Outer functions in Qp#,  Qp,0#

Abstract

For 0<p<∞,  Qp#  (Qp,0#) is the class of meromorphic functions f defined in the unit disk D={z:|z|<1} satisfying sup⁡w∈D∬D(f#(z))2gp(z,w)dσz<∞  (lim⁡w→∂D∬D(f#(z))2gp(z,w)dσz=0), where g(z,w) is Green's function of D. Criteria for funtions f to belong to Qp#  (Qp,0#) are given by the Ahlfors-Shimizu characteristic. Further, outer functions in Qp#  (Qp,0#) are characterized and shown that every function in Qp#  (Qp,0#) can be represented as the quotient of two functions in H∞∩Qp#  (H∞∩Qp,0#).