Estimates for truncated area functionals on the Bloch space

Recently, Kayumov [Lobachevskii J. Math. 38 (2017), pp. 466–468] obtained a sharp estimate for the n n -th truncated area functional normalized functions in Bloch space alttext="n less-than-or-equal-to 5"> ≤ 5 encoding="application/x-tex">n\le 5 and then, together with Wirths 40 (2019), 1319–1323], extended result equals 6"> = 6 encoding="application/x-tex">n=6 . We prove that non-negative Taylor coefficients, same is valid all For arbitrary functions, we obtain an asymptotically of order but slightly larger (roughly by factor alttext="4 slash e"> 4 / e encoding="application/x-tex">4/e ). also consider related weighted estimates functionals involving powers Superscript t"> t encoding="application/x-tex">n^t , alttext="t greater-than 0"> > 0 encoding="application/x-tex">t>0 show exponent 1"> 1 encoding="application/x-tex">t=1 represents critical case expected estimate.