arXiv subjects

Sanju Mandal

Publications and source records attributed to Sanju Mandal.

16 recordsLinked to original sources

arXiv2026

Sharp inequalities for Logarithmic Coefficients for Certain Classes of Univalent Functions

Let $\mathcal{S}$ denote the class of functions $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ that are analytic and univalent in the open unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$. In this paper, we determine the sharp bounds of the Toeplitz determinants whose entries are the logarithmic coefficients of $f \in \mathcal{S}$. Furthermore, we investigate the corresponding Toeplitz determinants for the logarithmic coefficients of the associated inverse functions. These sharp bounds are established for functions belonging to several well-known subclasses of $\mathcal{S}$, namely, the classes $\mathcal{S}^*(α)$ of starlike functions of order $α$, $\mathcal{C}(α)$ of convex functions of order $α$, $\mathcal{S}^*_α$ and $\mathcal{C}_α$ of strongly starlike and strongly convex functions of order $α$, and $\mathcal{R}(α)$ of functions with bounded turning. As special cases of our main results, we obtain the exact bounds of these determinants for the classical classes of starlike, convex, and bounded turning functions.

math.CV↗

arXiv2026

Sharp Coefficient and Inverse Problems for Holomorphic Semigroup Generators

In this paper, we study extremal problems for coefficient functionals associated with a distinguished subclass of holomorphic semigroup generators, denoted by $\mathcal{A}_β$ ($0 \le β\le 1$), defined on the unit disk $\mathbb{D}$. This class forms a natural filtration of the class $\mathcal{G}_0$ of infinitesimal generators, with the class $\mathcal{R}$ of functions of bounded turning arising as its minimal element. We obtain sharp bounds for the initial logarithmic coefficients $γ_n$, the inverse coefficients $A_n$, and the logarithmic inverse coefficients $Γ_n$ for $n = 1,2,3$ within the class $\mathcal{A}_β$. In addition, we address the successive coefficient problem by deriving sharp upper and lower estimates for the differences $|A_{n+1}| - |A_n|$ for $n = 1,2$. Furthermore, we establish sharp bounds for a generalized Fekete--Szegö functional in the class $\mathcal{R}$. The extremality of the obtained results is demonstrated by explicit constructions, including functions related to Gauss hypergeometric functions. Our results unify and extend several earlier contributions in geometric function theory and reveal a structural connection between coefficient problems for functions of bounded turning and the dynamics of holomorphic semigroup generators.

math.CV↗

arXiv2026

Coefficient estimates and Bohr phenomenon for analytic functions involving semigroup generator

This article investigates the Bohr phenomenon and sharp coefficient problems for the class $\mathcal{A}_β$, a subclass of analytic self-maps of the unit disk with the holomorphic generators of one-parameter continuous semigroups. By integrating concepts from complex dynamics and geometric function theory, we derive sharp improvements to the classical Bohr radius by incorporating multiple Schwarz functions and certain functional expressions. We establish generalized versions of the Bohr and Bohr-Rogosinski inequalities and determine the best possible radii for these refinements. Furthermore, we provide a sharp solution to the classical Fekete-Szegö problem for the class $\mathcal{A}_β$ by obtaining sharp bounds for the functional $|a_3 - μa_2^2|$ for all real values of $μ$. Additionally, we derive sharp inequalities for the moduli of differences of logarithmic coefficients for both the functions and their inverses in this class.

math.CV↗

arXiv2026

Sharp Estimates of Logarithmic Coefficients for a Certain Class of Starlike Functions

In this article, we investigate the extremal properties of logarithmic coefficients for the class $\mathcal{S}_{ch}^*$ of starlike functions associated with the hyperbolic cosine function. We establish the sharp upper bounds for the initial logarithmic coefficients $γ_n$ for $n=1, 2, 3$, and determine the precise bound for the second Hankel determinant $H_{2,1}(F_f/2)$ within this class. Furthermore, we extend our analysis to the inverse functions, deriving sharp estimates for the logarithmic inverse coefficients and the corresponding second Hankel determinant $|H_{2,1}(F_{f^{-1}}/2)|$. Additionally, we provide sharp bounds for the moduli differences of both logarithmic and inverse logarithmic coefficients. The sharpness of all obtained inequalities is verified through the construction of specific extremal functions.

math.CV↗

arXiv2026

Normality Criteria for Differential Monomials and the Sharpness of Lappan-type Theorems

A fundamental result of Lappan [Comment. Math. Helv. \textbf{49} (1974), 492-495.] states that a meromorphic function $f$ in the unit disk $\mathbb{D}$ is normal if and only if its spherical derivative is bounded on a five-point subset $E \subset \mathbb{C}$. In this paper, we establish new normality criteria that bridge this classical result with contemporary trends in value distribution theory. We demonstrate that the cardinality of the set $E$ can be reduced from five to as few as three, provided that the spherical derivatives of the function and its successive derivatives $f, f', \dots, f^{(k-1)}$ are bounded on the pre-image of $E$. This shift reveals that analytic data from higher-order derivatives can effectively compensate for a reduction in geometric information from the target set. Furthermore, we extend the Pang-Zalcman theorem to a general class of differential monomials $M[f]$. We prove that if $(M[f])^{\#}$ is bounded on the set of $a$-points ($a \neq 0$), the family $\mathcal{F}$ is normal, provided the degree $d_M$ satisfies a specific sharp threshold relative to the weight $D_M$ and order $k$. These results offer a refined perspective on the natural boundaries of normality and generalize several established findings in the field.

math.CV↗

arXiv2025

Existence of Solutions to Systems of General Quadratic Functional Equations in $\mathbb{C}^n$

The main objective of this study is to investigate the existence and forms of solutions of systems of general quadratic functional equations in $\mathbb{C}^n$. By utilizing Nevanlinna theory in $\mathbb{C}^n$, we explore the existence and form of solutions for the several systems of general quadratic difference and partial differential-difference equations of the form $af^2 + 2αfg + bg^2 + 2βf + 2γg + C=0$, where $f$ and $g$ are non-constant meromorphic functions in $\mathbb{C}^n$. The obtained results in this article are improvements and generalizations of several results from [\textit{RACSAM}, \textbf{116}(8) (2022)]. Furthermore, appropriate remarks and illustrative examples are provided to validate and demonstrate the applicability of the obtained results concerning the existence and forms of solutions for such systems of equations.

math.CV↗

arXiv2025

Normality criterion for logharmonic mapping

In this paper, we present several necessary and sufficient conditions for a logharmonic mapping to be normal i.e., we establish Marty's criterion, Zalcman-Pang lemma and the Lohwater-Pommerenke theorem for logharmonic mappings, along with an application of the Zalcman-Pang lemma.

math.CV↗

arXiv2025

Results on Logarithmic Coefficients for the Class of Bounded Turning Functions

It is crucial to explore the sharp bounds of logarithmic coefficients and the Hankel determinant involving logarithmic coefficients as part of coefficient problems in various function classes. Our primary objective in this study is to determine the sharp bounds for logarithmic coefficients as well as logarithmic inverse coefficients of bounded analytic functions associated with a bean-shaped domain in the class $\mathcal{BT_\mathfrak{B}}$. For this class, we also establish the sharp bounds for the second Hankel determinant involving logarithmic coefficients as well as logarithmic inverse coefficients. In addition, we establish sharp bounds for the generalized Zalcman conjecture inequality and the moduli differences of logarithmic coefficients for the class $\mathcal{BT_\mathfrak{B}}$.

math.CV↗

arXiv2024

Coefficient bounds for starlike functions associated with Gregory coefficients

It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions $ f $ which are holomorphic in the open unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ of the form \begin{align*} f(z)=z+\sum_{n=2}^{\infty}a_nz^n\; \mbox{for}\; z\in\mathbb{D} \end{align*} and suppose that \begin{align*} F_{f}(z):=\log\dfrac{f(z)}{z}=2\sum_{n=1}^{\infty}γ_{n}(f)z^n, \;\; z\in\mathbb{D},\;\;\log 1:=0, \end{align*} where $ γ_{n}(f) $ is the logarithmic coefficients. The second Hankel determinant of logarithmic coefficients $H_{2,1}(F_{f}/2)$ is defined as: $H_{2,1}(F_{f}/2) :=γ_{1}γ_{3} -γ^2_{2}$, where $γ_1, γ_2,$ and $γ_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we first establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/64$ with logarithmic coefficients for the classes of starlike functions associated with Gregory coefficients. In addition, we establish the sharpness of Fekete-Szeg$ \ddot{o} $ inequality, Zalcman functional and generalized Zalcman functional for the class starlike functions associated with Gregory coefficients.

math.CV↗

arXiv2023

Hankel and Toeplitz determinants of logarithmic coefficients of Inverse functions for certain classes of univalent functions

The Hankel and Toeplitz determinants $H_{2,1}(F_{f^{-1}}/2)$ and $T_{2,1}(F_{f^{-1}}/2)$ are defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix} \;\;\mbox{and} \;\; T_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_1 \end{vmatrix} \end{align*} where $Γ_1, Γ_2,$ and $Γ_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 1/4$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$, $|T_{2,1}(F_{f^{-1}}/2)|\leq 5/16$ and $|T_{2,1}(F_{f^{-1}}/2)|\leq 145/2304$ for the logarithmic coefficients of inverse functions for the classes starlike functions and convex functions with respect to symmetric points. In addition, our findings are substantiated further through the incorporation of illustrative examples, which support the strict inequality and lend credence to our conclusions.

math.CV↗

arXiv2023

Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix}=Γ_1Γ_3-Γ^2_2, \end{align*} where $Γ_1, Γ_2,$ and $Γ_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 19/288$, $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/144$, and $|H_{2,1}(F_{f^{-1}}/2)| \leq 1/36$ for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order $1/2$, respectively.

math.CV↗

arXiv2023

Entire Solutions for quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $

In this paper, utilizing Nevanlinna theory, we study existence and forms of the entire solutions $ f $ of the quadratic trinomial-type partial differential-difference equations in $ \mathbb{C}^n $ \begin{align*} a\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) f(z + c) + b f(z + c)^2 = e^{g(z)} \end{align*} and \begin{align*} a\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right)^2 & + 2 ω\left(α\dfrac{\partial f(z)}{\partial z_i} + β\dfrac{\partial f(z)}{\partial z_j}\right) Δ_cf(z) + b [Δ_cf(z)]^2 = e^{g(z)}, \end{align*} where $ a, ω, b\in\mathbb{C} $, $ g $ is a polynomial in $ \mathbb{C}^n $ and $ Δ_cf(z)=f(z+c)-f(z) $. The main results of the paper improve several existence results in $ \mathbb{C}^n $ for integer $ n\geq 2 $ and $ 1\leq i<j\leq n $ and their corollaries of the paper are an extension of the results of Xu \emph{et al. } for trinomial equation with arbitrary coefficient in $ \mathbb{C}^2 $. Moreover, examples are exhibited to validate the conclusion of the main results.

math.CV↗

arXiv2023

Characterization of solutions of refined Fermat-type functional equations in $ \mathbb{C}^n $

The main purpose of this article is concerned with the existence and the precise forms of the transcendental solutions of several refined versions of Fermat-type functional equations with polynomial coefficients in several complex variables by utilizing the Nevanlinna theory of meromorphic functions in several complex variables. In fact, we investigate the existence and forms of the transcendental solutions of non-linear quadratic trinomial equations in $\mathbb{C}^n$. As a consequence of our result, we show that solutions of binomial equations in $\mathbb{C}^n$ can be explored and this exploration broaden the scope of the study of functional equations in $ \mathbb{C}^n $. The results we obtained are improvements over certain recent findings, noted as remarks. In addition, some examples relevant to the content of the paper have been exhibited in support of the validation of each results. Further, we discuss situations when solutions of such equations does not exist.

math.CV↗

arXiv2023

Second Hankel determinant of Logarithmic coefficients for Starlike and Convex functions associated with lune

The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as: \begin{align*} H_{2,1}(F_{f}/2):= \begin{vmatrix} γ_1 & γ_2 γ_2 & γ_3 \end{vmatrix}, \end{align*} where $γ_1, γ_2,$ and $γ_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/16$ and $|H_{2,1}(F_{f}/2)| \leq 23/3264$ for the logarithmic coefficients of starlike and convex functions associated with lune.

math.CV↗

arXiv2023

Sharp bounds for second Hankel determinant of logarithmic coefficients for certain classes of univalent functions

The Hankel determinant $H_{2,2}(F_{f}/2)$ is defined as: \begin{align*} H_{2,2}(F_{f}/2):= \begin{vmatrix} γ_2 & γ_3 γ_3 & γ_4 \end{vmatrix}, \end{align*} where $γ_2, γ_3,$ and $γ_4$ are the second, third, and fourth logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,2}(F_{f}/2)|\leq (1272 + 113\sqrt{678})/32856$ and $|H_{2,2}(F_{f}/2)| \leq 13/1080$ for the logarithmic coefficients of starlike and convex functions with respect to symmetric points. Moreover, we provide examples that demonstrate the strict inequality holds.

math.CV↗

arXiv2022

Transcendental entire solutions of several general quadratic type PDEs and PDDEs in $ \mathbb{C}^2 $

The functional equations $ f^2+g^2=1 $ and $ f^2+2αfg+g^2=1 $ are respectively called Fermat-type binomial and trinomial equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. Utilizing Nevanlinna's theory for several complex variables, in this paper, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form $ af^2+2αfg+b g^2+2βf+2γg+C=0 $ in $ \mathbb{C}^2 $. Consequently, we obtain certain corollaries of the main results of this paper concerning binomial equations which generalize many results in [\textit{Rocky Mountain J. Math.} \textbf{51}(6) (2021), 2217-2235] in the sense of arbitrary coefficients.

math.CV↗