Searcharxiv⌕ Search

arXiv subjects

Gopal Datt

Publications and source records attributed to Gopal Datt.

At least 19 recordsLinked to original sources
arXiv2026

Extension of Lohwater-Pommerenke's Theorem for strongly-normal Maps

We introduce strong normality for holomorphic curves and logharmonic mappings, extending classical normality concepts. We establish an extension of the rescaling characterization due to Lohwater and Pommerenke for not strongly-normal maps. In addition, we also study the Bloch mappings, little-Bloch mappings and prove Zalcman-Pang type rescaling results for them. The framework is further extended to strongly $\varphi$-normal mappings, yielding a unified treatment across these settings.

math.CV↗
arXiv2026

Avoidance criteria for normality of quasiregular mappings

Peter Lappan in [9] proved that for each $n\in \mathbb{N}=\{1,2,3,\dots\}$, let $f_{1,n}, f_{2,n}$ and $f_{3,n}$ be three continuous functions on $\mathbb{D}:=\{z\in \mathbb{C} : |z| < 1\}$ such that for each $j=1,2,3,$ the sequence $(f_{j,n})$ converges locally uniformly to a function $f_j$ on $\mathbb{D}$. Suppose that the three functions $f_1, f_2,$ and $f_3$ avoid each other on $\mathbb{D}$. Let $\mathcal{F} =(g_n)$ be a sequence of meromorphic functions in $\mathbb{D}$ with the property that for each $n$, the four functions $g_n, f_{1,n}, f_{2,n},$ and $f_{3,n}$ avoid each other, then $\mathcal{F}$ is normal. We present here an analogue of this result in the setting of quasiregular mappings. We also obtain analogues of a few other results by Peter Lappan in [9] to quasiregular setting in the Euclidean space $\mathbb{R}^n$ for normal families and normal quasiregular mappings.

math.CV↗
arXiv2026

Avoidance Criteria for Normal Holomorphic Curves on Complex Projective Space

We establish an avoidance criterion for families of holomorphic curves from the unit disk in complex plane to the complex projective space that omit sufficiently many moving hypersurfaces in pointwise general position. Furthermore, we study families of holomorphic curves that share hyperplanes and derive analogous normality conditions in this context.

math.CV↗
arXiv2025

A Function-Sharing Criterion for Normal Functions

In this paper, we present a function-sharing criterion for the normality of meromorphic functions. Let $f$ be a meromorphic function in the unit disc $\mathbb{D}\subset \mathbb{C}$, $\psi_1$, $\psi_2$, and $\psi_3$ be three meromorphic functions in the unit disc $\mathbb{D}$, continuous on $ \partial{\mathbb{D}}:=\{z\in\mathbb{C}\,:\,|z|=1\}$, such that $\psi_i(z)\neq\psi_j(z)$ $(1\leq i<j\leq 3)$ $\partial\mathbb{D}$. We prove that, if $\psi_1$, $\psi_2$, and $\psi_3$ share the function $f$ on $\mathbb{D}$, then $f$ is normal. Building upon this, we further establish an additional criterion for normal functions.

math.CV↗
arXiv2024

Dynamics of complex harmonic mappings

This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a holomorphic function. The existence of harmonic mapping with a wandering domain is also shown.

math.CV↗
arXiv2024

Normal Families of Holomorphic Curves and Sharing of Moving Hyperplanes Wandering on $\mathbb{P}^n$

In this paper, we extend a result of Schwick concerning normality and sharing values in one complex variable for families of holomorphic curves taking values in $\mathbb{P}^n$. We consider wandering moving hyperplanes (i.e., depending on the respective holomorphic curve in the family under consideration), and establish a sufficient condition of normality concerning shared hyperplanes.

math.CV↗
arXiv2024

Lappan's five-point theorem for {\phi}-Normal Harmonic Mappings

A harmonic mapping $f=h+\overline{g}$ in $\mathbb{D}$ is $\varphi$-normal if $f^{\#}(z)=\mathcal{O}(|\varphi(z)|), \text{ as } |z|\to 1^-,$ where $f^{\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish several sufficient conditions for harmonic mappings to be $\varphi$-normal. We also extend the five-point theorem of Lappan for $\varphi$-normal harmonic mappings.

math.CV↗
arXiv2024

Ascent and Descent of Weighted Composition Operators on Lorentz spaces

The aim of this article is to detect the ascent and descent of weighted composition operators on Lorentz spaces. We investigate the conditions on the measurable transformation $T$ and the complex-valued measurable function $u$ defined on measure space $(X, \mathcal{A}, \mu)$ that cause the weighted composition operators on Lorentz space $L(p, q)$, $1 < p \leq \infty, 1 \leq q \leq \infty$ to have finite or infinite ascent (descent). We also give a number of examples in support of our findings.

math.FA↗
arXiv2024

Direct observation of nanometer-scale orbital angular momentum accumulation

Conversion of charge to orbital angular momentum through the orbital Hall effect (OHE) holds transformative potential for the development of orbital-based electronics, however, it is challenging to directly observe the electrically generated orbital accumulation. Here, we detect the OHE by directly quantifying the orbital accumulation along the edges of a titanium thin film using a scanning transmission electron microscope. We measure the Ti L-edge using electron energy-loss spectroscopy with nanometer resolution and find a sizable orbital accumulation at the sample's outer perimeters, consistent with all signatures expected for the OHE, and determine an orbital diffusion length $\ell_o \approx 7.3$ nm. Our data points to a surprising dependence of the orbital diffusion length on the nano-structural morphology.

cond-mat.mes-hall↗
arXiv2019

Normality and Montel's Theorem

In this article, we prove a normality criterion for a family of meromorphic functions having zeros with some multiplicity which involves sharing of a holomorphic function by the members of the family. Our result generalizes Montel's normality test in a certain sense.

math.CV↗
arXiv2019

A normality Criterion for a Family of Meromorphic Functions

Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in\mathcal{F}$, $(f^n)^{(k)}\neq 1$, for $z\in D$, where $n, k$ are positive integers such that $n\geq k+3$, then $\mathcal{F}$ is a normal family in $D$. In this paper, we investigate the opposite view that if for each $f\in\mathcal{F}$, $(f^n)^{(k)}(z)-\psi(z)$ has zeros in $D$, where $\psi(z)$ is a holomorphic function in $D$, then what can be said about the normality of the family $\mathcal{F}$?

math.CV↗
arXiv2018

Meromorphically normal families and a meromorphic Montel-Carath\'eodory theorem

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain $D\subset \mathbb{C}^m$ into $\mathbb{P}^n$ to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto's work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of $\mathbb{P}^n$-valued holomorphic mappings. We also establish a meromorphic version of the Montel-Carath\'eodory theorem.

math.CV↗
arXiv2015

Quasi-normal families of meromorphic mappings sharing hypersurfaces

In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion concerning shared moving hypersurfaces.

math.CV↗
arXiv2015

A Criterion for quasi normality in $\mathbb{C}^n$

In this article, we give a Zalcman type renormalization result for the quasinormality of a family of holomorphic functions on a domain in $\mathbb{C}^n$ that takes values in a complete complex Hermitian manifold.

math.CV↗
arXiv2014

Some Normality Criteria

In this article we prove some normality criteria for a family of meromorphic functions which involves sharing of a non-zero value by certain differential monomials generated by the members of the family. These results generalizes some of the results of Schwick.

math.CV↗
arXiv2014

On a Theorem of Schwick

Let D be a domain, n, k be positive integers and n >= K+3. Let F be a family of functions meromorphic in D. If each f in F satisfies (f^n)^(k) not equal to 1 for z in D, then F is normal family. This result was proved by Schwick. In this paper we extend this result.

math.CV↗