Note on polyharmonic functions
R. J. Duffin, Zeev Nehari · Proceedings of the American Mathematical Society · 1961
It is the purpose of this note to show that there exists an inequality of the same general character as (1) if the hypothesis that u(P) be harmonic is replaced by the considerably weaker assumption that u(P) be a polyharmonic function of order n (n> 1). The latter expression refers to a function which is continuous, has continuous derivatives up to the order 2n, and is a solution of ln)u=O, where A(k) is defined as A(A(k-1)). Our precise result is the following: If u(P) is a non-negative polyharmonic function of order n in the m-dimensional sphere OP = r < R, then