{"paper":{"title":"Univalence of the average of two analytic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"M. Obradovi\\'c, S.Ponnusamy","submitted_at":"2012-03-13T04:45:40Z","abstract_excerpt":"Let $\\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\\ID=\\{z:\\,|z|<1\\}$ of the form $f(z)=z+\\sum_{n=2}^{\\infty}a_nz^n.$ Let $\\mathcal{U}$ denote the set of all $f\\in \\mathcal{A}$, $f(z)/z\\neq 0$ and satisfying the condition $$ | f'(z) (\\frac{z}{f(z)})^{2}-1 | < 1 {for $z\\in \\ID$}. $$ Functions in ${\\mathcal U}$ are known to be univalent in $\\ID$. For $\\alpha \\in [0,1]$, let $$ \\mathcal{N}(\\alpha)= \\{f_\\alpha :\\, f_\\alpha (z)=(1-\\alpha)f(z)+\\alpha \\int_0^z\\frac{f(t)}{t}\\,dt, {$f\\in\\mathcal{A}$ with $|a_n|\\leq n$ for $n\\geq 2$}\\}. $$ In this paper, we first show that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.2713","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"}