PreprintCC BY 4.0arXiv
On φ-Normality of Harmonic Mappings and Their Families
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 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 [k/2]+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
@misc{datt2026φnormality,
title = {{On $φ$-Normality of Harmonic Mappings and Their Families}},
author = {Gopal Datt and Ritesh Pal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19829},
eprint = {2609.19829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On φ-Normality of Harmonic Mappings and Their Families
AU - Gopal Datt
AU - Ritesh Pal
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19829
UR - https://arxiv.org/abs/2609.19829
AB - 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 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 [k/2]+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.
ER -