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.