arXiv subjects

Bappaditya Bhowmik

Publications and source records attributed to Bappaditya Bhowmik.

At least 19 recordsLinked to original sources

arXiv2026

Radius of starlikeness of $\mathcal{S}\ast \mathcal {S}t(\alpha)$

Let $\mathcal{S}$ be the set of all analytic univalent functions $f$ defined in the open unit disc $\mathbb{D}$, with $f(0)=0=f'(0)-1$. For $\alpha\in[0,1)$, let $\mathcal{S}t(\alpha)$ be the set of all starlike functions of order $\alpha$ in $\mathcal{S}$. In this article, by applying duality technique we obtain the radius of a disc that is mapped onto a starlike domain with respect to the origin by the functions in the set $\mathcal{S}\ast \mathcal {S}t(\alpha):=\{f\ast g :f\in\mathcal{S},~g\in\mathcal{S}t(\alpha)\}$. Here, `$\ast$' denotes the convolution (or Hadamard product) of two analytic functions in $\mathbb{D}$.

math.CV↗

arXiv2026

Radius of convexity of certain classes of functions defined by convolution

Let $\mathcal{S}$ be the class of analytic univalent functions defined in the open unit disc $\mathbb{D}$ of the complex plane with the normalizations $f(0)=0$ and $f'(0)=1$. For $A\in (1,2]$, let $Co(A)$ denote the class of concave univalent functions defined in $\mathbb{D}$ with the opening angle $\pi A$ at infinity. In this article, by applying certain convolution techniques, we investigate the radius of convexity for the class $Co(A)\ast\mathcal{S}t(1/2)$, where $\mathcal{S}t(1/2)\subsetneq \mathcal{S}$ denotes the class of starlike functions of order $1/2$. Furthermore, we establish that the radius of convexity of the class $\mathcal{S}\ast\mathcal{S}t(1/2)$ is at least $0.19191$ (approximately). Here, `$\ast$' denotes the convolution (or Hadamard product) of two classes of functions.

math.CV↗

arXiv2025

Gehring-Hayman Inequality for Meromorphic Univalent Mappings

Let $f$ be a meromorphic univalent function on the open unit disk having a simple pole at $p\in (0,1)$ that extends continuously to the left half $\IT^{-}$ of the unit circle. In this article, we prove that the ratio of the length of the image of the vertical diameter $\IA$ of the unit disk to the length of the image of $\IT^{-}$ under the mapping $f$ is bounded by a constant depending only on $p.$ Next, we extend this result by considering any hyperbolic geodesic and any Jordan curve in $\D$ sharing the same endpoints. These results extend the classical Gehring-Hayman inequality to meromorphic univalent functions and also prove a conjecture posed by Bhowmik and Maity [Bull. Sci. Math. \textbf{199} (2025), \# 103583].

math.CV↗

arXiv2024

Length Distortion Of Curves Under Meromorphic Univalent Mappings

Let $f$ be a conformal (analytic and univalent) map defined on the open unit disk $\D$ of the complex plane $\IC$ that is continuous on the semi-circle $\partial \D^{+}=\{z\in\IC:|z|=1, {\rm{Im}}\,z>0\}$. The existence of a uniform upper bound for the ratio of the length of the image of the horizontal diameter $(-1,1)$ to the length of the image of $\partial \D^{+}$ under $f$ was proved by Gehring and Hayman. In this article, at first, we generalize this result by introducing a simple pole for $f$ in $\D$ and considering the ratio of the length of the image of the vertical diameter $I=\{z: {\rm{Re}}\,z=0; ~|{\rm{Im}}\,z|<1\}$ to the length of the image of the semi-circle $C'=\{z: |z|=1;~ {\rm{Re}}\,z<0\}$ under such $f$. Finally, we further generalize this result by replacing the vertical diameter $I$ with a hyperbolic geodesic symmetric with respect to the real line, and by replacing $C'$ with the corresponding arc of the unit circle passing through the point $-1$.

math.CV↗

arXiv2024

On the radius of concavity for certain classes of functions

Let $\mathcal{A}$ denote the class of all analytic functions $f$ defined in the open unit disc $\mathbb{D}$ with the normalization $f(0)=0=f'(0)-1$ and let $P'$ be the class of functions $f\in\mathcal{A}$ such that ${\rm{Re}}\,f'(z)>0$, $z\in\mathbb{D}$. In this article, we obtain radii of concavity of $P'$ and for the class $P'$ with the fixed second coefficient. After that, we consider linearly invariant family of functions, along with the class of starlike functions of order $1/2$ and investigate their radii of concavity. Next, we obtain a lower bound of radius of concavity for the class of functions $\mathcal{U}_0(λ)=~\{f\in\mathcal{U}(λ) : f''(0)=0\}$, where $$ \mathcal{U}(λ)=\left\{f\in\mathcal{A} : \left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<λ,~z\in \mathbb{D}\right\},\quad λ\in (0,1]. $$ We also investigate the meromorphic analogue of the class $\mathcal{U}(λ)$ and compute its radius of concavity.

math.CV↗

arXiv2024

Landau-type Theorem and Bloch constant for elliptic harmonic mappings

In this article, we obtain certain estimates for the Taylor coefficients of $(K,K')$-elliptic harmonic mappings and using these estimates, we prove a Landau-type theorem for these mappings. We also derive Bloch constant for the class of $(K, K')$-elliptic harmonic mappings.

math.CV↗

arXiv2020

On the Bohr phenomenon for complex valued and vector valued functions

We explore the Bohr inequality involving the Fourier transforms of complex valued integrable and square integrable functions defined on a second countable compact topological group. We also investigate the connection of the Bohr phenomenon with a modulus of convexity of the space of bounded linear operators defined on a complex Hilbert space.

math.FA↗

arXiv2020

Bohr phenomenon for operator valued functions with fixed initial coefficient

The purpose of this article is to study Bohr inequalities involving the absolute values of the coefficients of an operator valued function. To be more specific, we establish an operator valued analogue of a classical result regarding the Bohr phenomenon for scalar valued functions with fixed initial coefficient. Apart from that, operator valued versions of other related and well known results are obtained.

math.CV↗

arXiv2019

A note on the Bohr inequality

This article focuses on the Bohr radius problem for the derivatives of analytic functions, along with a technique of establishing Bohr inequalities in classical and generalized settings.

math.CV↗

arXiv2019

Bohr phenomenon for operator valued functions

In this article we establish Bohr inequalities for operator valued functions, which can be viewed as the analogues of a couple of interesting results from scalar valued settings. Some results of this paper are motivated by the classical flavor of Bohr inequality, while the others are based on a generalized concept of the Bohr radius problem.

math.CV↗

arXiv2019

Bohr phenomenon for locally univalent functions and logarithmic power series

In this article we prove Bohr inequalities for sense-preserving $K$-quasiconformal harmonic mappings defined in $\mathbb{D}$ and obtain the corresponding results for sense-preserving harmonic mappings by letting $K\to\infty$. One of the results includes the sharpened version of a theorem by Kayumov $\textit{et. al.}$ ($\textit{Math. Nachr.}$, 291 (2018), no. 11--12, 1757--1768). In addition Bohr inequalities have been established for uniformly locally univalent holomorphic functions, and for $\log(f(z)/z)$ where $f$ is univalent or inverse of a univalent function.

math.CV↗

arXiv2018

Loewner chain and quasiconformal extension of some classes of univalent functions

In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the extended complex plane. Some of these extensions have been obtained by constructing suitable Loewner chains and others have been obtained by applying a well-known result.

math.CV↗

arXiv2017

On the Taylor coefficients of a subclass of meromorphic univalent functions

Let $\mathcal{V}_p(λ)$ be the collection of all functions $f$ defined in the unit disc $\ID$ having a simple pole at $z=p$ where $0<p<1$ and analytic in $\ID\setminus\{p\}$ with $f(0)=0=f'(0)-1$ and satisfying the differential inequality $|(z/f(z))^2 f'(z)-1|< λ$ for $z\in \ID$, $0<λ\leq 1$. Each $f\in\mathcal{V}_p(λ)$ has the following Taylor expansion: $$ f(z)=z+\sum_{n=2}^{\infty}a_n(f) z^n, \quad |z|<p. $$ In \cite{BF-3}, we conjectured that $$ |a_n(f)|\leq \frac{1-(λp^2)^n}{p^{n-1}(1-λp^2)}\quad \mbox{for}\quad n\geq3. $$ In the present article, we first obtain a representation formula for functions in the class $\mathcal{V}_p(λ)$. Using this representation, we prove the aforementioned conjecture for $n=3,4,5$ whenever $p$ belongs to certain subintervals of $(0,1)$. Also we determine non sharp bounds for $|a_n(f)|,\,n\geq 3$ and for $|a_{n+1}(f)-a_n(f)/p|,\,n\geq 2$.

math.CV↗

arXiv2017

Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension

Let $Σ_k(p)$ be the class of univalent meromorphic functions defined on $\mathbb{D}$ with $k$-quasiconformal extension to the extended complex plane $\widehat{\mathbb{C}}$, where $0\leq k < 1$. Let $Σ_k^0(p)$ be the class of functions $f \in Σ_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb{D}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $Σ_k^0(p)$. As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.

math.CV↗

arXiv2017

Sufficient conditions for univalence and study of a class of meromorphic univalent functions

In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(λ)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(λ)\subsetneq \mathcal{V}_{p}(λ)$, where $\mathcal{U}_{p}(λ)$ was introduced and studied in \cite{BF-1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(λ)$ and end the article with a coefficient conjecture.

math.CV↗

arXiv2017

On some results for meromorphic univalent functions having quasiconformal extension

We consider the class $Σ(p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in[0,1)$ with residue 1. Let $Σ_k(p)$ be the class of functions in $Σ(p)$ which have $k$-quasiconformal extension to the extended complex plane $\sphere$ %with $q=\frac{1+k}{1-k}$ where $0\leq k < 1$. We first give a representation formula for functions in this class and using this formula we derive an asymptotic estimate of the Laurent coefficients for the functions in the class $Σ_k(p)$. Thereafter we give a sufficient condition for functions in $Σ(p)$ to belong in the class $Σ_k(p).$ Finally we obtain a sharp distortion result for functions in $Σ(p)$ and as a consequence, we get a distortion estimate for functions in $Σ_k(p).$

math.CV↗

arXiv2017

Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions

Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic in $\ID\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$. In this article, we first prove a sufficient condition for univalency for functions in $\mathcal{A}(p)$. Thereafter, we consider the class denoted by $Σ(p)$ that consists of functions $f \in \mathcal{A}(p)$ that are univalent in $\ID$. We obtain the exact value for $\ds\max_ {f\in Σ(p)}Δ(r,z/f)$, where the Dirichlet integral $Δ(r,z/f)$ is given by $$ Δ(r,z/f)=\ds\iint_{|z|<r} |\left(z/f(z)\right)'|^2 \,dx\, dy, \quad(z=x+iy),~0<r\leq 1. $$ We also obtain a sharp estimate for $Δ(r,z/f)$ whenever $f$ belongs to certain subclasses of $Σ(p)$. Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.

math.CV↗

arXiv2015

Regions of variability for a class of analytic and locally univalent functions defined by subordination

In this article we consider a family $\mathcal{C}(A, B)$ of analytic and locally univalent functions on the open unit disc $\ID=\{z :|z|<1\}$ in the complex plane that properly contains the well-known Janowski class of convex univalent functions. In this article, we determine the exact set of variability of $\log(f'(z_0))$ with fixed $z_0 \in \ID$ and $f"(0)$ whenever $f$ varies over the class $\mathcal{C}(A, B)$.

math.CV↗