Truncated Dirichlet estimates for Bloch functions

Abstract

We study the truncated Dirichlet functional \[ Φ_n (f) = \sum_{k = 1}^n k |a_k|^2, \quad f(z) = \sum_{k = 0}^\infty a_k z^k, \] when $f$ belongs to the Bloch space. If $n \geq 27$, we find a normalized Bloch function $f_n$ such that \[ Φ_n (f_n) > n B_n^2, \] where $B_n = \max \{ |a_n| : \| f \|_{\mathcal{B}} = 1 \}$, thus proving that an inequality by Kayumov and Wirths cannot be extended to large values of $n$. Nevertheless, we show that the inequality $Φ_n(f) \leq n B_n^2$ holds for all normalized functions when $n \leq 9$.