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Vasudevarao Allu

Publications and source records attributed to Vasudevarao Allu.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2026

On Landau-Type Theorems for Poly-Analytic Functions

In this paper, we establish three Landau-type theorems for certain bounded poly-analytic functions, which generalize the corresponding result for bi-analytic functions given by Liu and Ponnusamy [Canad. Math. Bull. 67(1): 2024, 152-165]. Further, we prove three bi-Lipschitz theorems for these subclasses of poly-analytic functions.

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↗
arXiv2025

Bohr inequalities for holomorphic mappings in higher-dimensional complex Banach spaces

In this paper, we investigates the Bohr phenomenon for holomorphic mappings $F$ from the unit ball $\mathbb{B}_X$ of a complex Banach space $X$ into the closure of the unit polydisc $\mathbb{D}^m$ within the space $\mathbb{C}^m$. First, we prove an improved Bohr inequality involving the squared norms of the mapping and its homogeneous expansions. Second, we derive a refined Bohr inequality that incorporates a combination of the coefficient norms and their squares. Finally, we obtain a refined Bohr inequality for compositions $F\circ ν$, where $ν$ is a Schwarz mapping with a zero of order $k$ at the origin. For each result, we demonstrate that the derived Bohr radius is sharp.

math.CV↗
arXiv2025

Rough isometry between Gromov hyperbolic spaces and unbounded uniformization

In a recent paper, Zhou, Ponnusamy, and Rasila [Math. Nachr. (2025)] have established that the conformal deformations, with parameter $ε>0$, of a Gromov hyperbolic space via Busemann functions are uniform spaces for sufficiently small $ε$. In this paper, we demonstrate that if two proper, roughly starlike Gromov hyperbolic spaces are roughly isometric, then the uniformity of their conformal deformations is a simultaneous property; that is, either both are uniform spaces or neither is. Our results provide a counterpart to the work of Shanmugalingam and Lindquist [Ann. Fenn. Math. (2021)].

math.MG↗
arXiv2025

On multidimensional Bohr radius of finite dimensional Banach spaces

In this paper, we improve the lower estimate of multidimensional Bohr radius for unit ball of $\ell^n_q$-spaces ($1\leq q\leq \infty$) for bounded holomorphic functions with values in finite dimensional complex Banach spaces. The new estimate provides the improved lower bound for the Bohr radius which was previously given by Defant, Maestre, and Schwarting [Adv. Math. 231 (2012), 2837--2857].

math.CV↗
arXiv2025

Generalized complex symmetric composition operators with applications

We characterize the weighted composition-differentiation operators $D_{\mfn,ψ,φ}$ acting on $\mathcal{H}_γ(\mathbb{D}^d)$ over the polydisk $\mathbb{D}^d$ which are complex symmetric with respect to the conjugation $\mathcal{J}$. We obtain necessary and sufficient conditions for $D_{\mfn,ψ,φ}$ to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators $M_{n, ψ, φ}=\displaystyle\sum_{j=1}^{n}a_jD_{j,ψ_j, φ},$ (where $a_j\in \mathbb{C}$ for $j=1, 2, \dots, n$) on the reproducing kernel Hilbert space $\mathcal{H}_γ(\mathbb{D})$ of analytic functions on the unit disk $\mathbb{D}$ with respect to a weighted composition conjugation $C_{μ, ξ}$. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on $\mathcal{H}_γ(\mathbb{D})$ are investigated. Additionally, geometrical interpretations have also been employed.

math.FA↗
arXiv2025

An estimation of the pre-Schwarzian norm for certain classes of analytic functions

The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions $f$ in the class $\mathcal{S}^*(φ)$ and $\mathcal{C}(φ)$ when $φ(z)=1/(1-z)^s$ with $0<s\leq 1$ and $φ(z)=(1+sz)^2$ with $0<s\leq 1/\sqrt{2}$, where $\mathcal{S}^*(φ)$ and $\mathcal{C}(φ)$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $φ$, respectively.

math.CV↗
arXiv2025

The Bohr's Phenomenon involving multiple Schwarz functions

The primary objective of this paper is to establish several sharp versions of Bohr inequalities for bounded analytic functions in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$ involving multiple Schwarz functions. Moreover, we obtain an improved version of the classical Rogosinski inequality for analytic functions in $\mathbb{D}$.

math.CV↗
arXiv2024

The Bohr's phenomenon for certain K-quasiconformal harmonic mappings

The primary objective of this paper is to establish several sharp versions of improved Bohr inequalities, refined Bohr inequalities, and Bohr-Rogosinski inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\Bbb{D}:=\{z\in\mathbb{C}: |z|<1\}$, where $f$ and $g$ are analytic functions in $\mathbb{D}$.

math.CV↗
arXiv2024

Visible quasihyperbolic geodesics

In this paper, motivated by the work of Bonk, Heinonen, and Koskela (Asterisque, 2001), we consider the problem of the equivalence of the Gromov boundary and Euclidean boundary. Our strategy to study this problem comes from the recent work of Bharali and Zimmer (Adv. Math., 2017) and Bracci, Nikolov, and Thomas (Math. Z., 2021). We present the concept of a quaihyperbolic visibility domain (QH-visibility domain) for domains that meet the visibility property in relation to the quasihyperbolic metric. By utilizing this visibility property, we offer a comprehensive solution to this problem. Indeed, we prove that such domains are precisely the QH-visibility domains that have no geodesic loops in the Euclidean closure. Furthermore, we establish a general criterion for a domain to be the QH-visibility domain. Using this criterion, one can determine that uniform domains, John domains, and domains that satisfy quasihyperbolic boundary conditions are QH-visibility domains. We also compare the visibility of hyperbolic and quasihyperbolic metrics for planar hyperbolic domains. As an application of the visibility property, we study the homeomorphic extension of quasiconformal maps. Moreover, we also study the QH-visibility of unbounded domains in $\mathbb{R}^n$. Finally, we present a few examples of QH-visibility domains that are not John domains or QHBC domains.

math.CV↗
arXiv2024

Operator valued analogues of multidimensional refined Bohr's inequalities

Let $\mathcal{B}(\mathcal{H})$ denote the Banach algebra of all bounded linear operators acting on complex Hilbert spaces $\mathcal{H}$. In this paper, we first establish several sharply refined versions of Bohr's inequality analogues with operator valued functions in the class $\mathcal{B}(\mathbb{D}, \mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}$ to $\mathcal{B}(\mathcal{H})$ with $\sup_{|z|<1}\Vert f(z)\leq 1$ by utilizing a certain power of the function's norm. Additionally, we establish several multidimensional analogues of refined Bohr's inequalities by using operator valued functions in the complete circular domain $Ω\subset\mathbb{C}^n$. All of the results are sharp.

math.CV↗
arXiv2024

Improved Bohr-type Inequalities for the Cesaro Operator

In this paper, we derive the sharp improved versions of Bohr-type inequalities for the Cesáro operator acting on the class of bounded analytic functions defined on the unit disk $\D=\left\{z\in\C:\left|z\right|<1\right\}$. In order to achieve these results, we utilize the principle of substituting the initial coefficients of the majorant series with the absolute values of the Cesáro operator associated with a bounded analytic function defined on $\D$ and its derivative, as well as for the Schwarz function.

math.CV↗
arXiv2024

Uniformization of intrinsic Gromov hyperbolic spaces with Busemann functions

For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any complete intrinsic hyperbolic space with atleast two points in the Gromov boundary can be uniformized by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification of the Gromov boundary of $X$ with the metric boundary of the deformed space.

math.CV↗
arXiv2024

Uniformization of intrinsic Gromov hyperbolic spaces

The purpose of this paper is to provide a uniformization procedure for Gromov hyperbolic spaces, which need not be geodesic or proper. We prove that the conformal deformation of a Gromov hyperbolic space is a bounded uniform space. Further, we show that there is a natural quasi-isometry between the Gromov boundary and the metric boundary of the deformed space. Our main results are a generalization of the results of Bonk, Heninonen, and Koskela [Proposition 4.5, Proposition 4.13, Astérisque 270 (2001)].

math.MG↗