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A little surprisingly, we will see that the set is described by the variability region of the quantity $zf'(z)/f(z),~|z|<1,$ for the class in most cases which we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spiralli"},"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":"1101.3832","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-01-20T07:15:01Z","cross_cats_sorted":[],"title_canon_sha256":"4c4d2637451dc19434a517709b06c4a7a17fd5883297848251e2464e76b96aba","abstract_canon_sha256":"4557907d5de7579bcad752dd5453762d6900b3fe3bbf18eeed5b31e89d1fd848"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:21.271257Z","signature_b64":"lFxyP77fjii/fOqqQD5Vh6Mx5DzaZm/qaZMqrH+4fnfZWFRO6C4XhxYOff5Yjq1FPN3Pm4KyCDIR98HRPwjnCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb63ca34d3bd35797337b89a632cf66027d0bb74d31e9888cc825ae537b8fd7f","last_reissued_at":"2026-05-18T04:31:21.270558Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:21.270558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On power deformations of univalent functions","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":"2011-01-20T07:15:01Z","abstract_excerpt":"For an analytic function $f(z)$ on the unit disk $|z|<1$ with $f(0)=f'(0)-1=0$ and $f(z)\\ne0, 0<|z|<1,$ we consider the power deformation $f_c(z)=z(f(z)/z)^c$ for a complex number $c.$ We determine those values $c$ for which the operator $f\\mapsto f_c$ maps a specified class of univalent functions into the class of univalent functions. 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