Estimates for truncated area functionals on the Bloch space

Recently, Kayumov \cite{K} obtained a sharp estimate for the $n$-th truncated area functional for normalized functions in the Bloch space for $n\le 5$ and then, together with Wirths \cite{KW1}, extended the result for $n=6$. We prove that for the functions with non-negative Taylor coefficients, the same sharp estimate is valid for all $n$. For arbitrary functions, we obtain an estimate that is asymptotically of the same order but slightly larger (roughly by a factor of $4\mathrm{/}e$). We also consider related weighted estimates for functionals involving the powers $n^t$, $t>0$, and show that the exponent $t=1$ represents the critical case for the expected sharp estimate.