International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 92176, 9 pages

Riesz-Martin representation for positive super-polyharmonic functions in A Riemannian manifold

V. Anandam and S. I. Othman

Department of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 19 December 2005; Revised 28 February 2006; Accepted 4 April 2006

Copyright © 2006 V. Anandam and S. I. Othman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let u be a super-biharmonic function, that is, Δ2u0, on the unit disc D in the complex plane, satisfying certain conditions. Then it has been shown that u has a representation analogous to the Poisson-Jensen representation for subharmonic functions on D. In the same vein, it is shown here that a function u on any Green domain Ω in a Riemannian manifold satisfying the conditions (Δ)iu0 for 0im has a representation analogous to the Riesz-Martin representation for positive superharmonic functions on Ω.