{"paper":{"title":"On univalence of the power deformation $z(f(z)/z)^c$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Toshiyuki Sugawa, Yong Chan Kim","submitted_at":"2011-12-29T06:51:30Z","abstract_excerpt":"In this note, we mainly concern the set $U_f$ of $c\\in\\mathbb{C}$ such that the power deformation $z(f(z)/z)^c$ is univalent in the unit disk $|z|<1$ for a given analytic univalent function $f(z)=z+a_2z^2+\\cdots$ in the unit disk. We will show that $U_f$ is a compact, polynomially convex subset of the complex plane $\\C$ unless $f$ is the identity function. In particular, the interior of $U_f$ is simply connected. This fact enables us to apply various versions of the $\\lambda$-lemma for the holomorphic family $z(f(z)/z)^c$ of injections parametrized over the interior of $U_f.$ We also give nece"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6237","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"}