arXiv subjects

Nabadwip Sarkar

Publications and source records attributed to Nabadwip Sarkar.

At least 19 recordsLinked to original sources

arXiv2026

Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces

In this paper, we investigate the second Hankel determinant, Toeplitz determinants, Hermitian--Toeplitz determinants and the generalized Zalcman functional for a class of normalized biholomorphic mappings on complex Banach spaces. Motivated by recent results for the corresponding class of analytic functions in the unit disk, we establish Banach space analogues of these determinant estimates. Our results extend the one-variable inequalities of Allu, Lecko and Thomas \cite{ALT2022} to the setting of complex Banach spaces, thereby providing higher-dimensional counterparts for the class corresponding to $\mathcal{F}_{O}(λ)$.

math.CV↗

arXiv2026

Improved Bohr inequalities involving Fréchet derivatives associated with Holomorphic mappings in Banach spaces

Motivated by recent advances in derivative Bohr inequalities and their refinements, we investigate the Bohr phenomenon for holomorphic mappings in complex Banach spaces associated with Schwarz functions. We establish sharp Bohr-type inequalities involving higher-order Fréchet derivatives for both $|F(z)|$ and its refined counterpart $|F(z)|^2$. The corresponding Bohr radii are determined and shown to be best possible. Our results provide affirmative answers to Questions 1.1 and 1.2 and extend several classical and recent Bohr inequalities from the unit disk to the setting of holomorphic mappings on Banach spaces.

math.CV↗

arXiv2026

Radii of Concavity for Subclasses of Univalent Functions Associated with the Exponential Mapping

We determine the radii of concavity for the classes $\mathcal{S}_e^*$ and $\mathcal{C}_e$ of univalent functions associated with the exponential mapping $e^z$. Under the geometric framework of Avkhadiev--Wirths for conformal mappings with unbounded convex complements of opening angle $πA$ ($A \in (1,2]$), the radii are characterized by explicit transcendental equations. Sharpness is established globally by constructing explicit univalent extremal functions related to the disk automorphism $ω_0(z) = -z$, and the strict monotonicity of these radii with respect to the parameter $A$ is verified numerically.

math.CV↗

arXiv2026

Coefficient Problems and Sharp Determinant Estimates for the Class $\mathcal{P}^{*}$

We study coefficient problems for the class $\mathcal{P}^{*}$ of normalized analytic functions $f$ in the unit disk satisfying the subordination condition \[ f'(z) \prec e^z(1+\sin z), \qquad z\in\mathbb{D}. \] For functions in this class, we determine sharp bounds for the logarithmic coefficients $γ_n$ for $n=1,2,3,4$ and for the inverse logarithmic coefficients $Γ_n$ for $n=1,2,3$. In particular, we prove \[ |γ_n| \le \frac{1}{n+1} \quad (n=1,2,3,4), \qquad |Γ_1|,|Γ_2| \le \frac12,\quad |Γ_3|\le \frac{35}{48}, \] with equality cases explicitly identified. Moreover, we establish sharp estimates for the moduli of the differences $|γ_2|-|γ_1|$ and $|Γ_2|-|Γ_1|$, yielding \[ -\frac12 \le |γ_2|-|γ_1| \le \frac13, \qquad -\frac{1}{\sqrt{10}} \le |Γ_2|-|Γ_1| \le \frac13. \] Turning to determinant problems, we obtain sharp upper bounds for the second-order Hankel determinants associated with both the logarithmic and inverse logarithmic coefficients: \[ \bigl|H_{2,1}(F_f/2)\bigr| \le \frac19,\qquad \bigl|H_{2,1}(F_{f^{-1}}/2)\bigr| \le \frac{11}{96}. \] Finally, we derive the sharp two-sided estimate \[ -\frac{9}{35} \le T_{3,1}(f) \le 1 \] for the third-order Hermitian--Toeplitz determinant. All bounds are sharp, and the extremal functions are given explicitly. Our results extend and refine several known coefficient estimates for related subclasses of starlike and convex functions.

math.CV↗

arXiv2026

Sharp Coefficient Estimates for the Exponential Starlike class $\mathcal{S}_{ex}^{\ast}$

Let $\mathcal{S}_{ex}^{\ast}$ denote the class of normalized analytic functions $f$ in the unit disk satisfying \[ \frac{zf'(z)}{f(z)} \prec e^{αz},\qquad 0<α\le 1. \] We obtain sharp bounds for the initial inverse logarithmic coefficients $Γ_1$, $Γ_2$, and $Γ_3$. In particular, the bound for $|Γ_3|$ has a four-branch structure with transition values \[ α_\star \approx 0.542712,\qquad α_b \approx 0.679103,\qquad α_c \approx 0.691095. \] We also establish sharp upper and lower bounds for the successive coefficient difference \[ |Γ_2|-|Γ_1|, \] for the inverse logarithmic Hankel determinant \[ H_{2,1}\bigl(F_{f^{-1}}/2\bigr), \] and for the third-order Hermitian--Toeplitz determinant $T_{3,1}(f)$, whose sharp lower bound changes at \[ α_H=\frac{-1+\sqrt{241}}{15}\approx0.9683. \] Furthermore, we completely solve the sharp generalized Fekete--Szegő problem for the functional \[ |a_3-λa_2^2|-μ|a_2|. \] All estimates are sharp, and the corresponding extremal functions are explicitly constructed by using Carathéodory function representations.

math.CV↗

arXiv2026

Sharp Hankel and Hermitian--Toeplitz Determinants Involving Logarithmic and Inverse Logarithmic Coefficients for the class $\mathcal{S}_{car}^*$

This paper focuses on coefficient problems for the class of starlike functions related to a cardioid region. We derive sharp bounds for the Hankel determinants of logarithmic coefficients and logarithmic inverse coefficients of order two. Additionally, we completely solve the extremal problem for the third-order Hermitian--Toeplitz determinant by providing its sharp upper and lower bounds. All results are supported by extremal examples demonstrating the sharpness of the obtained estimates.

math.CV↗

arXiv2026

Second Hankel Determinant for $β$-Spirallike Convex Mappings in Complex Banach Spaces

We establish the bound for the second-order Hankel determinant $H_{2,2}(F) = A_2 A_4 - A_3^2$ associated with the class $\mathcal{C}_{B}^β(\mathbb{B})$ of normalized $β$-spirallike quasi-convex mappings of type $B$ on the open unit ball $\mathbb{B}$ of a complex Banach space. By utilizing a generalized framework based on a directional slice homogeneous polynomial expansion, we eliminate the standard, restrictive assumption that the mapping is of the form $F(x) = g(x)x$. Under these weaker operational conditions, we parameterize the targeted scalar invariants $A_n$ via the classical Carathéodory functional parameters. A rigorous optimization analysis proves that the established upper bound is strictly sharp for the classical non-spirallike case $β= 0$, yielding a maximal value of $1/8$. This sharp bound is verified by constructing explicit multi-dimensional extremal mappings that lift the corresponding single-variable convex profile. Finally, an unresolved open question regarding the exact variational behavior for $β\neq 0$ is formulated.

math.CV↗

arXiv2026

Bohr-type inequalities with Fréchet derivative and Area terms in Complex Banach spaces

We establish new Bohr-type inequalities for holomorphic mappings from the unit ball of a complex Banach space into the closed unit disk. Using Fréchet derivatives, Schwarz mappings of prescribed orders and area functionals, we obtain sharp Bohr radii characterized by explicit equations. We further prove a refined Bohr inequality involving derivative, coefficient and area terms and show that the corresponding radius is the unique positive solution of $r^m=(\sqrt{17}-3)/4$. The sharpness of the obtained radii and constants is also established. Our results extend several recent Bohr-type inequalities for bounded analytic functions on the unit disk to the setting of holomorphic mappings on complex Banach spaces.

math.CV↗

arXiv2026

Higher-Order Schippers-Schwarzian Derivatives for Non-Circular Starlike Functions

We find sharp upper bounds for the initial higher-order Schippers-Schwarzian derivatives $σ_3(f)(0)$ and $σ_4(f)(0)$ for several subclasses of univalent and starlike functions in the unit disk. The functions in these classes are defined by subordination to domains bounded by an exponential-sine curve, a cardioid, or a petal-shaped curve. We determine the sharp bounds and find the corresponding extremal functions for each invariant in these subclasses. As an application, we also obtain sharp bounds for the initial Grunsky coefficients $ω_{1,1}$ and $ω_{1,2}$.

math.CV↗

arXiv2026

Sharp Estimates for Hankel, Fekete-Szegö and Zalcman Functionals for $\mathcal{S}_\mathbb{B}^{*}(α)$ in Complex Banach Spaces

Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $α$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $α$ defined on the unit ball of a complex Banach space. Employing Fréchet derivatives together with suitable auxiliary lemmas, we establish sharp upper bounds for the second-order Hankel determinant, the Fekete--Szegö functional, and the Zalcman functional associated with this class of mappings. In each case, the obtained estimates are shown to be sharp by identifying the corresponding extremal mappings. Furthermore, our results reduce to the known one-dimensional sharp estimates when the underlying Banach space is the complex plane, thereby extending several classical results of Cho \emph{et al.} \cite{CKKLS2018} to the setting of complex Banach spaces.

math.CV↗

arXiv2026

On Hankel Determinants of $β$-Spirallike Mappings in Complex Banach Spaces

This paper is devoted to the study of the second Hankel determinant for normalized $β$-spirallike mappings on the unit ball of a complex Banach space. Using the relationship between Banach-space holomorphic mappings and corresponding one-variable $β$-spirallike functions, we first establish sharp estimates for the initial homogeneous expansion coefficients. These coefficient inequalities are subsequently applied to derive a sharp bound for the second Hankel determinant.

math.CV↗

arXiv2026

Sharp coefficient Estimates for the class $\mathcal{S}_{\mathcal{AP}}^{*}$

We investigate several classic coefficient problems for the Ma--Minda starlike subclass $\mathcal{S}_{\mathcal{AP}}^{*}$ defined by the apple-like subordination function $ψ_{\mathcal{AP}}(z)=e^{z}\sqrt{1+z}$. Sharp bounds are derived for the initial inverse logarithmic coefficients $Γ_1$, $Γ_2$, $Γ_3$, and the successive modulus difference $|Γ_2|-|Γ_1|$. In addition, we evaluate the second-order inverse logarithmic Hankel determinant, the generalized Fekete--Szegö functional over all real parameter domains, and the third-order Hermitian--Toeplitz determinant. The corresponding extremal functions are explicitly determined for each functional.

math.CV↗

arXiv2026

Inverse Logarithmic Coefficients and Associated Sharp Estimates for the Apple-Like Convex Subclass $\mathcal{C}_{\mathcal{AP}}$

This paper addresses coefficient problems for the apple-like convex subclass $\mathcal{C}_{\mathcal{AP}}$, defined by the subordination relation $1+zf''(z)/f'(z) \prec e^z\sqrt{1+z}$. We determine sharp bounds for the initial inverse logarithmic coefficients $Γ_1$, $Γ_2$, $Γ_3$, and the consecutive modulus difference $|Γ_2|-|Γ_1|$. We also obtain the sharp upper bound for the second-order inverse logarithmic Hankel determinant, characterize the generalized Fekete--Szegő functional for all real parameters, and compute sharp bounds for the third-order Hermitian--Toeplitz determinant. Extremal functions are given for each case.

math.CV↗

arXiv2026

Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szegö Functionals for the Class $\mathcal{C}_e$

In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w) = \log\!\left(\frac{f^{-1}(w)}{w}\right) = 2\sum_{n=1}^{\infty}Γ_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that \[ |Γ_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |Γ_2|-|Γ_1| \le \frac1{12}, \] and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: \[ \left| H_{2,1}\!\left(F_{f^{-1}}/2\right) \right| \le \frac{85}{12096}. \] Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete--Szegö functional $F_{λ, μ}(f) = \big| a_3(f) - λa_2(f)^2 \big| - μ|a_2(f)|$ within this setting and establish relationships associated with the starlike class $\mathcal{S}^{\ast}_ρ$. The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results.

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

Meromorphic solutions of a certain type of nonlinear differential equation

Our paper focuses on investigating the existence and possible forms of solutions to the nonlinear differential equation \beas f^m+\big(Rf^{(k)}\big)^n=Qe^α,\eeas where where $k$, $m$ and $n$ are three positive integers, $Q$ and $R$ are non-zero rational functions, and $α$ is a polynomial. We examine this equation systematically for all positive integral values of $k$, $m$ and $n$, providing comprehensive conditions under which entire or meromorphic solutions may exist. Our results offer substantial improvements over those of Tang and Liao \cite{TL1} and Han and Lü \cite{HL1}, particularly with respect to the existence criteria and the explicit analytic forms of admissible solutions. These findings contribute meaningfully to the broader study of nonlinear differential polynomials and Fermat-type functional equations in complex analysis.

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↗