NORMAL FUNCTIONS - L(P) ESTIMATES

For a meromorphic (or harmonic) function f, let us call the dilation o f f at z the ratio of the (spherical) metric at f(z) and the (hyperbol ic) metric at z. Inequalities are known which estimate the sup norm of the dilation in terms of its L(P) norm, for p > 2, while capitalizing on the symmetries of f. In the present paper we weaken the hypothesis by showing that such estimates persist even if the L(P) norms are tak en only over the set of z on which f takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harm onic functions in the case that p = 2.