INTEGRAL MEANS OF THE POISSON INTEGRAL OF A DISCRETE MEASURE

It is proved that a function u, harmonic in the unit disc, can be repr esented in the form u(z)=Sigma lambda(j)1-\z\(2)/\1-w(j)z\(2), \z\<1, with \w(j)\=1(j=1, 2, ...), Sigma\lambda(j)\(p)<infinity, and 1/2<p<1, if and only if M(p)(u, r)=0(1-r)(1/p-1) (r-->1(-)). The discussion of the case p less than or equal to 1/2 involves the integral means of d erivatives of u. (C) 1994 Academic Press, Inc.