Bloch constants for meromorphic functions

Abstract

The classical Bloch constant is defined for the family of functions f which are holomorphic on the open unit disk IB and normalized by f ' ( 0 ) = 1. The Bloch constant involves the size of unramified disks which lie on RI, the Riemann surface of f viewed as spread over the complex plane C. The radius of the unramified disk is measured relative to the euclidean distance function. We shall be concerned with various Bloch constants associated with families of meromorphic functions which are defined on either C or a compact Riemann surface. These meromorphic functions are not required to be normalized at some fixed point. For a meromorphic function f we view the Riemann surface R I as being spread over the Riemann sphere IP and measure the radius of unramified disks relative to the spherical distance. Let us state our main results. We show that the Bloch constant for the family of locally schlicht meromorphic functions on C is ~/2. The Bloch constant for the family of all nonconstant meromorphic functions on II? lies between ~/3 and 2 arctan (1/l~). In addition, we consider Bloch constants for other families of meromorphic functions on ~2. Our second group of results concerns the Bloch constant for the family of nonconstant meromorphic functions on a compact Riemann surface of genus g. We show that there is a lower bound N(g) for this Bloch constant that depends only on the genus g and not on the conformal type of the compact surface. This is a consequence of a theorem about branched coverings of IP that might be of independent interest. We discuss upper and lower bounds for ~(g). Finally, we outline an extension of our results to compact bordered Riemann surfaces that involves the family of holomorphic functions which have modulus one on the boundary.