Year: 2026
Type: preprint
Abstract: In this paper, we present Lappan's five-point type criteria for $φ$-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 suitable conditions. We further reduce the number of test values to $\left[\frac{k}{2}\right]+2$ under an additional differential condition involving the analytic and co-analytic parts. We also prove a generalization in which the spherical derivative is replaced by a polynomial with non-negative real coefficients.
Source: arXiv (Cornell University)
Authors: Datt, Gopal, Pal, Ritesh
Institution: Babasaheb Bhimrao Ambedkar University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Human Resource Development Group
OpenAlex ID: W7213611532