{"paper":{"title":"Analytic and geometric properties of open door functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Ming Li, Toshiyuki Sugawa","submitted_at":"2015-12-10T06:36:25Z","abstract_excerpt":"In this paper, we study analytic and geometric properties of the solution $q(z)$ to the differential equation $q(z)+zq'(z)/q(z)=h(z)$ with the initial condition $q(0)=1$ for a given analytic function $h(z)$ on the unit disk $|z|<1$ in the complex plane with $h(0)=1.$ In particular, we investigate the possible largest constant $c>0$ such that the condition $|\\Im[zf\"(z)/f'(z)]|<c$ on $|z|<1$ implies starlikeness of an analytic function $f(z)$ on $|z|<1$ with $f(0)=f'(0)-1=0.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03153","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"}