L. Sario has extended R.
Nevanlinna’s concept of defect to functions defined on Wp Riemann surfaces. He has
shown that for a large class of functions the defect sum ∑δ(a) is bounded above by
2 + η, where η is a number depending on the topological complexity of the surface
and the rate of growth of the function under study.
By studying the relation between the rate of growth of a meromorphic function
w and its P-derivative wp= wz∕(dp∕dz + idp∗∕dz), where p is a capacity
function, and z is a local variable, we are able to establish a bound that
implies that of Sario and can be smaller than 2 + η. It is also shown that the
classical theorem of Picard holds unchanged for the meromorphic function wp
provided that w has maximum defect. In the concluding section a version of
Milloux’s extension of Nevanlinna’s second main theorem is given for Wp
surfaces.
