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Raju Biswas

Publications and source records attributed to Raju Biswas.

At least 19 recordsLinked to original sources
arXiv2026

Landau-type theorems for $K$-quasiregular harmonic mappings

In this paper, our aim is to establish several sharp and improved Landau-type theorems for $K$-quasiregular harmonic mappings $f=h+\overline{g}$ in the unit disk $\Bbb{D} = \{z\in\Bbb{C}: |z|<1\}$. Under various boundedness assumptions on the analytic part $h$ or its derivative, we obtain explicit univalence radii and corresponding schlicht disk radii that significantly improve upon existing estimates in the literature. We also establish new Landau-type theorems under novel hypotheses. We provide examples to illustrate our results, and comprehensive numerical tables present quantitative values of the radii for various parameter choices, demonstrating the effectiveness of our results.

math.CV↗
arXiv2026

Geometric analysis of a class of harmonic mappings defined by a differential inequality

In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the differential inequality \[ \text{Re}\left((1-α)h'(z) + αz h''(z)\right) > -M + \left|(1-α)g'(z) + αz g''(z)\right|\quad\text{for}\quad z\in\Bbb{D}, \] where $M > 0$, $α\in (0,1]$, and $g'(0) = 0$. This class extends the harmonic analogue of functions with positive real part and offers a unified framework for analyzing their geometric characteristics. We obtain sharp coefficient bounds for both the analytic and co-analytic parts, establish sharp growth bounds, and determine the radii of univalency, starlikeness, and convexity. Furthermore, we show that $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ is closed under convex combinations, and under suitable restrictions on the parameters, it is also closed under convolution. Our findings generalize and extend several known results in the theory of harmonic mappings.

math.CV↗
arXiv2026

Bohr phenomenon for certain integral operators and transforms in complex Banach spaces

In this paper, we investigate several Bohr radii associated with the Cesáro operator, Bernardi integral operator, $β$-Cesáro operator, and discrete Fourier transform, all defined on a set of holomorphic mappings from the unit ball of a complex Banach space into the closure of the unit polydisc $\mathbb{D}^n$ within the space $\mathbb{C}^n$.

math.CV↗
arXiv2026

Pre-Schwarzian and Schwarzian norm estimates for certain classes of analytic and harmonic mappings

Let $\mathcal{A}$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $f(0)=f'(0)-1=0$. In this paper, we introduce a new subclass $\mathcal{C}_θ(γ)$ of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation \[ \textrm{Re}\left(e^{iθ}\left(1+\frac{zf''(z)}{f'(z)}\right)\right)<\left(1+\fracγ{2}\right)\cosθ,~ z\in\mathbb{D},~ γ>0, ~\text{and}~|θ|<\fracπ{2},\] and investigate the Schwarzian derivative and Schwarzian norm for functions $f$ belonging to the class $\mathcal{C}_θ(γ)$. We establish sharp estimates for the Schwarzian norm $\|S_f\|$ of functions $f$ in the class $\mathcal{C}_θ(γ)$ and derive univalence criteria using both pre-Schwarzian and Schwarzian norm estimates. We also introduce a corresponding harmonic class $\mathcal{HC}_θ(γ)$ consisting of mappings $f = h+\overline{g}$ with $h\in\mathcal{C}_θ(γ)$ and dilatation $ω=g'/h'\in\mathrm{Aut}(\mathbb{D})$. For this harmonic class, we derive bounds for both the pre-Schwarzian and Schwarzian norms, including sharp results in special cases.

math.CV↗
arXiv2026

Bohr phenomenon for analytic and harmonic mappings on shifted disks

The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on the shifted disks \[ Ω_γ=\left\{z\in\mathbb{C}:\left|z+\fracγ{1-γ}\right|<\frac{1}{1-γ}\right\}\quad\text{for}\quadγ\in[0,1).\]

math.CV↗
arXiv2025

Bohr inequalities for holomorphic mappings in higher-dimensional complex Banach spaces

In this paper, we investigates the Bohr phenomenon for holomorphic mappings $F$ from the unit ball $\mathbb{B}_X$ of a complex Banach space $X$ into the closure of the unit polydisc $\mathbb{D}^m$ within the space $\mathbb{C}^m$. First, we prove an improved Bohr inequality involving the squared norms of the mapping and its homogeneous expansions. Second, we derive a refined Bohr inequality that incorporates a combination of the coefficient norms and their squares. Finally, we obtain a refined Bohr inequality for compositions $F\circ ν$, where $ν$ is a Schwarz mapping with a zero of order $k$ at the origin. For each result, we demonstrate that the derived Bohr radius is sharp.

math.CV↗
arXiv2025

On the pre-Schwarzian and Schwarzian derivatives of log-harmonic mappings

In this paper, we introduce definitions of the pre-Schwarzian and the Schwarzian derivatives for any locally univalent log-harmonic mappings defined in the unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$. We explore the properties and applications of these concepts in the context of geometric function theory, and we also establish a necessary and sufficient condition for a non-vanishing log-harmonic mapping having a finite pre-Schwarzian norm. Additionally, we establish a relationship between the pre-Schwarzian norm of a non-vanishing log-harmonic mapping and that of a certain analytic function in $\mathbb{D}$.

math.CV↗
arXiv2025

A unified framework for Bohr-type inequalities using multiple Schwarz functions

This paper introduces a unified framework for Bohr-type inequalities by incorporating multiple Schwarz functions into the majorant series for $K$-quasiconformal harmonic mappings in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$. In this study, we establish several improved and refined versions of the Bohr inequality that generalize and interconnect numerous known results. Our approach not only systematically recovers the existing theorems as special cases but also generates new results that are inaccessible through single-function methods.

math.CV↗
arXiv2025

On certain subclasses of analytic and harmonic mappings

Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{D}_{\mathcal{H}}^0(α, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-α)h'(z)+αzh''(z)-1+α\right|\leq M+\left|(1-α)g'(z)+αzg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $α\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(α, M)$, such as the coefficient bounds, growth estimates, starlikeness and some other properties. Furthermore, we obtain the sharp bound of the second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions $f\in\mathcal{P}(M)$ in $\mathbb{D}$ satisfying the condition $\text{Re}\left(zf''(z)\right)>-M$ for $0<M\leq 1/\log4$ and $z\in\mathbb{D}$.

math.CV↗
arXiv2025

Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes

In this paper, we establish the sharp estimates of the pre-Schwarzian norm of functions $f$ in the Ma-Minda type starlike and convex classes $\mathcal{S}^*(φ)$ and $\mathcal{C}(φ)$, respectively, whenever $φ(z)=3/\left(3+(α-3)z-αz^2\right)$ with $-3<α\leq 1$, $φ(z)=(1+z)(1-s z)$ with $-1/3\leq s\leq 1/3$ and $φ(z)=1+z/\left((1-z) (1+αz)\right)$ with $0\leq α\leq 1/2$.

math.CV↗
arXiv2025

An estimation of the pre-Schwarzian norm for certain classes of analytic functions

The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions $f$ in the class $\mathcal{S}^*(φ)$ and $\mathcal{C}(φ)$ when $φ(z)=1/(1-z)^s$ with $0<s\leq 1$ and $φ(z)=(1+sz)^2$ with $0<s\leq 1/\sqrt{2}$, where $\mathcal{S}^*(φ)$ and $\mathcal{C}(φ)$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $φ$, respectively.

math.CV↗
arXiv2025

The Bohr's Phenomenon involving multiple Schwarz functions

The primary objective of this paper is to establish several sharp versions of Bohr inequalities for bounded analytic functions in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$ involving multiple Schwarz functions. Moreover, we obtain an improved version of the classical Rogosinski inequality for analytic functions in $\mathbb{D}$.

math.CV↗
arXiv2025

On meromorphic solutions of certain Fermat-type difference and analogues equations concerning open problems

In this paper, we have found that some certain Fermat-type shift and difference equations have the meromorphic solutions generated by Riccati type functions. Also we have solved the open problems posed by Liu and Yang (A note on meromorphic solutions of Fermat types equations, An. Stiint. Univ. Al. I. Cuza Lasi Mat. (N. S.), 62(2)(1), 317-325 (2016)). We have fortified the claims by some examples.

math.CV↗
arXiv2025

Solutions of systems of certain Fermat-type PDDEs

The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex variables. These results represent some refinements and generalizations of the earlier findings, especially the results due to Xu {\it et al.} (J. Math. Anal. Appl. 483(2) (2020)). We provide some examples to support the results.

math.CV↗
arXiv2024

Solutions for certain Fermat-type PDDEs concerning an open problem of Xu and Wang

The objective of this study is to ascertain the existence and forms of the finite order meromorphic and entire functions of several complex variables satisfying some certain Fermat-type partial differential-difference equations by considering the more general forms of the PDDEs in an open problem on $\mathbb{C}^2$ due to Xu and Wang (Notes on the existence of entire solutions for several partial differential-difference equations, Bull. Iran. Math. Soc., 47, 1477-1489 (2020)). We provide examples to illustrate the results.

math.CV↗
arXiv2024

Entire functions of several complex variables satisfying certain Fermat-type PDDEs

In this paper, we solve certain Fermat-type partial differential-difference equations for finite order entire functions of several complex variables. These results are significant generalizations of some earlier findings, especially those of Haldar and Ahamed (Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $\mathbb{C}^2$, Anal. Math. Phys., 12 (2022)). In addition, the results improve the previous results from the situation with two complex variables to the situation with several complex variables. To support our results, we have included several examples.

math.CV↗
arXiv2024

A geometric investigation of a certain subclass of univalent functions

Let $\mathcal{H}$ be the space of all functions that are analytic in $\mathbb{D}$. Let $\mathcal{A}$ denote the family of all functions $f\in\mathcal{H}$ and normalized by the conditions $f(0)=0=f'(0)-1$. Obradović and Ponnusamy have introduced the class $\mathcal{M}(λ)$ such that the functions in $\mathcal{M}(λ)$ are univalent in $\mathbb{D}$ whenever $0<λ\leq 1$. In this paper, we address a radius property of the class $\mathcal{M}(λ)$ and a number of associated results pertaining to $\mathcal{M}$. The main objective of this paper is to examine the largest disks with sharp radius for which the functions $F$ defined by the relations $g(z)h(z)/z$, $z^2/g(z)$, and $z^2/\int_0^z (t/g(t))dt$ belong to the class $\mathcal{M}$, where $g$ and $h$ belong to some suitable subclasses of $\mathcal{S}$, the class of univalent functions from $\mathcal{A}$. In the final analysis, we obtain the sharp Bohr radius, Bohr-Rogosinski radius and improved Bohr radius for a certain subclass of starlike functions.

math.CV↗