Searcharxiv⌕ Search

arXiv subjects

Nikhil Bharti

Publications and source records attributed to Nikhil Bharti.

7 recordsLinked to original sources
arXiv2026

A Zalcman-Pang type rescaling result for $ϕ$-normal harmonic mappings and applications

We investigate normality and $ϕ$-normality criteria for harmonic mappings in the unit disk. A new sufficient condition for normality involving extended spherical derivatives is established. We further prove a Zalcman-Pang type rescaling lemma for $ϕ$-normal harmonic mappings and derive new $ϕ$-normality criteria as applications. In addition, we introduce $ϕ$-normal families of harmonic mappings and obtain corresponding Lappan-type characterizations. In particular, we show that for sense-preserving harmonic mappings the relevant test set may be taken to consist of only three points.

math.CV↗
arXiv2024

Results on normal harmonic and $φ$-normal harmonic mappings

In this paper, we study the concepts of normal functions and $φ$-normal functions in the framework of planar harmonic mappings. We establish the harmonic mapping counterpart of the well-known Zalcman-Pang lemma and as a consequence, we prove that a harmonic mapping whose spherical derivative is bounded away from zero is normal. Furthermore, we introduce the concept of the extended spherical derivative for harmonic mappings and obtain several sufficient conditions for a harmonic mapping to be $φ$-normal.

math.CV↗
arXiv2024

Weighted Yosida Mappings of Several Complex Variables

Let $M$ be a complete complex Hermitian manifold with metric $E_{M}$ and let $φ: [0,\infty)\rightarrow (0,\infty)$ be positive function such that $$γ_r=\sup\limits_{r\leq a<b}\left|(φ(a)-φ(b))/(a-b)\right|\leq C,~r\in (0,\infty),$$ for some $C\in (0,1],$ and $\lim_{r\rightarrow\infty}γ_r=0.$ A holomorphic mapping $f:\mathbb{C}^{m}\rightarrow M$ is said to be a weighted Yosida mapping if for any $z,~ξ\in\mathbb{C}^{m}$ with $\|ξ\|=1,$ the quantity $φ(\|z\|)E_{M}(f(z), df(z)(ξ))$ remains bounded above, where $df(z)$ is the map from $T_z(\mathbb{C}^{m})$ to $T_{f(z)}(M)$ induced by $f.$ We present several criteria of holomorphic mappings belonging to the class of all weighted Yosida mappings.

math.CV↗
arXiv2023

A Normal Criterion Concerning Sequence of Functions and their Differential Polynomials

In this paper, a normality criterion concerning a sequence of meromorphic functions and their differential polynomials is obtained. Precisely, we have proved: Let $\left\{f_j\right\}$ be a sequence of meromorphic functions in the open unit disk $\mathbb{D}$ such that, for each $j,$ $f_j$ has poles of multiplicity at least $m,~m\in\mathbb{N}.$ Let $\left\{h_j\right\}$ be a sequence of holomorphic functions in $\mathbb{D}$ such that $h_j\rightarrow h$ locally uniformly in $\mathbb{D},$ where $h$ is holomorphic in $\mathbb{D}$ and $h\not\equiv 0.$ Let $Q[f_j]$ be a differential polynomial of $f_j$ having degree $λ_Q$ and weight $μ_Q.$ If, for each $j,$ $f_j(z)\neq 0$ and $Q[f_j]-h_j$ has at most $μ_Q + λ_Q(m-1)-1$ zeros, ignoring multiplicities, in $\mathbb{D},$ then $\left\{f_j\right\}$ is normal in $\mathbb{D}.$

math.CV↗
arXiv2023

Value distribution of certain differential polynomials leading to some normality criteria

In this paper, we prove some normality criteria concerning transitivity of normality from one family of meromorphic functions to another which improve and generalize some recent results. We also prove some value distribution results for certain differential polynomials which lead to some normality criteria involving sharing of holomorphic functions with certain differential polynomials. As a consequence, a counterexample to the converse of the Bloch's principle is also given.

math.CV↗
arXiv2022

Normality through partial sharing of sets with differential polynomials

This article aims at finding sufficient conditions for a family of meromorphic functions to be normal by involving partial sharing of sets with differential polynomials. Moreover, corresponding results for normal meromorphic functions are also established which improve and generalize many known results.

math.CV↗