Estimates for truncated area functionals on the Bloch space

Efraimidis, Iason; Mas, Alejandro; Vukotić, Dragan

doi:10.1090/proc/16382

Estimates for truncated area functionals on the Bloch space
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by Iason Efraimidis, Alejandro Mas and Dragan Vukotić

Proc. Amer. Math. Soc. 151 (2023), 3845-3854

DOI: https://doi.org/10.1090/proc/16382

Published electronically: June 6, 2023

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Abstract:

Recently, Kayumov [Lobachevskii J. Math. 38 (2017), pp. 466–468] 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 [Lobachevskii J. Math. 40 (2019), pp. 1319–1323], 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 $4/e$). 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.

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