(1) Department of Pure Mathematics, Imam Khomeini International University, Qazvin, 34194, Iran;(2) Mathematics Department, Institute for Studies in Theoretical Physics and Mathematics, Tehran, 19395-5746, Iran
Abstract:
We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation
formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already
found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights
are superbiharmonic and satisfy the stated growth condition near the boundary.
Research supported in part by IPM under the grant number 83310011.