Geodesic
A Double Inequality for the Combination of Toader Mean and the Arithmetic Mean in Terms of the Contraharmonic Mean
Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 237 .

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We find the greatest value $\lambda$ and the least value $\mu$ such that the double inequality \begin{align*} C(ambda a+(1-ambda)b,ambda b+(1-ambda)a)(a,b)+(1-lpha)T(a,b) (\mu a+(1-\mu)b,\mu b+(1-\mu)a) \end{align*} holds for all $\alpha\in(0,1)$ and $a,b>0$ with $a\neq b$, where $C(a,b)$, $A(a,b)$, and $T(a,b)$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$.
DOI : 10.2298/PIM141026001J
Classification : 26E60 33E05
Keywords: bound, contraharmonic mean, arithmetic mean, Toader mean, complete elliptic integrals 
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     author = {Wei-Dong Jiang and Feng Qi},
     title = {A {Double} {Inequality} for the {Combination} of {Toader} {Mean} and the {Arithmetic} {Mean} in {Terms} of the {Contraharmonic} {Mean}},
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Wei-Dong Jiang; Feng Qi. A Double Inequality for the Combination of Toader Mean and the Arithmetic Mean in Terms of the Contraharmonic Mean. Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 237 . doi : 10.2298/PIM141026001J. http://geodesic.mathdoc.fr/articles/10.2298/PIM141026001J/

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