Optimal inequalities for a Toader-type mean by quadratic and contraharmonic means

Volume 11, Issue 1, pp 150--160 http://dx.doi.org/10.22436/jnsa.011.01.11 Publication Date: January 20, 2018       Article History


Zhengchao Ji - Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Qing Ding - College of Mathematics and Statistics, Hunan University of Finance and Economics, Changsha 410205, China. Tiehong Zhao - Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China.


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 \[\alpha_1Q(a,b)+(1-\alpha_1)C(a,b) <T_{Q,C}(a,b)<\beta_1Q(a,b)+(1-\beta_1)C(a,b),\] \[\qquad\ Q^{\alpha_2}(a,b)C^{1-\alpha_2}(a,b) <T_{Q,C}(a,b)<Q^{\beta_2}(a,b)C^{1-\beta_2}(a,b),\] \[\frac{Q(a,b)C(a,b)}{\alpha_3Q(a,b)+(1-\alpha_3)C(a,b)} <T_{Q,C}(a,b)<\frac{Q(a,b)C(a,b)}{\beta_3Q(a,b)+(1-\beta_3)C(a,b)},\] \[C\left(\sqrt{\alpha_4a^2+(1-\alpha_4)b^2},\sqrt{(1-\alpha_4)a^2+\alpha_4b^2}\right) <T_{Q,C}(a,b)<C\left(\sqrt{\beta_4a^2+(1-\beta_4)b^2},\sqrt{(1-\beta_4)a^2+\beta_4b^2}\right) \] hold for all \(a, b>0\) 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.