Estimates for truncated area functionals on the Bloch space

TL;DR

This paper extends sharp estimates for truncated area functionals on the Bloch space to functions with non-negative Taylor coefficients for all n, and explores related weighted estimates and critical exponents.

Contribution

It proves the sharp estimate for all n for functions with non-negative Taylor coefficients and analyzes asymptotic behavior and weighted estimates for general functions.

Findings

Sharp estimates valid for all n with non-negative coefficients

Asymptotic estimates for arbitrary functions

Critical exponent t=1 for weighted estimates

Abstract

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 $4/e$). 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.