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A Note on Hyperplane mean values of the Modulus of a Harmonic Function

A Note on Hyperplane mean values of the Modulus of a Harmonic Function A NOTE ON HYPERPLANE MEAN VALUES OF THE MODULUS OF A HARMONIC FUNCTION D. H. ARMITAGE 1. Let D denote the open Euclidean half-space R"x(a, +00), where n ^ 1 an and aeR . An arbitrary point of D is denoted by (X,y), where XeR" and an y e (a, + 00). If s is a non-negative subharmonic function in D „, we define a function M{s, •) on (a, + 00) by writing Jl{s,y)= [s(X,y)dX, the integral being taken in the sense of Lebesgue. Thus Jt{s, •) exists and is non-negative (the value +00 being permitted) in (a, +00). In [4, Theorem 1], Nualtaranee proved: THEOREM A. If s is non-negative and subharmonic in D „ and Jl{s, •) is locally integrable in (0, +00) and +l ) (1) as y-> +00, then Jf{s,-) is real-valued, decreasing, convex and continuous in (0,+00) . This result has recently been improved in two respects by Rippon [5]. (I am grateful to Dr. Rippon for a pre-print of his paper.) The purpose of this note is to show that the hypothesis (1) can be greatly relaxed in the special case where s = \h\ for some harmonic function h in D http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Note on Hyperplane mean values of the Modulus of a Harmonic Function

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.1.33
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Abstract

A NOTE ON HYPERPLANE MEAN VALUES OF THE MODULUS OF A HARMONIC FUNCTION D. H. ARMITAGE 1. Let D denote the open Euclidean half-space R"x(a, +00), where n ^ 1 an and aeR . An arbitrary point of D is denoted by (X,y), where XeR" and an y e (a, + 00). If s is a non-negative subharmonic function in D „, we define a function M{s, •) on (a, + 00) by writing Jl{s,y)= [s(X,y)dX, the integral being taken in the sense of Lebesgue. Thus Jt{s, •) exists and is non-negative (the value +00 being permitted) in (a, +00). In [4, Theorem 1], Nualtaranee proved: THEOREM A. If s is non-negative and subharmonic in D „ and Jl{s, •) is locally integrable in (0, +00) and +l ) (1) as y-> +00, then Jf{s,-) is real-valued, decreasing, convex and continuous in (0,+00) . This result has recently been improved in two respects by Rippon [5]. (I am grateful to Dr. Rippon for a pre-print of his paper.) The purpose of this note is to show that the hypothesis (1) can be greatly relaxed in the special case where s = \h\ for some harmonic function h in D

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1984

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