arXiv subjects

Sujoy Majumder

Publications and source records attributed to Sujoy Majumder.

At least 37 records · Page 2Linked to original sources

arXiv2026

On the structure and classification of solutions to certain nonlinear differential equations

This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation \[ (f^n)^{(k)}(g^n)^{(k)} = α^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $α$ is a common small function of $f$ and $g$. Our main results provide a detailed characterization of the solutions, improving upon earlier works by Fang-Qiu [5], Fang [4], Zhang-Xu [19], and Li-Yi [9]. Notably, we identify and correct significant errors in the proof of Lemma 2.11 [13], which represents the most recent contribution in this area and provide a resolved and rigorous treatment of the problem. Equations of this type arise naturally in various areas of mathematics and applied sciences such as in the study of complex dynamical systems, integrable systems and value distribution theory in complex analysis. Moreover, understanding the meromorphic solutions helps to realize the growth behavior of solutions, stability analysis and modeling of phenomena in physics and engineering. By characterizing these solutions, one can develop methods to solve broader classes of nonlinear differential equations and explore their qualitative properties, which are essential for both theoretical studies and practical applications.

math.CV↗

arXiv2026

Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}Δ(0;1_n)$. We provide a definitive resolution to the Bohr phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $ω_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the directional derivative operator $\partial_uf(z) = \sum_{k=1}^{n} u_k \frac{\partial f(z)}{\partial z_k}$, where $u=(u_1,u_2,\ldots,u_n)\in\mathbb{C}^n$ such that $|u_1|+|u_2|+\ldots+|u_n|=1$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.

math.CV↗

arXiv2026

Borel exceptional values in several complex variables and their applications to shared values of shifts and difference operators

In this paper, we investigate shared value problems for shifts and higher-order difference operators of meromorphic and entire functions in several complex variables. Using Nevanlinna theory in $\mathbb{C}^n$, we obtain new uniqueness theorems when functions share values counting or ignoring multiplicities, extending several classical one-variable results to higher dimensions. A key contribution of this work appears in Section 2, where we establish fundamental results on Borel exceptional values in several complex variables. These propositions provide the main tools for proving our principal theorems. As applications, we derive conditions ensuring that a transcendental entire function satisfies $Δ_c^{k} \equiv d\hspace{.05cc} f$ and we study meromorphic solutions of certain partial differential-difference equations, obtaining growth estimates and structural descriptions of entire solutions. To the best of our knowledge, this is the first systematic study of such shared value problems for higher-order difference operators in several complex variables.

math.CV↗

arXiv2026

On Stable Univalence and Coefficient Estimates for a Class of Pluriharmonic Mappings in Convex Reinhardt Domains

In this paper, we investigate the geometric properties of complex-valued pluriharmonic mappings defined over convex Reinhardt domains in $\mathbb{C}^n$. We first establish a multidimensional analogue of the Noshiro-Warschawski Theorem, providing sufficient conditions for the univalence of pluriharmonic mappings based on the real part of their partial derivatives. Furthermore, we introduce and study the class $\mathcal{B}_{\mathcal{H}_{n}^{0}}(M)$ of normalized pluriharmonic mappings, characterized by a specific bound on the sum of their second-order partial derivatives. We prove a one-to-one correspondence between this pluriharmonic class and a corresponding class of holomorphic functions, extending known results from the planar harmonic case to higher dimensions. Specifically, we show that a pluriharmonic mapping $f=h+\overline{g}$ is stable pluriharmonic univalent if and only if its holomorphic counterpart $F=h+g$ is stable holomorphic univalent on the unit polydisk $\mathbb{P}Δ(0;1)$. Finally, we provide sharp coefficient estimates and sufficient conditions for functions to belong to the class $\mathcal{B}_{\mathcal{H}_{n}^{0}}(M)$. Our results generalize several classical theorems in the theory of univalent harmonic functions to the setting of several complex variables.

math.CV↗

arXiv2026

Bruck conjecture for solutions of first-order partial differential equations in Cm

In this paper, we study the Brück conjecture \cite{Bruck-1996} by interpreting it through solutions of first-order partial differential equations in several complex variables. Our results show that the Brück conjecture \cite{Bruck-1996} in $\mathbb{C}^m$ holds under certain additional conditions. In pursuit of this objective, we also establish a Borel-Caratheodory theorem in $\mathbb{C}^m$ and derive several fundamental results on the order and hyper-order of entire functions in higher dimensions.

math.CV↗

arXiv2026

Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{D}^n$. We establish a sharp extension of the classical Bohr inequality, proving that the Bohr radius remains $R_n = 1/(3n)$ for the family of holomorphic functions bounded by unity in the multivariate setting. Further, we provide a definitive resolution to the Bohr-Rogosinski phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $ω_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the radial (Euler) derivative operator $Df(z) = \sum_{k=1}^{n} z_k \frac{\partial f(z)}{\partial z_k}$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. Finally, a multidimensional version of the area-based Bohr inequality is established. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.

math.CV↗

arXiv2025

Multidimensional analogues of the refined Bohr type inequalities

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{D}^n$.We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables: one by replacing the constant term with the absolute value of the function, and another by replacing it with the square of the absolute value of the function.Furthermore, we establish multidimensional analogues of known results concerning the modulus of the derivative of analytic functions in the unit disk $\mathbb{D}$, replacing the derivative with the radial derivative of holomorphic functions in $\mathbb{D}^n$.All of the established results are shown to be sharp.

math.CV↗

arXiv2025

Relationship between the value-sharing behavior of an entire function and its derivative, and the analytic structure of a nonlinear differential equation

In this paper, we study uniqueness problems for entire functions that partially share two values with their higher-order derivatives. The results obtained here both improve and generalize the related results of Li and Yi \cite{LYi}, Lü et al. \cite{LXY1} and Sauer and Schweizer \cite{SS1}. Furthermore, we show that our results reveal a deep relationship between the value-sharing behavior of an entire function $f$ and its $k$-th derivative $f^{(k)}$, and the analytic structure of a particular type of nonlinear differential equation. Several examples are provided to illustrate the necessity of the conditions used in our results.

math.CV↗

arXiv2025

Multidimensional analogues of the improved Bohr's inequality

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{P}Δ(0;1_n)$. We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables by replacing the constant term with the absolute value of the function and the square of the absolute value of the function, respectively. All the results are shown to be sharp.

math.CV↗

arXiv2025

On the existence of entire solutions to a system of nonlinear Fermat-type partial differential-difference equations

The aim of this study is to investigate the precise form of finite-order entire solutions to the following system of Fermat-type partial differential-difference equations: \beas \begin{cases} \left(\frac{\partial f_1\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_1} + f_2^{m_1} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1,\\ \left(\frac{\partial f_2\left(z_1, z_2, \ldots, z_m \right)}{\partial z_1}\right)^{n_2} + f_1^{m_2} \left(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m \right) = 1 \end{cases} \eeas for various combinations of the positive integers $n_1$, $n_2$, $m_1$ and $m_2$. Our results extend the work of Xu et al. (Entire solutions for several systems of non-linear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483(2), 2020), generalizing the setting $\mathbb{C}^2$ to $\mathbb{C}^m$. Several examples are provided to illustrate the applicability and sharpness of the obtained results.

math.CV↗

arXiv2025

Solutions of certain Fermat-type partial differential-difference equations

The purpose of this paper is to investigate the non-constant entire as well as meromorphic solutions of the Fermat-type partial differential-difference equation: \[\left(\sum_{j=1}^m\frac{\partial f(z_1, z_2, \ldots, z_m)}{\partial z_j}\right)^{m_1} + f^{m_2}(z_1 + c_1, z_2 + c_2, \ldots, z_m + c_m ) = 1,\] where $m_1$ and $m_2$ are positive integers such that $m_1+m_2>2$ and $(c_1, c_2, \ldots, c_m)\in \mathbb{C}^m$. The results of our paper generalize the result of Xu and Wang \cite {XW1} from $\mathbb{C}^2$ to $\mathbb{C}^m$. Also in the paper we give positive answer of the open problem addressed by Xu and Wang \cite {XW1}. Moreover plenty of examples are provided to illustrate our findings.

math.CV↗

arXiv2025

Differential Operators, Multiple Schwarz Functions, and the Bohr Radius of Stable Harmonic Maps

In this paper, we study the Bohr phenomenon for differential operators $D$ and $\mathscr{D}$ of stable harmonic mappings involving multiple Schwarz functions in $\mathcal{B}_n$, using distance formulations. By constructing suitable combinations of multiple Schwarz functions, we establish sharp and improved Bohr-type inequalities for these mappings. The corresponding Bohr radii are also determined for certain subclasses of stable harmonic functions and their associated differential operators. Moreover, Bohr-Rogosinski-type inequalities are derived, which highlights the influence of multiple Schwarz functions on the geometric properties of stable harmonic mappings. All the radii are determined, and we prove that each one is the best possible.

math.CV↗

arXiv2025

Entire solution of a partial differential equation

In this paper, using Nevanlinna's value distribution theory of meromorphic functions in several complex variables, we study for the existence of entire solutions $f$ in $\mathbb{C}^2$ of the following partial differential equation \[a_1\left(\frac{\partial f(z_1,z_2)}{\partial z_1}\right)^n+a_2f^n(z_1,z_2)=p_1e^{r(z_1,z_2)}+p_2e^{s(z_1,z_2)},\] where $n$ is a positive integer such that $n\geq 3$, $a_1,a_2,p_1,p_2$ are non-zero constants and $r(z_1,z_2), s(z_1,z_2)$ are arbitrary polynomials in $\mathbb{C}^2$.

math.CV↗

arXiv2025

The Yang-Hua theorems in several complex variables

In this paper, we investigate meromorphic solutions in $\mathbb{C}^m$ of the nonlinear differential equation \[\displaystyle f^n\partial_u(f)g^n\partial_u(g)=1,\] where $\partial_u(f)=\sum_{j=1}^mu_j\partial_j(f)$ and $\sum_{j=1}^m u_j\neq 0$. Our results extend those of Yang and Hua [{\sc C. C. Yang} and {\sc X. H. Hua}, Uniqueness and value sharing of meromorphic functions, \textit{Ann. Acad. Sci. Fenn. Math.}, \textbf{22} (1997), 395-406.] to the framework of several complex variables. Moreover, we establish new uniqueness theorems that further generalize their conclusions to higher dimensions. As an application, explicit solutions of certain nonlinear partial differential equations in several variables are derived, and their physical interpretations are summarized in tabular form.

math.CV↗

arXiv2025

On Coefficient problems for classes $\mathcal{S}_e^{\ast}$ and $\mathcal{C}_e$

Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study,we focus on the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ of starlike and convex functions, respectively, \begin{align*} \mathcal{S}_e^\ast := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec e^z, \ z \in \mathbb{D} \right\}, \end{align*} and \begin{align*} \mathcal{C}_e := \left\{ f \in \mathcal{S} : 1 + \frac{z f''(z)}{f'(z)} \prec e^z, \ z \in \mathbb{D} \right\}. \end{align*} This paper investigates the sharp bounds of the logarithmic coefficients and the Hermitian-Toeplitz determinant of these coefficients for the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$. Additionally, we examine the generalized Zalcman conjecture and the generalized Fekete-Szegö inequality for these classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ and show that the inequalities are sharp.

math.CV↗

arXiv2025

Power of a meromorphic function sharing value with its k-th order directional derivative in C^m

In the context of several complex variables, we investigate the uniqueness problem for a power of a meromorphic function that shares a value with its $k$-th order directional derivative in $\mathbb{C}^m$. Our results extend previous uniqueness theorems from the one-variable case to higher dimensions. Furthermore, we provide numerous examples to demonstrate that our results are, in certain senses, best possible.

math.CV↗

arXiv2025

A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk

We investigate improved forms of the Bohr inequality, using the quantity $S_r/π$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of $\mathbb{D}_r$. We then generalize the inequality for harmonic mappings ($\mathcal{P}^0_{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(α)$ of the form $f = h + \overline{g}$) by introducing a sequence $\{φ_n(r)\}_{n=0}^\infty$ of differentiable, increasing functions on $[0, 1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp.

math.CV↗

arXiv2025

Uniqueness of first derivatives and differences in meromorphic functions and the characterization of entire function periodicity

The objective of the paper is twofold. The first objective is to study the uniqueness problem of meromorphic function $f(z)$ when $f^{(1)}(z)$ shares two distinct finite values $a_1$, $a_2$ and $\infty$ CM with $Δ_cf(z)$. In this context, we provide a result that resolves the open problem posed by Qi et al. [Comput. Methods Funct. Theory, 18 (2018), 567-582] for the case when hyper order of the function is less than $\infty$. The second objective is to establish sufficient conditions for the periodicity of transcendental entire functions. In this direction, we obtain a result that affirms the question raised by Wei et al. [Anal. Math., 47 (2021), 695-708.]

math.CV↗