Searcharxiv⌕ Search

arXiv subjects

Debabrata Pramanik

Publications and source records attributed to Debabrata Pramanik.

At least 19 recordsLinked to original sources
arXiv2026

Coefficient estimates and Bohr phenomenon for pluriharmonic mappings in the polydisc

We introduce the class $\mathscr{P}_{\mathcal{H}_n^0}(α)$ $(0\leq α<1)$ of normalized pluriharmonic mappings in the setting of several complex variables. This class extends the harmonic family $\mathscr{P}_{\mathcal{H}}^{0}(α)$ to the multidimensional framework. We establish sharp coefficient estimates and growth theorems for functions in $\mathscr{P}_{\mathcal{H}_n^0}(α)$, thereby generalizing the corresponding results of Li and Ponnusamy \cite{Li-Ponnusamy-2013a} and Allu and Halder \cite{Allu-Halder-2021}. We further determine the associated Bohr radius and investigate the sections (partial sums) of functions in this class, obtaining quantitative results that describe the behavior of their truncated expansions.

math.CV↗
arXiv2026

Coefficient bounds and growth estimates for a class of pluriharmonic mappings in unit polydisk

In this paper, we first introduce and study the class $\mathcal{P}_{\mathcal{H}_n^0}(M)$ 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 class of holomorphic functions, extending the known result of Ghosh and Vasudevarao \cite{Ghosh-Allu-2020} to the setting of several complex variables. Finally, we provide sharp coefficient bounds and growth estimates for functions in the class $\mathcal{P}_{\mathcal{H}_n^0}(M)$.

math.CV↗
arXiv2026

Meromorphic functions and linearization phenomena in partial differential equations

In this paper, we investigate meromorphic solutions of certain nonlinear partial differential equations in several complex variables involving differential and functional operators. Let $f$ be a non-constant meromorphic function in $\mathbb{C}$, $g$ an entire function in $\mathbb{C}^n$, and $h(z)=f(z_1+z_2+\ldots+z_n)$. We study the equations \begin{align*} \frac{\partial h(z)}{\partial z_i}=a G^g_{h}(z)+bh(z)+c\;\;\text{and}\;\;\frac{\partial h(z)}{\partial z_i}=a(z)G^g_{h}(z)+b(z)h(z)+c(z), \end{align*} where $z\in\mathbb{C}^n$, $i\in\{1,2,\ldots,n\}$, $a(\neq 0), b, c\in\mathbb{C}$ or $a(z)(\not\equiv 0), b(z),c(z)$ are polynomials in $\mathbb{C}^n$, and $G^g_h(z)=h(g(z),g(z),\ldots,g(z))$. The results obtained in the paper, extend previous studies on meromorphic solutions of functional-differential equations to the setting of several complex variables, and further illustrate the rigidity imposed by value distribution properties on nonlinear functional equations.

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

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↗
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

On the two conjectures

In this paper, we investigate the uniqueness problem of entire functions that share an entire function with their higher-order difference operators. We obtain two results that confirm the conjectures posed by Liu and Laine \cite{LL1} and by Zhang et al. \cite{ZKL1}, respectively. In addition, we present several relevant examples to further illustrate and support our findings

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

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

Partial sharing and cross sharing of entire function with its derivative

In the paper, we investigate the uniqueness problem of a power of an entire function that share one value partially with it's linear differential polynomial and obtain a result, which improves several previous results in a large scale. Also in the paper we include some applications of our main result. Moreover, we solve the question raised by Wang and Liu (Value cross-sharing of meromorphic functions, Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00481-9) for a special case related to value cross-sharing.

math.CV↗
arXiv2025

Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture

In the paper, we find out the precise form of the finite order entire solutions of the following differential-difference equation \[f^{(k)}(z)=\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc),\] where $a_0, a_1,\ldots,a_n(\neq 0)\in\mathbb{C}$. Also in the paper we study the differential-difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function $f(z)$ having a Borel exceptional small function of $f(z)$, when $f^{(k)}(z)$ and $\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc)$ share a small function of $f(z)$. The obtained results, significantly generalize and improve the results due to Liu and Dong (Some results related to complex differential-difference equations of certain types, Bull. Korean Math. Soc., 51 (5) (2014), 1453-1467). Some examples are given to ensure the necessity of the condition (s) of our main results.

math.CV↗
arXiv2025

Study on the certain type of nonlinear algebraic partial differential equation in $\mathbb{C}^m$

In the paper, using Nevanlinna's value distribution theory of meromorphic functions in $\mathbb{C}^m$, we study for the existence of entire solutions $f$ in $\mathbb{C}^m$ of the following algebraic partial differential equation \[f^n(z)+P_d(f(z))=p(z)e^{\langle α,\ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial in $f$ of degree $d \leq n-2$, $n \geq 3$ is an integer, $p$ is a non-zero polynomial, $α=(α_{1},\ldots,α_{m})\neq (0,\ldots,0)$ and $\langle α,\ol z\rangle=\sideset{}{_{k=1}^{m}}{\sum}α_{1k} z_k$. Also in the paper, we study for the non-existence of entire solutions $f$ in $\mathbb{C}^m$ of the following algebraic partial differential equation \[f^n(z)+P_d(f(z))=p_1(z)e^{\langle α, \ol z\rangle}+p_2(z)e^{\langle β, \ol z\rangle},\] where $P_d(f)$ is an algebraic differential polynomial of degree $d \leq n-3$, $n \geq 4$ is an integer, $p_1$ and $p_2$ are two non-zero polynomials, $α=(α_{11},\ldots,α_{1m})\neq (0,\ldots,0)$ and $β=(α_{21},\ldots,α_{2m})\neq (0,\ldots,0)$ such that $α_{1i}\neq 0$, $α_{2i}\neq 0$ and $α_{1i}/α_{2i}\not\in\mathbb{Q}$ for all $i\in\mathbb{Z}[1,m]$. Our findings extend and improve the results of Li and Yang (J. Math. Anal. Appl., 320 (2006) 827-835) and Zhang and Liao (Taiwanese J. Math., 15 (5) (2011), 2145-2157) into higher dimensions.

math.CV↗
arXiv2021

On the phase behaviour of pure PSPC and PEGylated multi-component lipid and their interaction with Paclitaxel: An all-atom MD study

The study of the structural behaviour of pure and multi-component lipids at various temperatures and the interaction of these multi-component lipids with pharmaceutically important drugs carry huge importance. Here, we investigated the phase behaviour of the pure PSPC (1-palmitoyl-2-stearoyl-sn-glycero-3-phosphocholine), and multicomponent PSPC and DSPE-PEG2000(1,2-distearoyl-sn-glycero-3-phosphoethanolamine-N[amino(polyethylene glycol)-2000]) membranes at seven different temperatures ranging from 280 K to 360 K, and calculated their structural properties. We observe a transition from the gel phase to the liquid crystalline phase between 320 K and 330 K in agreement with experimental reports for pure PSPC. PSPC remained in the tilted gel phase L\b{eta}$'$ at 320 K and 310 K, entered the 'mixed ordered' domain with a partially interdigitated region at 300 K, and finally formed the sub gel phase at 280 K. We studied the self-assembly for the multicomponent PSPC and DSPE-PEG2000 membranes and found the coexistence of ordered and disordered phases at 320 K. In comparison to the pure PSPC, for multicomponent system, this transition was gradual, and a complete liquid crystalline to gel phase transformation occurred between 320 K and 310 K. We further studied the interaction of Paclitaxel with pure PSPC and PEGylated multicomponent lipid bilayers using umbrella sampling technique and observed PEG promotes the interaction of Paclitaxel with the later one in comparison to the former. Above the bilayer transition temperature, Paclitaxel interacts more with the bilayer and enters inside the bilayer easily for both systems. Understanding of structural and interaction behaviour of the PEGylated multicomponent lipid bilayers with Paclitaxel will help explore Paclitaxel based drug applications in the future.

cond-mat.soft↗
arXiv2018

Ligand dissociation mechanisms from all-atom simulations: Are we there yet?

Large parallel gains in the development of both computational resources as well as sampling methods have now made it possible to simulate dissociation events in ligand-protein complexes with all--atom resolution. Such encouraging progress, together with the inherent spatiotemporal resolution associated with molecular simulations, has left their use for investigating dissociation processes brimming with potential, both in rational drug design, where it can be an invaluable tool for determining the mechanistic driving forces behind dissociation rate constants, as well as in force-field development, where it can provide a catalog of transient molecular structures on which to refine force-fields. Although much progress has been made in making force-fields more accurate, reducing their error for transient structures along a transition path could yet prove to be a critical development helping to make kinetic predictions much more accurate. In what follows we will provide a state-of-the-art compilation of the molecular dynamics (MD) methods used to investigate the kinetics and mechanisms of ligand-protein dissociation processes. Due to the timescales of such processes being slower than what is accessible using straightforward MD simulations, several ingenious schemes are being devised at a rapid rate to overcome this obstacle. Here we provide an up-to-date compendium of such methods and their achievements/shortcomings in extracting mechanistic insight into ligand-protein dissociation. We conclude with a critical and provocative appraisal attempting to answer the title of this review.

physics.bio-ph↗