Abstract
The aim of this paper is to establish properties of solutions to the α-harmonic equations: (Formula presented.), where (Formula presented.) is a continuous function and (Formula presented.) denotes the closure of the unit disc (Formula presented.) in the complex plane (Formula presented.). We obtain Schwarz type and Schwarz-Pick type inequalities for solutions to the α-harmonic equation. In particular, for (Formula presented.), solutions to the above equation are called α-harmonic functions. We determine the necessary and sufficient conditions for an analytic function ψ to have the property that (Formula presented.) is α-harmonic function for any α-harmonic function f. Furthermore, we discuss the Bergman-type spaces on α-harmonic functions.
| Original language | English |
|---|---|
| Pages (from-to) | 1981-1997 |
| Number of pages | 17 |
| Journal | Complex Variables and Elliptic Equations |
| Volume | 65 |
| Issue number | 12 |
| DOIs | |
| State | Published - 1 Dec 2020 |
Keywords
- 35J25
- Bergman-type spaces
- Primary 31A05
- Schwarz lemma
- Secondary 35C15
- composition
- α-harmonic equation
- α-harmonic function