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.
In this paper, we present Lappan's five-point type criteria for $\varphi$-normality of sense-preserving harmonic mappings and families of harmonic mappings in the unit disc $\mathbb{D}$. We establish a normality criterion involving the higher-order spherical derivative of order $k$, and $k+2$ distinct test values under...
Recently, Li and Ponnusamy~\cite{LiPonnusamy2025} established the coefficient conjecture proposed by Wang et al.~\cite{Wang2024} for several prominent geometric subclasses of $\mathcal{S}^0_H(K)$, the class of sense-preserving $K$-quasiconformal univalent harmonic mappings in the unit disk. In this paper, we show that...
Jasbir Parashar, Saminathan Ponnusamy, A. Kaliraj· 0 citations
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 $$ g'(z)=\omega(z)h'(z)+\psi(z),$$ where $\omega$ and $\psi$ are analytic functions in $\mathbb{D}$ with $\|\omega\|_{\inft...
We determine the radii of concavity for the classes $\mathcal{S}_e^*$ and $\mathcal{C}_e$ of univalent functions associated with the exponential mapping $e^z$. Under the geometric framework of Avkhadiev--Wirths for conformal mappings with unbounded convex complements of opening angle $\pi A$ ($A \in (1,2]$), the radii...
In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouv...
Let $n \ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematic...
Yi-Ru-Ya Zhang· 0 citations
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