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Vasudevarao Allu

Publications and source records attributed to Vasudevarao Allu.

At least 19 recordsLinked to original sources
arXiv2026

Composition operators and Carleson embeddings for the Nevanlinna class of Dirichlet series

This paper systematically investigates the structural, analytical, and operator-theoretic properties of the Nevanlinna class $\mathcal{N}_u$ of Dirichlet series, introduced by Brevig and Perfekt [Adv.\ Math., 2021] and further developed by Guo \textit{et al.}\ [Ann.\ Inst.\ Fourier (Grenoble), 2025]. First, we examine the topological structure of $\mathcal{N}_u$. We then prove a Littlewood--Paley type identity for functions in $\mathcal{N}_u$, establish an equivalent characterization via vertical limit functions, and demonstrate that the interchange of limits in this identity is permissible. In addition, we provide an alternative proof of this identity using potential-theoretic approach. Applying these analytical tools, we study composition operators $C_Φ$ acting on $\mathcal{N}_u$ and characterize those symbols $Φ$ that induce bounded composition operators $C_Φ$. Utilizing Carleson measure techniques on half-planes and infinite-dimensional tori, we characterize the symbols $Φ$ that generate bounded and compact composition operators. In particular, we prove the complete equivalence between the (vanishing) Carleson embedding condition and the geometric (vanishing) Carleson condition on Carleson squares. We conclude with two function-theoretic applications to functions in $\mathcal{N}_u$.

math.FA↗
arXiv2026

Bohr-Type Inequalities for Shifted Disks via Optimal $H^2$-Embeddings

The primary objective of this paper is to systematically generalize this phenomenon by replacing the standard unit disk with a family of nested, internally tangent shifted disks $Ω_γ$ parameterized by $γ\in [0, 1)$, defined by$$Ω_γ= \left\{ z \in \mathbb{C} : \left| z + \fracγ{1 - γ} \right| < \frac{1}{1 - γ},\; γ\in [0, 1) \right\}.$$ By exploiting the geometric characteristics of $Ω_γ$ and evaluating the limiting behavior as $γ\to 1^-$, we establish a novel framework to determine the Bohr radius for the unbounded half-plane $\mathbb{H}_1 = \{z \in \mathbb{C} : \text{Re}(z) < 1\}$. Furthermore, we prove several sharp variations of the Bohr inequality within these domains, including refined and improved formulations for unimodular bounded analytic functions. The results obtained herein not only extend classical radius problems to unbounded regions but also illuminate the delicate interplay between domain deformation and coefficient estimates.

math.CV↗
arXiv2026

Bloch and Landau Type Theorems for Harmonic Mappings with Inhomogeneous Analytic Dilatation

We study Bloch and Landau type theorems for a class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying the inhomogeneous analytic dilatation equation $$ g'(z)=ω(z)h'(z)+ψ(z),$$ where $ω$ and $ψ$ are analytic functions in $\mathbb{D}$ with $\|ω\|_{\infty}\leq k<1$ and $\|ψ\|_{\infty}\leq M$. Here $h$ and $g$ are called analytic and co-analytic part of $f$, respectively. We first establish a Bloch type theorem for certain normalized class of harmonic functions under the condition $k+M<1$. Finally, we obtain two versions of the Landau theorem under additional assumptions: one for bounded harmonic mappings and another under the assumption that the analytic part of a harmonic functions has bounded Bloch seminorm.

math.CV↗
arXiv2026

Landau-type theorems for $K$-quasiregular harmonic mappings

In this paper, our aim is to establish several sharp and improved Landau-type theorems for $K$-quasiregular harmonic mappings $f=h+\overline{g}$ in the unit disk $\Bbb{D} = \{z\in\Bbb{C}: |z|<1\}$. Under various boundedness assumptions on the analytic part $h$ or its derivative, we obtain explicit univalence radii and corresponding schlicht disk radii that significantly improve upon existing estimates in the literature. We also establish new Landau-type theorems under novel hypotheses. We provide examples to illustrate our results, and comprehensive numerical tables present quantitative values of the radii for various parameter choices, demonstrating the effectiveness of our results.

math.CV↗
arXiv2026

On the Third Hankel Determinant for Inverse Coefficients of Starlike Functions: A Bernstein Polynomial Approach

Let $\mathcal{A}$ denote the class of normalized analytic functions $f$ in the open unit disk defined as $ \mathbb{D}:=\{z\in\mathbb{C}:|z|<1\} $ with $f(0)=0$ and $f'(0)=1$. A function $f\in\mathcal{A}$ is said to be starlike if $f(\mathbb{D})$ is starlike domain. By using the Bernstein polynomial method to obtain the required maximum estimate, we establish sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of starlike univalent ({\it i.e.}, one-to-one) functions in the unit disk $\mathbb{D}$.

math.CV↗
arXiv2026

Sharp Estimates of Hankel Determinants for certain classes of convex univalent functions

Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ We examine the properties of the class $\mathcal{C}(φ)$ defined as $\mathcal{C}(φ) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec φ(z):=1+z+ m/n\, \, z^2, \text{ with } 2m \le n,\text{ for } m, n \in \mathbb{N} \right\},$ and compute the sharp second and third Hankel determinants for the functions in $\mathcal{C}(φ)$. Furthermore, we determine the extremal functions for the sharp estimates of the Hankel determinants.

math.CV↗
arXiv2026

Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces

The main aim of this paper is to study multidimensional Bohr radii for holomorphic functions defined in complete Reinhardt domains in $\mathbb{C}^n$ with values in complex Banach spaces. More specifically, for holomorphic functions with values in arbitrary complex Banach spaces, we explore the asymptotic estimates of the classical Bohr radius and arithmetic Bohr radius in the unit ball of $\ell^n_q$ $(1\leq q\leq \infty)$ spaces. Further, we study a mixed version of Bohr radii for vector-valued holomorphic functions and as a consequence we obtain the exact value of mixed arithmetic Bohr radius.

math.CV↗
arXiv2026

On multidimensional Bohr radii for Banach spaces

In this paper, we study a more general version of multidimensional Bohr radii for the holomorphic functions defined on unit ball of $\ell^n_q\,\,(1\leq q\leq \infty)$ spaces with values in arbitrary complex Banach spaces. More precisely, we study the multidimensional Bohr radii for bounded linear operators between complex Banach spaces, primarily motivated by the work of A. Defant, M. Maestre, and U. Schwarting [Adv. Math. 231 (2012), pp. 2837--2857]. We obtain the exact asymptotic estimates of multidimensional Bohr radius for both finite and infinite dimensional Banach spaces. As an application, we find the lower bound of arithmetic Bohr radius.

math.FA↗
arXiv2026

Arithmetic Bohr radius for the Minkowski space

The main aim of this paper is to study the arithmetic Bohr radius for holomophic functions defined on a Reinhardt domain in $\mathbb{C}^n$ with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611--2619]. A part of our study in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space $\ell^n_{q}\, , 1\leq q\leq \infty$. Further, we determine the exact value of a Bohr radius in terms of arithmetric Bohr radius.

math.CV↗
arXiv2026

The second and third Hankel determinants for certain classes of functions

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ In this paper, we consider $\mathcal{S}^*(φ) := \left\{ f \in \mathcal{A} : zf'(z)/f(z) \prec φ(z):=(1+z/2)^2 \right\}$, a subclass of starlike functions and we compute the sharp second and third Hankel determinants for the functions in $\mathcal{S}^*(φ)$. Furthermore, we determine the extremal functions for the coefficient bounds of the functions belonging to $\mathcal{S}^*(φ)$.

math.CV↗
arXiv2026

The second and third Hankel determinants for certain convex subclass of functions

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ In the present paper, we consider $\mathcal{C}(φ) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec φ(z):=(1+z/2)^2 \right\}$, as subclass of convex functions and compute the sharp second and third Hankel determinants for functions in $\mathcal{C}(φ)$.

math.CV↗
arXiv2026

Geometric analysis of a class of harmonic mappings defined by a differential inequality

In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the differential inequality \[ \text{Re}\left((1-α)h'(z) + αz h''(z)\right) > -M + \left|(1-α)g'(z) + αz g''(z)\right|\quad\text{for}\quad z\in\Bbb{D}, \] where $M > 0$, $α\in (0,1]$, and $g'(0) = 0$. This class extends the harmonic analogue of functions with positive real part and offers a unified framework for analyzing their geometric characteristics. We obtain sharp coefficient bounds for both the analytic and co-analytic parts, establish sharp growth bounds, and determine the radii of univalency, starlikeness, and convexity. Furthermore, we show that $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ is closed under convex combinations, and under suitable restrictions on the parameters, it is also closed under convolution. Our findings generalize and extend several known results in the theory of harmonic mappings.

math.CV↗
arXiv2026

The second and third Hankel determinants for starlike MA--Minda subclass associated to quadratic polynomials

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$. In this paper, we discuss the properties of a starlike subclass and compute its second and third Hankel determinants; where the class is defined as $\mathcal{S}^*(φ):=\{f\in\mathcal{A}:{zf'(z)}/{f(z)}\prec φ(z):=1+z+{m}/{n}\,\, z^2,\text{ such that } 2m \le n, \text{ where } m,n\in\mathbb{N}\}.$ Furthermore, we show that the bounds are sharp by determining the extremal functions for the Hankel determinants.

math.CV↗
arXiv2026

Meromorphic Solutions of Difference Equations Involving Borel and Nevanlinna Exceptional Values

The existence of meromorphic solutions to various difference equations has been extensively studied in recent years, the precise functional forms of such solutions -- particularly when the function and its difference operators share values -- remain largely unexplored. This paper addresses this research gap by investigating the sharing value problem between finite-order meromorphic functions $f(z)$ and their linear difference operators $L_{c}^{n}(f)$. Specifically, we consider functions having Borel or Nevanlinna exceptional values. We prove not only the existence but also characterize the explicit general meromorphic solutions to the difference equation $L_{c}^{n}(f)\equiv Af$ for $A\in\mathbb{C}\backslash\{0\}$. To validate our main results and demonstrate the necessity of our conditions, we provide several concrete examples. Furthermore, we investigate the existence and nature of both rational and transcendental meromorphic solutions for the second-order difference equation $b_{2}(z)f(z+2η)+b_{1}(z)f(z+η)+b_{0}(z)f(z)=b(z)$ with polynomial coefficients.

math.CV↗
arXiv2026

Bohr phenomenon for certain integral operators and transforms in complex Banach spaces

In this paper, we investigate several Bohr radii associated with the Cesáro operator, Bernardi integral operator, $β$-Cesáro operator, and discrete Fourier transform, all defined on a set of holomorphic mappings from the unit ball of a complex Banach space into the closure of the unit polydisc $\mathbb{D}^n$ within the space $\mathbb{C}^n$.

math.CV↗
arXiv2026

Pre-Schwarzian and Schwarzian norm estimates for certain classes of analytic and harmonic mappings

Let $\mathcal{A}$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $f(0)=f'(0)-1=0$. In this paper, we introduce a new subclass $\mathcal{C}_θ(γ)$ of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation \[ \textrm{Re}\left(e^{iθ}\left(1+\frac{zf''(z)}{f'(z)}\right)\right)<\left(1+\fracγ{2}\right)\cosθ,~ z\in\mathbb{D},~ γ>0, ~\text{and}~|θ|<\fracπ{2},\] and investigate the Schwarzian derivative and Schwarzian norm for functions $f$ belonging to the class $\mathcal{C}_θ(γ)$. We establish sharp estimates for the Schwarzian norm $\|S_f\|$ of functions $f$ in the class $\mathcal{C}_θ(γ)$ and derive univalence criteria using both pre-Schwarzian and Schwarzian norm estimates. We also introduce a corresponding harmonic class $\mathcal{HC}_θ(γ)$ consisting of mappings $f = h+\overline{g}$ with $h\in\mathcal{C}_θ(γ)$ and dilatation $ω=g'/h'\in\mathrm{Aut}(\mathbb{D})$. For this harmonic class, we derive bounds for both the pre-Schwarzian and Schwarzian norms, including sharp results in special cases.

math.CV↗
arXiv2026

Bohr phenomenon for analytic and harmonic mappings on shifted disks

The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on the shifted disks \[ Ω_γ=\left\{z\in\mathbb{C}:\left|z+\fracγ{1-γ}\right|<\frac{1}{1-γ}\right\}\quad\text{for}\quadγ\in[0,1).\]

math.CV↗