arXiv subjects

Molla Basir Ahamed

Publications and source records attributed to Molla Basir Ahamed.

At least 37 records · Page 2Linked to original sources

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

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↗

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

Value Distribution and Picard-type Theorems for Total Differential Polynomials in $\mathbb{C}^n$

This paper investigates the value distribution and growth properties of linear total differential polynomials $\mathcal{L}_k[D]f$ for meromorphic functions in several complex variables $\mathbb{C}^n$. By extending the classical Milloux inequality to the framework of total derivatives, we derive a series of fundamental growth estimates for the Nevanlinna characteristic function $T(r, \mathcal{L}_k[D]f)$. We address the value-sharing problem for meromorphic functions $f$ and $g$ sharing values with their differential polynomials. Under the condition $2δ(0,f)+(k+4)Θ(\infty,f)>k+5$, we establish that $\frac{\mathcal{L}_k[D]f-1}{\mathcal{L}_k[D]g-1}$ is a non-zero constant for non-transcendental meromorphic functions. Furthermore, we provide an affirmative answer to several Picard-type inquiries, proving that if an entire function $f$ in $\mathbb{C}^n$ omits a value $a$ while its linear total differential polynomial $\mathcal{L}_k[D]f$ omits a non-zero value $b$, then $f$ must be constant. Our results generalize and extend several existing uniqueness and Picard-type theorems from the classical one-variable setting to the higher-dimensional complex space $\mathbb{C}^n$.

math.CV↗

arXiv2026

Geometric subfamily of locally univalent functions, Blaschke products and quasidisk

In this article, we consider the family $\mathcal{F}(α)$ defined for $α\in (0, 3]$ by \begin{align*} {\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \fracα{2} \quad \text{for } z \in \mathbb{D}. \end{align*} Our primary objective is to show that this family possesses significant geometric and analytic properties, including connections with Blaschke products and the Schwarzian derivative, as well as its sharp bounds. Furthermore, we prove that if $f \in \mathcal{F}(α)$, then the image $f(\mathbb{D})$ is a quasidisk. We also show that if $f \in \mathcal{F}(α)$, then $\|S_f\| = 2α(2-α)$. Moreover, we establish the sharp estimate $\|P_{f}\| \leq 2α+1$ for the pre-Schwarzian derivative of harmonic mappings $f = h + \bar{g} \in \mathcal{F}_{\mathcal{H}}(α)$, where the analytic part $h$ belongs to $\mathcal{F}(α)$.

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

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↗

arXiv2025

Growth, Distortion, Pre-Schwarzian and Schwarzian norm estimates for Generalized Robertson class

This paper investigates the geometric properties of functions within the generalized Robertson class which consists of alpha-starlike functions of order beta. The study's significance lies in providing a deeper understanding of the univalence and geometric behavior of these functions, which are fundamental in complex analysis and geometric function theory. The primary objective is to derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in this class. These bounds are expressed in terms of the initial coefficient of the function, specifically focusing on the important case where this initial coefficient is zero. Additionally, the paper establishes sharp distortion and growth theorems for the functions belonging to this generalized class. Finally, the research addresses the radius problem for this function class by determining the sharp radius of concavity and the sharp radius of convexity.

math.CV↗

arXiv2025

Multidimensional analogues of the improved Bohr's inequality

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{P}Δ(0;1_n)$. We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables by replacing the constant term with the absolute value of the function and the square of the absolute value of the function, respectively. All the results are shown to be sharp.

math.CV↗

arXiv2025

Bohr's theorem for Cesáro operator and certain integral transforms over octonions

In this paper, we first establish the Bohr's theorem for Cesáro operator defined for $f\in \mathcal{SRB}(\mathbb{B})$ of slice regular functions in the open unit ball $\mathbb{B}$ of the largest alternative division algebras of octonions $\mathbb{O}$, such that $|f(x)| \leq 1$ for all $x \in \mathbb{B}$. Next, we establish Bohr type inequalities for Bernardi operator for the functions $f\in \mathcal{SRB}(\mathbb{B})$, and with the help of this, we obtain Bohr type inequality for Libera operator and Alexander operator. Finally, we obtain Bohr-type inequalities for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms for $f\in \mathcal{SRB}(\mathbb{B}).$ All the results are proven to be sharp.

math.CV↗

arXiv2025

Pre-Schwarzian and Schwarzian norm Estimates for Robertson class

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-π/2<α<π/2$, let $\mathcal{S}_α$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{iα}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)\}>0$ for $z\in\mathbb{D}$. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass $\mathcal{S}_α$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$.

math.CV↗

arXiv2025

Differential Operators, Multiple Schwarz Functions, and the Bohr Radius of Stable Harmonic Maps

In this paper, we study the Bohr phenomenon for differential operators $D$ and $\mathscr{D}$ of stable harmonic mappings involving multiple Schwarz functions in $\mathcal{B}_n$, using distance formulations. By constructing suitable combinations of multiple Schwarz functions, we establish sharp and improved Bohr-type inequalities for these mappings. The corresponding Bohr radii are also determined for certain subclasses of stable harmonic functions and their associated differential operators. Moreover, Bohr-Rogosinski-type inequalities are derived, which highlights the influence of multiple Schwarz functions on the geometric properties of stable harmonic mappings. All the radii are determined, and we prove that each one is the best possible.

math.CV↗

arXiv2025

Bohr phenomena for slice regular functions over Quaternions

Slice regular functions are a generalization of holomorphic functions to the setting of quaternions (and more generally, Clifford algebras). In this paper, we first establish the Bohr inequality for slice starlike functions and slice close-to-convex functions over quaternions $\mathbb{H}$. Next, we present a generalization of the Bohr inequality, and improved versions of the Bohr inequality for slice regular functions on the open unit ball $\mathbb{B}$ of $\mathbb{H}$. Finally, we provide a refined version of the Bohr inequality for slice regular functions $f$ on $\mathbb{B}$ such that $ {\rm Re}(f(q)) \leq 1 $ for all $q \in \mathbb{B}$. All the results are demonstrated to be sharp.

math.CV↗

arXiv2025

Quasiconformal Extension of Meromorphic Functions with High-Order Poles

In this paper, we study the class ${Σ^{(m)}(p)}$ of meromorphic univalent functions $f$ in $\mathbb{D}$ with a pole of order ${m \geq 1}$ at $p \in (0,1)$, admitting a $k$-quasiconformal extension ($0 \leq k < 1$) to $\widehat{\mathbb{C}}$. Using the Area Theorem and convolution methods, we establish a generalized area-type inequality and derive explicit analytic membership conditions for $Σ^{(m)}(p)$. We also extend the convolution theorem to a modified Hadamard product of $m$ functions, $f_j \in Σ^{(m)}_{k_j}(p)$, determining sufficient conditions for the product to be in ${Σ^{(m)}_α(p)}$, with $α$ defined by $k_j$ and $p$. Further results include a sufficient criterion for sense-preserving harmonic mappings on convex domains to admit quasiconformal extensions, and the sharp Schwarzian norm for $f \in Σ_k(p)$ (the $m=1$ case). These findings improve upon existing results of [{\em Proc. Amer. Math. Soc.}, {144}(6) (2016), 2593--2601].

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

On Coefficient problems for classes $\mathcal{S}_e^{\ast}$ and $\mathcal{C}_e$

Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study,we focus on the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ of starlike and convex functions, respectively, \begin{align*} \mathcal{S}_e^\ast := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec e^z, \ z \in \mathbb{D} \right\}, \end{align*} and \begin{align*} \mathcal{C}_e := \left\{ f \in \mathcal{S} : 1 + \frac{z f''(z)}{f'(z)} \prec e^z, \ z \in \mathbb{D} \right\}. \end{align*} This paper investigates the sharp bounds of the logarithmic coefficients and the Hermitian-Toeplitz determinant of these coefficients for the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$. Additionally, we examine the generalized Zalcman conjecture and the generalized Fekete-Szegö inequality for these classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ and show that the inequalities are sharp.

math.CV↗

arXiv2025

A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk

We investigate improved forms of the Bohr inequality, using the quantity $S_r/π$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of $\mathbb{D}_r$. We then generalize the inequality for harmonic mappings ($\mathcal{P}^0_{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(α)$ of the form $f = h + \overline{g}$) by introducing a sequence $\{φ_n(r)\}_{n=0}^\infty$ of differentiable, increasing functions on $[0, 1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp.

math.CV↗

arXiv2025

Bohr Phenomenon for $K$-quasiconformal Harmonic Mappings Involving One Parameter

In this article, we study Bohr-type inequalities involving a parameter or convex combinations for $K$-quasiconformal, sense-preserving harmonic mappings in $\mathbb{D}$, where the analytic part is subordinate to a convex function. Moreover, we establish similar inequalities when the subordinating function is chosen from the class of concave univalent functions with pole $p$, as well as from the family of concave univalent functions with opening angle $πα$. The results generalize several existing results.

math.CV↗