{"paper":{"title":"Weighted Integral Means of Mixed Areas and Lengths under Holomorphic Mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Jie Xiao, Wen Xu","submitted_at":"2011-05-30T17:03:30Z","abstract_excerpt":"This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\\alpha dt^2$, $-\\infty<\\alpha<\\infty$, $0<t<1$) integral means $\\mathsf{A}_{\\alpha,\\beta}(f,\\cdot)$ and $\\mathsf{L}_{\\alpha,\\beta}(f,\\cdot)$ of the mixed area $(\\pi r^2)^{-\\beta}A(f,r)$ and the mixed length $(2\\pi r)^{-\\beta}L(f,r)$ ($0\\le\\beta\\le 1$ and $0<r<1$) of $f(r\\mathbb D)$ and $\\partial f(r\\mathbb D)$ under a holomorphic map $f$ from the unit disk $\\mathbb D$ into the finite complex plane $\\mathbb C$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.6042","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"}