A Note on a Maximum Theorem for Harmonic Functions

Abstract

Lowater, Bruckner, Ryan (1) 이 발표한 단위원내에서 조화인 함수의 최대치 정리와 관련된 결과를 복소수함수론의 cluster set을 써서 직접 간결하게 증명했다.

Let u(r,θ) be harmonic in D=｛ㅣzㅣ<1｝ and let lim inf u(r,θ) be non-negative for almost all θin 0 This Result is known; the proof, howeverm, given by Lowater, Bruckner, and Ryan (1), uses deep results from the theory of real functions; a firect proof by complex analysis is given.

Let u(r,θ) be harmonic in D=｛ㅣzㅣ<1｝ and let lim inf u(r,θ) be non-negative for almost all θin 0 This Result is known; the proof, howeverm, given by Lowater, Bruckner, and Ryan (1), uses deep results from the theory of real functions; a firect proof by complex analysis is given.