Abstract
In this paper, the concepts of p-harmonic mappings and log-p-harmonic mappings in the unit disk have been introduced and studied by many researchers. We proved the normality of the p-harmonic mappings and log-p-harmonic mappings, which extend the related results of harmonic mappings of earlier authors.
Highlights
Introduction and PreliminariesFor real-valued harmonic functions defined in D, Lappan [1] established that φ is normal if sup z∈D À 1 − jzj2 Á∣gradφðzÞ 1 + φ2ðzÞ ∣< ∞, ð1Þ where grad φ is the gradient vector of φ
We are motivated to establish the topic of normality for complex-valued p-harmonic mappings and log-p-harmonic mappings defined in the unit disk
The definition of Bloch harmonic function given by Colonna [14], we will prove that the polyharmonic mapping F and log-pharmonic mapping f defined in the unit disk D are normal if they satisfy a Lipschitz type condition
Summary
We are motivated to establish the topic of normality for complex-valued p-harmonic mappings and log-p-harmonic mappings defined in the unit disk. The definition of Bloch harmonic function given by Colonna [14], we will prove that the polyharmonic mapping F and log-pharmonic mapping f defined in the unit disk D are normal if they satisfy a Lipschitz type condition. When p = 1, the mapping f is called log-harmonic. In virtue of being inspired by these results, we establish the normality of polyharmonic mappings and log-p-harmonic mappings.
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