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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1112.6237","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-12-29T06:51:30Z","cross_cats_sorted":[],"title_canon_sha256":"55225929ca7484e1823d90ab34874fb6663d0a37f329e1bf820d3ff65ec3863c","abstract_canon_sha256":"0152f067eea8f558ce3fe7cc0e8f357b81d92dd30f581b1036f82f147a7f8fa0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:36.434492Z","signature_b64":"G/lRg4Sru9w6Na1s/rtThv8JRSmfNDehSBQObj3DBk+7VKcZE/bmkZvScBmwAG/V0bq+xUB3u1SY0l4YJaXKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b94b914f5b06399d85ef37ed7c849f6292e4caeb4506ca256e9528110685f9b4","last_reissued_at":"2026-05-18T04:05:36.433779Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:36.433779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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