%0 Journal Article %T Optimal inequalities for a Toader-type mean by quadratic and contraharmonic means %A Ji, Zhengchao %A Ding, Qing %A Zhao, Tiehong %J Journal of Nonlinear Sciences and Applications %D 2018 %V 11 %N 1 %@ ISSN 2008-1901 %F Ji2018 %X In this paper, we present the best possible parameters \(\alpha_i, \beta_i\ (i=1,2,3)\) and \(\alpha_4,\beta_4\in(1/2,1)\) such that the double inequalities \[\alpha_1Q(a,b)+(1-\alpha_1)C(a,b) 0\) with \(a\neq b\), where \(Q(a,b)\), \(C(a,b)\), and \(T(a,b)\) are the quadratic, contraharmonic, and Toader means, respectively, and \(T_{Q,C}(a,b)=T[Q(a,b),C(a,b)]\). As consequences, we provide new bounds for the complete elliptic integral of the second kind. %9 journal article %R 10.22436/jnsa.011.01.11 %U http://dx.doi.org/10.22436/jnsa.011.01.11 %P 150--160