{"paper":{"title":"Characterization and the pre-Schwarzian norm estimate for concave univalent functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"B. Bhowmik, K-J. Wirths, S. Ponnusamy","submitted_at":"2010-08-28T12:50:46Z","abstract_excerpt":"Let $Co(\\alpha)$ denote the class of concave univalent functions in the unit disk $\\ID$. Each function $f\\in Co(\\alpha)$ maps the unit disk $\\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\\left ( f''(z)/f'(z)\\right)$, $f\\in Co(\\alpha)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\\left(f''(z)/f'(z)\\right)$, $f\\in Co(\\alpha)$ whenever $f''(0)$ is fixed. We also give a cha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4861","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"}