A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function

2019/05/08 by Jianfeng Lu, Stefan Steinerberger, Lu, Jianfeng +1 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1905.03216

openalex publication_date 2019/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ ℝⁿ be a convex domain and let f:Ω→ ℝ be a subharmonic function, Δf ≥ 0, which satisfies f ≥ 0 on the boundary ∂ Ω. Then ∫_Ωf ~dx ≤ |Ω|^(1)/(n) ∫_{∂ Ω}f ~dσ. Our proof is based on a new gradient estimate for the torsion function, Δu = -1 with Dirichlet boundary conditions, which is of independent interest.

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