, quantitatively.
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 for the essential norm of Volterra type operators. Finally, we give a complete characterization of spectrum of Volterra type operators.
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 open question posed by Heinonen (Rev.~Math.~Iber, 1989), which has been studied by Gehring et al. (Math.~Scand, 1989).
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 open question proposed by Heinonen (Rev.~Math.~Iber, 1989), which was studied by Gehring et al. (Math.~Scand, 1989). As a tool, we study its connection between several geometric conditions.
Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $λ$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $λ$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily spirallike functions and uniform boundedness of hereditarily strongly starlike functions. Some coefficient conditions are given for hereditary strong starlikeness and hereditary spirallikeness. As a simple application, we consider a special form of harmonic functions.
Assume that $f$ lies in the class of starlike functions of order $α\in [0,1)$, that is, which are regular and univalent for $|z|<1$ and such that $${\rm Re} \left (\frac{zf'(z)}{f(z)} \right ) > α~\mbox{ for } |z|<1. $$ In this paper we show that for each $α\in [0,1)$, the following sharp inequality holds: $$ |f(re^{iθ})|^{-1} \int_{0}^{r}|f'(ue^{iθ})| du \leq \frac{Γ(\frac12)Γ(2-α)}{Γ(\frac32-α)} ~\mbox {for every $r<1$ and $θ$}. $$ This settles the conjecture of Hall (1980).
This paper mainly uses the nonnegative continuous function $\{ζ_n(r)\}_{n=0}^{\infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $\real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of the alternating series $A_f(r)$ with analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ in $|z|<1$. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series $M_f(r)$ and the odd and even bits of $f(z)$ are also established. We will prove that most of results are sharp.
In this paper, firstly we prove two refined Bohr-type inequalities associated with area for bounded analytic functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in the unit disk. Later, we establish the Bohr-type operator on analytic functions and sections.
In this paper, we give a new generalization of the Bohr inequality in refined form both for bounded analytic functions, and for sense-preserving harmonic functions with analytic part being bounded.
The concept of Bohr radius for the class of bounded analytic functions was introduced by Harald Bohr in 1914. His initial result received greater interest and was sharpened-refined-generalized by several mathematicians in various settings--which is now called Bohr phenomenon. Various generalization of Bohr's classical theorem is now an active area of research and has been a source of investigation in numerous other function spaces and including holomorphic functions of several complex variables. Recently, a new generalization of Bohr's ideas was introduced and investigated by Kayumov et al.. In this note, we investigate and refine generalized Bohr's inequality for the class of quasi-subordinations.
In this article, we consider the family of functions $f$ analytic in the unit disk $|z|<1$ with the normalization $f(0)=0=f'(0)-1$ and satisfying the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. We denote this class by $\mathcal{U}(λ)$ and we are interested in the relations between $\mathcal{U}(λ)$ and other families of functions holomorphic or harmonic in the unit disk. Our first example in this direction is the family of functions convex in one direction. Then we are concerned with the subordinates to the function $1/((1-z)(1-λz))$. We prove that not all functions $f(z)/z$ $(f \in \mathcal{U}(λ))$ belong to this family. This disproves an assertion from \cite{OPW}. Further, we disprove a related coefficient conjecture for $\mathcal{U}(λ)$. We consider the intersection of the class of the above subordinates and $\mathcal{U}(λ)$ concerning the boundary behaviour of its functions. At last, with the help of functions from $\mathcal{U}(λ)$, we construct functions harmonic and close-to-convex in the unit disk.
The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Cesáro operator.
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 $f=h+\overline{g}$ such that $f^{(n_p)}=h^{(n_p)}+\overline{g^{(n_p)}}$ are univalent in the unit disk, where $\{n_p\}_{p=1}^{\infty}$ be a strictly increasing sequence of nonnegative integers. In terms of the sequence $\{n_p\}_{p=1}^{\infty}$, we derive several necessary conditions for these mappings to be entire and also establish an upper bound for the order of these mappings.
Based on improving the classical Bohr inequality, we get in this paper some refined versions for a quasi-subordination family of functions, one of which is key to build our results. By means of these investigations, for a family of harmonic mappings defined in the unit disk $\D$, we establish an improved Bohr inequality with refined Bohr radius under particular conditions. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. % in a logical way. Here the family of harmonic mappings have the form $f=h+\overline{g}$, where $g(0)=0$, the analytic part $h$ is bounded by 1 and that $|g'(z)|\leq k|h'(z)|$ in $\D$ and for some $k\in[0,1]$.
In this paper, we establish Lipschitz conditions for the norm of holomorphic mappings between the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ and $X,$ a complex normed space. This extends the work of Djordjević and Pavlović.
Recently the present authors established refined versions of Bohr's inequality in the case of bounded analytic functions. In this article, we state and prove a generalization of these results in a reformulated "distance form" version and thereby we extend the refined versions of the Bohr inequality for the class of the quasi-subordinations which contains both the classes of majorization and subordination as special cases. As a consequence, we obtain several new results.
In this paper, we first establish a version of multidimensional analogues of the refined Bohr's inequality. Then we establish two versions of multidimensional analogues of improved Bohr's inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr's inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp.
In this paper, we first determine Bohr's inequality for the class of harmonic mappings $f=h+\overline{g}$ in the unit disk $\ID$, where either both $h(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ and $g(z)=\sum_{n=0}^{\infty}b_{pn+m}z^{pn+m}$ are analytic and bounded in $\ID$, or satisfies the condition $|g'(z)|\leq d|h'(z)|$ in $\ID\backslash \{0\}$ for some $d\in [0,1]$ and $h$ is bounded. In particular, we obtain Bohr's inequality for the class of harmonic $p$-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp.