VALIRON-TYPE AND VALIRON-TITCHMARSH-TYPE THEOREMS FOR SUBHARMONIC FUNCTIONS OF SLOW GROWTH

Let u be a subharmonic function of order zero in R-m, m >= 2, with Riesz measure mu on the negative semiaxis Ox(1), n(r, u) = mu({x is an element of R-m : vertical bar x vertical bar <= r}), d(m) = m - 2 for m >= 3, d(2) = 1, and N(r, u) = d(m) integral(r)(1) n(r, u)/t(m-1). Under the condition of slow growth of N(r, u), we determine the asymptotics of u(x) as vertical bar x vertical bar = r -> + infinity. We also study the inverse relationship between the regular growth of u and the behavior of N(r, u) as r -> + infinity.<bold> </bold>