{"paper":{"title":"Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Judit Mak\\'o, Zsolt P\\'ales","submitted_at":"2012-12-04T21:10:52Z","abstract_excerpt":"In this paper, the connection between the functional inequalities $$ f\\Big(\\frac{x+y}{2}\\Big)\\leq\\frac{f(x)+f(y)}{2}+\\alpha_J(x-y) \\qquad (x,y\\in D)$$ and $$ \\int_0^1f\\big(tx+(1-t)y\\big)\\rho(t)dt \\leq\\lambda f(x)+(1-\\lambda)f(y) +\\alpha_H(x-y) \\qquad (x,y\\in D)$$ is investigated, where $D$ is a convex subset of a linear space, $f:D\\to\\R$, $\\alpha_H,\\alpha_J:D-D\\to\\R$ are even functions, $\\lambda\\in[0,1]$, and $\\rho:[0,1]\\to\\R_+$ is an integrable nonnegative function with $\\int_0^1\\rho(t)dt=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0874","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}