Work

The sharp constant in the Chang--Marshall inequality

Year: 2026

Type: preprint

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 $m\geq1$ at the origin, and has normalized Dirichlet integral $d\leq1$, 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.15378

Authors: Frank, Rupert L., Ivanisvili, Paata

Institutions: Ludwig-Maximilians-Universität München, University of California, Irvine, Munich Center for Quantum Science and Technology

Language: English

Topic: Analytic and geometric function theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Deutsche Forschungsgemeinschaft

Awards: G1768993395, G1889978188, G5141357660, TRR 352: Mathematik der VielteilchenQuantensysteme und ihrer kollektiven Phänomene, CAREER: Discrete Structures and Orthogonal Systems, EXC 2111: Munich Center for Quantum Science and Technology (MCQST)

OpenAlex ID: W7221806536

Cites:

Cited by:

Related to: 10