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Abhijit Banerjee

Publications and source records attributed to Abhijit Banerjee.

At least 19 recordsLinked to original sources
arXiv2026

Entire solutions of Fermat-Type Partial Differential-Difference equations in higher dimensions

The purpose of this paper is to investigate the finite-order entire solutions of the following Fermat-type partial differential-difference equations: \begin{align*} (P_1(z)f(z+c))^{m_1}+\left(P_2(z)f(z)+\sideset{}{_{j=1}^m}{\sum} Q_j(z)\partial_{z_j}(f(z))\right)^{m_2}=P(z), \end{align*} \begin{align*} (P_1(z)f(z))^{m_1}+\left(P_2(z)f(z+c)+\sideset{}{_{j=1}^m}{\sum} Q_j(z)\partial_{z_j}(f(z))\right)^{m_2}=P(z) \end{align*} and \begin{align*} \left(\sideset{}{_{j=1}^m}{\sum} Q_j(z)\partial_{z_j}(f(z))\right)^{m_1}+\left(P_1(z)f(z+c)+P_2(z)f(z)\right)^{m_2}=P(z), \end{align*} in $\mathbb{C}^m$, where $m_1,m_2\in\mathbb{N}$ such that $m_1+m_2>2$, $c=(c_1,c_2,\ldots,c_m)\in\mathbb{C}^m$, $P(\not\equiv 0)$, $P_1$, $P_2$ and $Q_i\;(i=1,2,\ldots,m)$ are polynomials in $\mathbb{C}^m$ such that $P(z)$ is irreducible in $\mathbb{C}^m$. The results established in this paper generalize those of Xu et al. \cite{Xu-Luo-Cao-2025} from $\mathbb{C}$ to $\mathbb{C}^m$.

math.CV↗
arXiv2026

Estimates for a Parameterized Hankel Functional of Starlike Mappings of Order $α$ and Quasi-Convex Mappings of Type $B$ in Complex Banach Spaces

In this paper, we introduce and investigate a parameterized Hankel-type functional $$H_{2,2}^μ(f)=μa_2a_4-a_3^2,\qquad μ\in[0,1], $$ which reduces to the classical second-order Hankel determinant $H_{2,2}(f)$ when $μ=1$. We obtain estimates for $H_{2,2}^μ(f)$ over the class of starlike mappings of order $α$ on the unit ball of a complex Banach space and on the unit polydisk $\mathbb{U}^n$ in $\mathbb{C}^n$. We also establish estimates for $H_{2,2}^μ(f)$ over the class of quasi-convex mappings of type $B$ on the unit ball of a complex Banach space and on the unit polydisk $\mathbb{U}^n$. For $μ=1$, our results recover the corresponding estimates for the classical Hankel determinant and extend the known one-dimensional results of Cho et al. \cite{Cho-2018} and Thomas et al. \cite{Thomas-Tuneski} to higher-dimensional settings.

math.CV↗
arXiv2026

The Fekete-Szegö problem for starlike mappings of order $α$ and quasi-convex mappings of type $B$ in the unit balls of complex Banach spaces

We establish sharp estimates for the generalized Zalcman functional corresponding to the index $n=2$ for normalized starlike mappings of order $α$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space and on the unit polydisk in $\mathbb{C}^n$. We also derive sharp Fekete-Szegö inequalities for normalized starlike mappings of order $α$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space. Our results extend classical coefficient estimates to higher-dimensional holomorphic mappings and provide a partial confirmation of the generalized Zalcman conjecture in higher dimensions.

math.CV↗
arXiv2026

Toeplitz determinants for starlike mappings of order α in several complex variables

Let $F(z)=zf(z)$, where $f$ is a normalized scalar holomorphic function on the unit ball $\mathbb B$ of a complex Banach space $X$ or on the unit polydisk $\mathbb U^n$ of $\mathbb{C}^n$. We establish sharp bounds for two Toeplitz determinants involving the second, third and fourth Fréchet derivatives of $F$ when $F$ is starlike of order $α$, $0\leqα<1$. The first bound is sharp for the full range $0\leqα<1$, while the second is sharp for $0\leqα\leq 2/3$. The extremal mappings are explicitly identified, confirming the sharpness of the estimates.

math.CV↗
arXiv2026

Sharp Robertson-type coefficient estimates for radial starlike and quasi-convex of type B mappings of order α in complex Banach spaces

In this paper, we study biholomorphic mappings $F: \mathbb{B} \to X$ of the form $F(z) = f(z)z$ for $z \in \mathbb{B}$, where $f: \mathbb{B} \to \mathbb{C}$ is a holomorphic function satisfying $f(0) = 1$. Our main contributions are as follows: We establish sharp estimates for the coefficients of starlike functions of order $α$ as well as quasi-convex functions of type $B$ and order $α$ by employing the notion of majorization. The results established here directly extend the results due to Robertson \cite{Robertson-1936} to the broader setting of complex Banach spaces.

math.CV↗
arXiv2026

Logarithmic coefficients for exponential classes of starlike and convex functions

In this paper, we investigate two subclasses of univalent functions associated with the exponential mapping $φ(z)=e^{αz}, \ 0<α\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec e^{αz}\quad \text{and} \quad 1+\frac{zf''(z)}{f'(z)}\prec e^{αz}.$ These classes provide a natural exponential analogue of several classical subclasses arising in geometric function theory. We obtain sharp coefficient estimates, logarithmic coefficient inequalities and sharp bounds for the associated Hankel determinants. In particular, explicit estimates are derived for $H_{2,1}(F_f/2)$ for functions belonging to the exponential subclasses introduced above of starlike and convex functions. Our results extend and unify several earlier works on exponential subclasses and highlight connections with logarithmic coefficients and determinant functionals.

math.CV↗
arXiv2026

Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in $\mathbb{C}^n$

The primary objective of this paper is to determine the existence and explicit form of finite-order entire solutions in $\mathbb{C}^n$ of the following system of Fermat-type partial differential-difference equations: \[\begin{cases} \left(\frac{\partial f_1\left(z\right)}{\partial z_1}\right)^{n_1} + (f_2 \left(z+c\right)-f_1(z) )^{m_1}= 1, \medskip \left(\frac{\partial f_2\left(z\right)}{\partial z_1}\right)^{n_2} + (f_1 \left(z+c \right)-f_2(z) )^{m_2}= 1, \end{cases}\] for several choices of the positive integers $n_1$, $n_2$, $m_1$, and $m_2$, where $c=(c_1,c_2,\ldots,c_n)$. We obtain structural classifications and nonexistence results in the exponent regimes specified in the main theorems, extending results of Xu et al. \cite{XLL1} from $\mathbb{C}^2$ to $\mathbb{C}^n$. Several examples illustrate the resulting solution families.

math.CV↗
arXiv2026

Coefficient Problems for a Ma-Minda Convex Class Associated with the Normalized Arcsine Mapping

Let $\mathcal{C}_{\arcsin}$ denote the Ma--Minda subclass of convex functions generated by the normalized arcsine mapping $φ(z)=1+\frac{2}π\arcsin z.$ For this family, we develop a unified coefficient analysis based on subordination techniques, Carathéodory functions and sharp estimates for Schwarz functions. As consequences, we derive sharp estimates for the initial Taylor coefficients, logarithmic coefficients and certain differences involving the logarithmic and inverse logarithmic coefficients. We further determine the exact bounds for the second Hankel determinant $H_{2,2}(f)$ together with the Hankel determinants $H_{2,1}(F_f/2)$ and $H_{2,1}(F_{f^{-1}}/2)$ associated with the logarithmic coefficients of a function and its inverse. Moreover, sharp estimates are obtained for the initial generalized Zalcman functional and the generalized Fekete--Szegö functional. In every case, the corresponding extremal functions are identified, showing that all of the obtained inequalities are best possible.

math.CV↗
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 estimates and Hankel determinant for the class of leaf shaped starlike function

In this paper, we study a Ma--Minda type subclass of starlike functions associated with the normalized arcsine function $φ(z)=1+\frac{2}π\arcsin z.$ Using the Carathéodory function approach together with sharp coefficient estimates for Schwarz functions, we obtain explicit sharp bounds for the initial Taylor coefficients and logarithmic coefficients of functions in this class. We also derive a sharp estimate for the second Hankel determinant \(H_{2,2}(f)\), as well as for the Hankel determinants \(H_{2,1}(F_f/2)\) and \(H_{2,1}(F_{f^{-1}}/2)\) associated with the logarithmic coefficients of a function and its inverse. In addition, we establish a sharp bound for a difference involving logarithmic coefficients. The extremal functions for all the main results are identified, confirming the sharpness of the estimates.

math.CV↗
arXiv2026

Multidimensional analogues of the improved Bohr's inequality for shifted polydisks

In this article, we investigate the Bhor phenomenon for holomorphic functions defined on a general simply connected domain in $\mathbb{C}^n$. We improve the existing results Evdordis et al. (Improved Bohr's inequality for shifted disks, Results in Mathematics, 76, 14 (2021)) for a broader class of holomorphic functions in $\mathbb{C}^n$. Furthermore, we consider pluriharmonic mappings defined on a polydisk containing the unit polydisk $\mathbb{P}Δ(0_n, 1_n)$ and establish a Bohr-type inequality for this class of mappings.

math.CV↗
arXiv2026

On a class of pluriharmonic mappings in the unit polydisk

In this paper, we introduce and study the class $\mathcal{W}_{\mathcal{H}_n^0}(α)$ of normalized pluriharmonic mappings, characterized by a suitable bound on their second-order partial derivatives. We establish a one-to-one correspondence between this pluriharmonic class and an associated class of holomorphic functions, thereby extending a result of Ghosh and Vasudevarao \cite{Ghosh-Allu-2019} to the setting of several complex variables. Furthermore, we obtain sharp coefficient bounds, growth estimates and a convex combination theorem for functions in $\mathcal{W}_{\mathcal{H}_n^0}(α)$. Finally, we introduce sections (partial sums) of pluriharmonic mappings and investigate their properties for functions belonging to $\mathcal{W}_{\mathcal{H}_n^0}(α)$.

math.CV↗
arXiv2026

Vernier-assisted mode-selective PT symmetry in optoelectronic oscillators

Mode-selective PT-symmetry is the manifestation of Vernier-assisted cross-injection between oscillators with unequal delays. PT-symmetry has been proposed as a mechanism for achieving low-noise single-mode operation in optoelectronic oscillators, but existing formulations are typically based on matched delay loops and frequency-independent coupling, leading to symmetry transitions that are global in frequency and therefore not intrinsically mode selective. A more general formulation of PT-symmetric time-delay oscillators is developed in which the coupling operator is allowed to be dispersive. This permits frequency-selective PT-symmetry transitions and removes the requirement for matched delay loops. Two mathematically equivalent but physically distinct realisations are derived. The first corresponds to coupled gain-loss loops connected by a dispersive coupler, while the second corresponds to a pair of equal-gain oscillators coupled through symmetric cross-injection with unequal delays. Numerical simulations confirm the predicted behaviour and demonstrate strong sidemode suppression while preserving the low phase-noise characteristics of large-delay oscillators. The results establish a direct connection between PT-symmetry, cross-injection architectures, and Vernier oscillator design.

physics.optics↗
arXiv2026

Finite Order Transcendental Entire Solutions of Coupled Fermat-Type Difference Equations in Several Complex Variables

Motivated by recent developments in complex difference equations and Nevanlinna theory in several complex variables, we investigate finite-order transcendental entire solutions of the coupled Fermat-type difference system: \beas \begin{cases} f_1^{n_1}(z)+ f_2^{m_1} \left(z+c \right) = 1,\\ f_2^{n_2}(z) + f_1^{m_2} \left(z+c\right) = 1, \end{cases} \eeas where $z,c=(c_1,c_2,\ldots,c_n) \in \mathbb{C}^n$ for various choices of $n_i,m_i$, $i=1,2$. where $n_i,m_i\in\mathbb N$ and $n_i+m_i\ge2$ $(i=1,2)$. Extending the classical investigations of Gross--Yang, Liu, Liu--Cao--Cao and more recently, Xu \emph{et al.} in one and two complex variables, to a general coupled system in $\mathbb C^n$ we establish a complete characterization of all finite-order transcendental entire solutions. We have determined that the solution structure is completely determined by the relative sizes of the exponents.

math.CV↗
arXiv2026

PT-symmetric time delay oscillator modelling beyond the weak coupling limit via a scattering matrix formulation

Parity-time (PT) symmetry in time-delay oscillators such as lasers and optoelectronic oscillators provides a potential route to enhanced spectral purity, including reduced phase noise and improved sidemode suppression. Existing theoretical descriptions are typically based on coupled-mode formulations derived under slowly varying envelope and near-degeneracy assumptions, which restrict their validity to weak coupling, small gain/loss contrast, and small detuning. In this work, a non-perturbative formulation of PT symmetric time-delay oscillators is developed based on a delay-difference equation and a scattering matrix representation of the coupling network. The approach treats propagation delay explicitly and does not rely on modal truncation, remaining valid for arbitrary coupling strength, gain/loss imbalance, and resonance detuning. The exact eigenvalue structure of the system is obtained in closed form, yielding a complete characterization of the unbroken and broken PT symmetric regimes as well as the associated exceptional points. A dimensionless order parameter is introduced that governs the symmetry transition over the full parameter space. It is further shown that conventional coupled-mode theory is recovered as an asymptotic limit of the exact formulation for small parameters. The results provide a unified and physically transparent framework for analysing PT symmetric delay systems beyond the weak-coupling limit, with direct implications for the design and optimisation of low-noise oscillators and photonic systems.

physics.optics↗
arXiv2026

Multiscale phase dynamics and $2π$ phase kinks in injection-locked optoelectronic oscillators with large delay

Injection locking of optoelectronic oscillators (OEOs) with large delay gives rise to phase dynamics that lie beyond the scope of classical single mode locking theory, including the spontaneous formation of persistent $2π$ phase kinks. In this work, a multiscale theoretical framework is developed that explains the origin, structure, and stability of these phase slip phenomena in injection locked (IL) OEOs operating in large-delay regime. Starting from a complex envelope delay differential equation that explicitly incorporates hard-limiting gain saturation and RF BPF dynamics, a reduced phase-only description valid for nearly constant oscillation amplitude is derived. Exploiting the separation between fast round-trip dynamics and slow inter-round-trip evolution, a two-timescale reduction yields a continuum of Adler equations governing phase difference between injected signal and oscillator as a function of round-trip time. Analytical solutions obtained for weak injection and small detuning show how initially smooth phase profiles sharpen into localized $2π$ phase kinks, with p kinks appearing per round trip when the injection is tuned near the pth adjacent mode. Time-domain simulations of full complex-envelope model validate the predicted phase kink formation mechanism and reveal essential role of RF resonator dynamics in determining their persistence. While the reduced phase-only model correctly predicts kink sharpening, resonator-induced amplitude excursions associated with steep phase gradients can erase the kinks when the complex-envelope trajectory fails to encircle the origin, restoring conventional phase locking. These results provide a unified physical interpretation of $2π$phase kinks in IL-OEOs and delineate the limits of phase-only models in the presence of large, fast phase transients, identifying a regime of frequency locking without phase locking in large delay oscillators.

physics.optics↗
arXiv2026

Identification of L-functions in the extended Selberg class by preimages of finite sets

In 2023, Li, Du, Yi proved a uniqueness theorem for L functions in the extended Selberg class under the assumptions of positive degree, a shared functional equation, and the sharing of three complex values. This was later strengthened by the present authors, who showed that sharing an arbitrary finite set of complex values, counted with multiplicities, still forces equality of the two L functions, again under the assumption that they satisfy the same functional equation. In this paper, we significantly improve all these results. We completely remove the requirement that the two L functions satisfy the same functional equation, yet we still obtain the same strong uniqueness conclusion under far weaker hypotheses. As a major consequence, we prove that every polynomial with distinct zeros is a strong uniqueness polynomial for L functions.

math.CV↗