{"paper":{"title":"Convexity in one direction of convolutions and linear combination of harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Subzar Beig, V. Ravichandran","submitted_at":"2017-03-10T09:45:52Z","abstract_excerpt":"We show that the convolution of the harmonic function $f=h+\\bar{g}$, where $h(z)+{e}^{-2{i}\\gamma}g(z)=z/(1-{e}^{{i}\\gamma}z)$ having analytic dilatation ${e}^{{i}\\theta} z^n (0\\leq\\theta<2\\pi)$, with the mapping $f_{a,\\alpha}=h_{a,\\alpha}+\\overline{g}_{a,\\alpha}$, where $h_{a,\\alpha}(z)=(z/(1+a)-{e}^{{i}\\alpha}z^2/2)/(1-{e}^{{i}\\alpha}z)^2$, $g_{a,\\alpha}(z)=(a {e}^{2{i}\\alpha}z/(1+a)-{e}^{3{i}\\alpha}z^2/2)/(1-{e}^{{i}\\alpha}z)^2$ is convex in the direction $-(\\alpha+\\gamma)$. We also show that the convolution of $f_{a,\\alpha}$ with the right half-plane mapping having dilatation $(a-z^2)/(1-a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03599","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"}