On the relation between the Lebesgue integral means and Nevanlinna characteristic of analytic functions in the unit disc

The best possible asymptotic estimates for Lebesgue integral means $m_{p}(r,log f), 1 leq p$ of logarithms of analytic functions $f(z)$ in the unit disc in terms of their Nevanlinna characteristic $T(r,f)$ are obtained. We get sharp relation between the order of $T(r,f)$ and the order of $m_{p}(r,log f)$ for an analytic function $f(z)$ of finite order $alpha(f).$ This generalizes well-known results of L.~R.~Sons and C.~N.~Linden.