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Shantanu Panja

Publications and source records attributed to Shantanu Panja.

8 recordsLinked to original sources
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 $\varphi(z)=1+\frac{2}{\pi}\arcsin z.$ For this family, we develop a unified coefficient analysis based on subordination techniques, Carath\'eodory 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--Szeg\"o functional. In every case, the corresponding extremal functions are identified, showing that all of the obtained inequalities are best possible.

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 $\mathbb{P}\Delta(0_n, 1_n)$ 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 $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$ 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 $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$. Finally, we introduce sections (partial sums) of pluriharmonic mappings and investigate their properties for functions belonging to $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$.

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

Logarithmic coefficients for exponential classes of starlike and convex functions

In this paper, we investigate two subclasses of analytic and univalent functions associated with the exponential mapping $\varphi(z)=e^{\alpha z},\qquad 0<\alpha\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec e^{\alpha z} \quad \text{and} \quad 1+\frac{zf''(z)}{f'(z)}\prec e^{\alpha z}$. 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 and upper bounds for Toeplitz determinants. In particular, explicit estimates are derived for $$ |H_{2,1}(F_f/2)|, \quad |T_{2,1}(F_f/2)|, $$ for functions belonging to the introduced exponential subclasses 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

Rigidity of entire functions sharing a finite set with their partial derivatives in C^n

This paper investigates certain classes of entire functions in C^n that, together with their partial derivatives, share a finite set consisting of three elements. By employing normality criteria, we study the behaviour of such functions and derive the necessary conditions governing their existence. Our results extend those of [4], originally established for functions of a single complex variable, to the setting of several complex variables, thereby providing a comprehensive generalization of the earlier result in a direction not previously explored.

math.CV↗
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 \[ (f^n)^{(k)}(g^n)^{(k)} = \alpha^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $\alpha$ 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↗