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Swadesh Kumar Sahoo

Publications and source records attributed to Swadesh Kumar Sahoo.

At least 19 recordsLinked to original sources
arXiv2025

Modified Distance Ratio Metrics via Domain Diameter and their geometric implications

Let $D\subsetneq\mathbb{R}^n,~n\ge 2$, be a domain. In this manuscript, a new version of the Vuorinen's distance ratio metric $j_D$ [{\tt J. Analyse Math.} {\bf 45} (1985), 69--115], denoted by $ζ_D$, and a version of Gehring-Osgood's distance ratio metric $j_D'$ [{\tt J. Analyse Math.} {\bf 36} (1979), 50--74], denoted by $ζ_D'$, are introduced to better understand how quasihyperbolic geometry interacts with bounded uniform domains in $\mathbb{R}^n$. We show that the metric $m_D$, introduced in [{\tt arXiv:2505.10964v2}], is the inner metric of $ζ_D$ and explore their relations to several well-known hyperbolic-type metrics. The paper includes ball inclusion properties of these metrics associated with the metric $m_D$ and other hyperbolic-type metrics. The distortion properties of them are also considered under several important classes of mappings. Furthermore, as an application, we demonstrate that uniform domains can be characterized in terms of metrics $ζ_D$ and $m_D$.

math.MG↗
arXiv2025

A naive generalization of the hyperbolic and the quasihyperbolic metrics

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to this limitation. The proposed metric coincides with the hyperbolic metric on balls and half-spaces, and, quite unexpectedly, agrees with the quasihyperbolic metric in unbounded domains. We compute the density of this metric in several classical domains and discuss aspects of its curvature. Furthermore, we establish characterizations of uniform domains and John disks in terms of the newly defined metric. In addition, we investigate several geometric properties of the metric, including the existence of geodesics and the minimal length of non-trivial closed curves in multiply connected domains.

math.MG↗
arXiv2023

Univalence of horizontal shear of Cesàro type transforms

This manuscript investigates the classical problem of determining conditions on the parameters $α,β\in \mathbb{C}$ for which the integral transform $$C_{αβ}[φ](z):=\int_{0}^{z} \bigg(\frac{φ(ζ)}{ζ(1-ζ)^β}\bigg)^α\,dζ $$ is also univalent in the unit disk, where $φ$ is a normalized univalent function. Additionally, whenever $φ$ belongs to some subclasses of the class of univalent functions, the univalence features of the harmonic mappings corresponding to $C_{αβ}[φ]$ and its rotations are derived. As applications to our primary findings, a few non-trivial univalent harmonic mappings are also provided. The primary tools employed in this manuscript are Becker's univalence criteria and the shear construction developed by Clunie and Sheil-Small.

math.CV↗
arXiv2021

Continuity and bi-Lipschitz properties of the Hurwitz and its invariant metrics

This paper attempts to study the continuity of the Hurwitz metric in arbitrary proper subdomains of the complex plane and to introduce a new invariant metric bi-Lipschitz equivalent to the Hurwitz metric in hyperbolic domains. The lower semi-continuity and other basic properties of this invariant metric are also presented.

math.CV↗
arXiv2020

Properties of $β$-Cesàro operators on $α$-Bloch space

For each $ α> 0 $, the $α$-Bloch space is consisting of all analytic functions $f$ on the unit disk satisfying $ \sup_{|z|<1} (1-|z|^2)^α|f'(z)| < + \infty.$ In this paper, we consider the following complex integral operator, namely the $β$-Cesàro operator \begin{equation} C_β(f)(z)=\int_{0}^{z}\frac{f(w)}{w(1-w)^β}dw \nonumber \end{equation} and its generalization, acting from the $α$-Bloch space to itself, where $f(0)=0$ and $β\in\mathbb{R}$. We investigate the boundedness and compactness of the $β$-Cesàro operators and their generalization. Also we calculate the essential norm and spectrum of these operators.

math.FA↗
arXiv2020

Nehari's univalence criteria, pre-Schwarzian derivative and applications

In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain.

math.CV↗
arXiv2020

Carathéodory density of the Hurwitz metric on plane domains

It is well-known that the Carathéodory metric is a natural generalization of the Poincaré metric, namely, the hyperbolic metric of the unit disk. In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the hyperbolic metric when the domains are simply connected. In this paper, we define a new metric which generalizes the Hurwitz metric in the sense of Carathéodory. Our main focus is to study its various basic properties in connection with the Hurwitz metric.

math.CV↗
arXiv2020

A generalized Hurwitz metric

In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincaré's hyperbolic metric when the domains are simply connected. In this paper, we provide an alternate definition of the Hurwitz metric through which we could define a generalized Hurwitz metric in arbitrary subdomains of the complex plane. This paper mainly highlights various important properties of the Hurwitz metric and the generalized metric including the situations where they coincide with each other.

math.CV↗
arXiv2019

Successive coefficients for spirallike and related functions

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 $γ$-spirallike functions of order $α$. Our particular results include the case of starlike and convex functions of order $α$ and other related class of functions.

math.CV↗
arXiv2017

Meromorphic functions with small Schwarzian derivative

We consider the family of all meromorphic functions $f$ of the form $$ f(z)=\frac{1}{z}+b_0+b_1z+b_2z^2+\cdots $$ analytic and locally univalent in the puncture disk $\mathbb{D}_0:=\{z\in\mathbb{C}:\,0<|z|<1\}$. Our first objective in this paper is to find a sufficient condition for $f$ to be meromorphically convex of order $α$, $0\le α<1$, in terms of the fact that the absolute value of the well-known Schwarzian derivative $S_f (z)$ of $f$ is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions $g$ of the form $g(z)=z+a_2z^2+a_3z^3+\cdots$ analytic and locally univalent in the open unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\,|z|<1\}$, and show that $g$ is belonging to a family of functions convex in one direction if $|S_g(z)|$ is bounded above by a small positive constant depending on the second coefficient $a_2$. In particular, we show that such functions $g$ are also contained in the starlike and close-to-convex family.

math.CV↗
arXiv2017

Interpolation on Gauss hypergeometric functions with an application

In this paper, we use some standard numerical techniques to approximate the hypergeometric function $$ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots $$ for a range of parameter triples $(a,b,c)$ on the interval $0<x<1$. Some of the familiar hypergeometric functional identities and asymptotic behavior of the hypergeometric function at $x=1$ play crucial roles in deriving the formula for such approximations. We also focus on error analysis of the numerical approximations leading to monotone properties of quotient of gamma functions in parameter triples $(a,b,c)$. Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.

math.NA↗
arXiv2017

Mapping properties of a scale invariant Cassinian metric and a Gromov hyperbolic metric

In this paper, we consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. The quasi-invariance properties of both the metrics under quasiconformal maps are also established.

math.MG↗
arXiv2017

A Gromov hyperbolic metric vs the hyperbolic and other related metrics

We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls.

math.MG↗
arXiv2017

Geometric properties of $φ$-uniform domains

We consider proper subdomains $G$ of $\mathbb{R}^n$ and their images $G'=f(G)$ under quasiconformal mappings $f$ of $\mathbb{R}^n$. We compare the distance ratio metrics of $G$ and $G'$; as an application we show that $φ$-uniform domains are preserved under quasiconformal mappings of $\mathbb{R}^n$. A sufficient condition for $φ$-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If $G\subset\mathbb{R}^n$ is $ϕ$-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar $ϕ$-uniform domain whose complement is not $ψ$-uniform for any $ψ$.

math.MG↗
arXiv2016

On coefficient functionals associated with the Zalcman conjecture

We consider certain subfamilies, of the family of univalent functions in the open unit disk, defined by means of sufficient coefficient conditions for univalency. This article is devoted to studying the problem of the well-known conjecture of Zalcman consisting of a generalized coefficient functional, the so-called generalized Zalcman conjecture problem, for functions belonging to those subfamilies. We estimate the bounds associated with the generalized coefficient functional and show that the estimates are sharp.

math.CV↗
arXiv2016

Radius of convexity of partial sums of odd functions in the close-to-convex family

We consider the class of all analytic and locally univalent functions $f$ of the form $f(z)=z+\sum_{n=2}^\infty a_{2n-1} z^{2n-1}$, $|z|<1$, satisfying the condition $$ {\rm Re}\,\left(1+\frac{zf^{\prime\prime}(z)}{f^\prime (z)}\right)>-\frac{1}{2}. $$ We show that every section $s_{2n-1}(z)=z+\sum_{k=2}^na_{2k-1}z^{2k-1}$, of $f$, is convex in the disk $|z|<\sqrt{2}/3$. We also prove that the radius $\sqrt{2}/3$ is best possible, i.e. the number $\sqrt{2}/3$ cannot be replaced by a larger one.

math.CV↗