On $φ$-Normality of Harmonic Mappings and Their Families

Gopal Datt; Ritesh Pal

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arXiv · 2609.19829

On $φ$-Normality of Harmonic Mappings and Their Families

Gopal Datt·Ritesh Pal

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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.

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BibTeX RIS

Gopal Datt, Ritesh Pal. 2026-09-17. On $φ$-Normality of Harmonic Mappings and Their Families. https://arxiv.org/abs/2609.19829

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