Estimates for truncated area functionals on the Bloch space

Abstract

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