title: Optimal inequalities for bounding Toader mean by arithmetic and quadratic means.
abstract: In this paper, we present the best possible parameters [Formula: see text] and [Formula: see text] such that the double inequality [Formula: see text] holds for all [Formula: see text] and [Formula: see text] with [Formula: see text], and we provide new bounds for the complete elliptic integral [Formula: see text] [Formula: see text] of the second kind, where [Formula: see text], [Formula: see text] and [Formula: see text] are the Toader, arithmetic, and quadratic means ofaandb, respectively.
authors.affiliation: School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000 China.
authors.forename: Tie-Hong
authors.initials: TH
authors.lastname: Zhao
authors.affiliation: School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000 China.
authors.forename: Yu-Ming
authors.initials: YM
authors.lastname: Chu
authors.affiliation: Icahn School of Medicine at Mount Sinai, Friedman Brain Institute, New York, 10033 United States.
authors.forename: Wen
authors.initials: W
authors.lastname: Zhang
keywords: Toader mean; arithmetic mean; complete elliptic integral; quadratic mean