{"paper":{"title":"Order of convexity of Integral Transforms and Duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Sarika Verma, Sukhjit Singh, Sushma Gupta","submitted_at":"2013-05-03T14:47:56Z","abstract_excerpt":"Recently, Ali et al defined the class $\\mathcal{W}_{\\beta}(\\alpha, \\gamma)$ consisting of functions $f$ which satisfy $$\\Re e^{i\\phi}\\left((1-\\alpha+2\\gamma)\\frac{f(z)}{z}+(\\alpha-2\\gamma)f'(z)+\\gamma zf''(z)-\\beta\\right)>0,$$ for all $z\\in E=\\left\\{z : |z|<1\\right\\}$ and for $\\alpha, \\gamma\\geq0$ and $\\beta<1$, $\\phi\\in \\mathbb{R}$ (the set of reals). For $f\\in{\\mathcal{W}_{\\beta}(\\alpha, \\gamma)}$, they discussed the convexity of the integral transform $$V_{\\lambda}(f)(z):=\\int_{0}^{1}\\lambda(t)\\frac{f(tz)}{t}dt,$$ where $\\lambda$ is a non-negative real-valued integrable function satisfying "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0732","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"}