{"paper":{"title":"Differential Inequalities and Univalent Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Milutin Obradovi\\'c, Rosihan M. Ali, Saminathan Ponnusamy","submitted_at":"2019-05-05T14:38:06Z","abstract_excerpt":"Let ${\\mathcal M}$ be the class of analytic functions in the unit disk $\\ID$ with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition $$\\left |z^2\\left (\\frac{z}{f(z)}\\right )''+ f'(z)\\left(\\frac{z}{f(z)} \\right)^{2}-1\\right |\\leq 1, \\quad z\\in \\ID. $$ Functions in $\\mathcal{M}$ are known to be univalent in $\\ID$. In this paper, it is shown that the harmonic mean of two functions in ${\\mathcal M}$ are closed, that is, it belongs again to ${\\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.01694","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"}