PreprintCC BY 4.0arXiv
Truncated Dirichlet estimates for Bloch functions
Abstract
We study the truncated Dirichlet functional Φn (f) = ∑k = 1n k |ak|2, f(z) = ∑k = 0∞ ak zk, when f belongs to the Bloch space. If n ≥ 27, we find a normalized Bloch function fn such that Φn (fn) > n Bn2, where Bn = max { |an| : | f |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 Φn(f) ≤ n Bn2 holds for all normalized functions when n ≤ 9.
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BibTeX
@misc{efraimidis2026truncated,
title = {{Truncated Dirichlet estimates for Bloch functions}},
author = {Iason Efraimidis and Adrián Llinares and Alejandro Mas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41534},
eprint = {2610.41534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41534},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Truncated Dirichlet estimates for Bloch functions
AU - Iason Efraimidis
AU - Adrián Llinares
AU - Alejandro Mas
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41534
UR - https://arxiv.org/abs/2610.41534
AB - We study the truncated Dirichlet functional Φ_n (f) = ∑_k = 1^n k |a_k|^2, f(z) = ∑_k = 0^∞ a_k z^k, when f belongs to the Bloch space. If n ≥ 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 |_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) ≤ n B_n^2 holds for all normalized functions when n ≤ 9.
ER -