LOGARITHMIC CONVEXITY FOR SUPREMUM NORMS OF HARMONIC-FUNCTIONS

We prove the following convexity property for supremum norms of harmonic functions. Let Omega be a domain in R(n), Omega, and E a subdomain and a compact subset of Omega, respectively. Then there exists a constant alpha = alpha(E,Omega(0),Omega)is an element of(0,1] such that for all harmonic functions u on Omega, the inequality \\u\\(E) less than or equal to \\u\\(alpha)(Omega 0) \\u\\(1-alpha)(Omega) is valid. The case of concentric balls Omega(0) subset of E subset of Omega plays a key role in the proof. For positive harmonic functions on such balls, we determine the sharp constant alpha in the inequality.