Weighted biharmonic Green functions for rational weights

We present an algorithm for computing the Green function of the weighted biharmonic operator Delta\P'\(-2)Delta on the unit disc (with Dirichlet boundary conditions) for rational functions P. As an application, we show that if P is a Blaschke product with two zeros alpha(1), alpha(2) the Green function is positive if and only if \(alpha(1) - alpha(2))/(1 - <(alpha)over bar>(1)alpha(2))\ less than or equal to 2/7 root 10, and also obtain an explicit formula for the Green function of the operator Delta\G\(-2)Delta, where G is the canonical zero-divisor of a finite zero set on the Bergman space.