PreprintCC BY 4.0arXiv
The sharp constant in the Chang--Marshall inequality
Abstract
We prove that the optimal constant in the Chang--Marshall inequality is e, as conjectured by Andreev and Matheson. More precisely, if f is analytic in the unit disk, vanishes to order m≥1 at the origin, and has normalized Dirichlet integral d≤1, then the boundary mean of e|f|2 is at most ed/m. Equality holds precisely for monomials. The proof uses an auxiliary closed planar curve and the isoperimetric inequality to establish monotonicity of the exponential functional along normalized radial dilations.
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BibTeX
@misc{frank2026sharp,
title = {{The sharp constant in the Chang--Marshall inequality}},
author = {Rupert L. Frank and Paata Ivanisvili},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15378},
eprint = {2610.15378},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15378},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The sharp constant in the Chang--Marshall inequality
AU - Rupert L. Frank
AU - Paata Ivanisvili
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.15378
UR - https://arxiv.org/abs/2610.15378
AB - We prove that the optimal constant in the Chang--Marshall inequality is e, as conjectured by Andreev and Matheson. More precisely, if f is analytic in the unit disk, vanishes to order m≥1 at the origin, and has normalized Dirichlet integral d≤1, then the boundary mean of e^|f|^2 is at most e^d/m. Equality holds precisely for monomials. The proof uses an auxiliary closed planar curve and the isoperimetric inequality to establish monotonicity of the exponential functional along normalized radial dilations.
ER -