{"paper":{"title":"The proof of three power-exponential inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anibal Coronel, Fernando Huancas","submitted_at":"2014-09-06T00:02:38Z","abstract_excerpt":"In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{\\^i}rtoaje in the following two works:\n  \"{\\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009\" and \"{\\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011\". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\\in [0,e]$. In addition, three new conjectures are presented."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.1968","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"}