{"paper":{"title":"Directional convexity of harmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Subzar Beig, V. Ravichandran","submitted_at":"2017-03-10T09:32:02Z","abstract_excerpt":"The convolution properties are discussed for the complex-valued harmonic functions in the unit disk $\\mathbb{D}$ constructed from the harmonic shearing of the analytic function $\\phi(z):=\\int_0^z (1/(1-2\\xi\\textit{e}^{\\textit{i}\\mu}\\cos\\nu+\\xi^2\\textit{e}^{2\\textit{i}\\mu}))\\textit{d}\\xi$, where $\\mu$ and $\\nu$ are real numbers. For any real number $\\alpha$ and harmonic function $f=h+\\overline{g}$, define an analytic function $f_{\\alpha}:=h+\\textit{e}^{-2\\textit{i}\\alpha}g$. Let $\\mu_1$ and $\\mu_2$ $(\\mu_1+\\mu_2=\\mu)$ be real numbers, and $f=h+\\overline{g}$ and $F=H+\\overline{G}$ be locally-uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03593","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"}