Vol. 17, No. 1, 1966

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Remarks on the defect sum for a function meromorphic on an open Riemann surface

Phillip Emig

Vol. 17 (1966), No. 1, 45–55

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.

Mathematical Subject Classification
Primary: 30.60
Received: 6 May 1964
Published: 1 April 1966
Phillip Emig