MULTI-POINT VARIATIONS OF THE SCHWARZ LEMMA WITH DIAMETER AND WIDTH CONDITIONS

Suppose that f is holomorphic ill the unit disk D and f(D) subset of D, f(0) = 0. A classical inequality due to Littlewood generalizes the Schwarz lemma and asserts that for every w is an element of f(D), we have vertical bar w vertical bar <= Pi(j) vertical bar z(j)(w)vertical bar, where z(j)(w) is the sequence of pre-images of w. We prove a similar inequality by replacing the assumption f(D) subset of D with the weaker assumption Diam f(D) = 2. This inequality generalizes a growth bound involving only one pre-image, proven recently by Burckel et al. We also prove growth bounds for holomorphic f mapping D onto a region having fixed horizontal width. We give a complete characterization of the equality cases. The main tools in the proofs are the Green function and the Steiner symmetrization.