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Several properties of a class of generalized harmonic mappings

Bo-Yong Long, Qi-Han Wang

Abstract

We call the solution of a kind of second order homogeneous partial differential equation as real kernel alpha-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of the real kernel alpha-harmonic mappings are explored.

Several properties of a class of generalized harmonic mappings

Abstract

We call the solution of a kind of second order homogeneous partial differential equation as real kernel alpha-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of the real kernel alpha-harmonic mappings are explored.
Paper Structure (5 sections, 11 theorems, 68 equations)

This paper contains 5 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Representation theorem
  3. Lipschitz continuity
  4. Univalency of a subclass of real kernel $\alpha-$harmonic mappings
  5. Area $S_{u}$

Key Result

Lemma 1.1

MR3233580 Let $c>0$, $a\leq c$, $b\leq c$ and $ab\leq 0$$(ab\geq 0)$. Then the function $F(a,b;c;x)$ is decreasing (increasing) on $x\in (0, 1)$.

Theorems & Definitions (18)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Example 2.1
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • ...and 8 more