{"paper":{"title":"A monotonicity result for the $q-$fractional operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bahaaeldin Abdalla, Juan J. Nieto, Thabet Abdeljawad","submitted_at":"2016-02-24T21:12:06Z","abstract_excerpt":"In this article we prove that if the $q-$fractional operator $(~_{q}\\nabla_{qa}^\\alpha y)(t)$ of order $0<\\alpha\\leq 1$ , $0<q<1$ and starting at some $qa \\in T_q=\\{q^k: k \\in \\mathbb{Z}\\}\\cup \\{0\\},~~a>0$ is positive such that $y(a) \\geq 0$, then $y(t)$ is $c_q(\\alpha)-$increasing, $c_q(\\alpha)=\\frac{1-q^\\alpha}{1-q}q^{1-\\alpha}$. Conversely, if y(t) is increasing and $y(a)\\geq 0$, then $(~_{q}\\nabla_{qa}^\\alpha y)(t)\\geq 0$. As an application, we proved a $q-$fractional version of the Mean-Value Theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07713","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"}