Then

For a measurable function u(r)> 0 defined in the unit disc D: lzl < 1, we introduce a characteristic function which in the case of a meromorphic function becomes its Nevanlinna characteristic function in the Ahlfors-Shimizu form. In terms of new characteristic functions we prove necessary and suffi.cient conditions for a function to belong to the Nevanlinna class.l[, to the Hardy classes Ifp, 0 <p < foo, and to the hyperbolic Hardy classes Hi,0 < p< +m. 1. For a measurable function u(r)> 0 defined in the unit disc D: lzl < 1, on the complex z-plane, we introduce the characteristic function P(r,u) in the form P(r,u) : [ ' s(t'u) or, o < r < 1, \ / ' Jo t where and put P(1, u) : lim"*l P(r, u). If f (z) is a meromorphic function in D and f{(,):lnlf'e)l'r'-'lf'e)l1r+ lr{,)l')-', 0 (p ( *oo, then P(r, ft)