Bounds for Toader Mean in Terms of Arithmetic and Second Seiffert Means - He, Zai-Yin; Jiang, Yue-Ping; Chu, Yu-Ming (Communications in Mathematics and Applications, 2019)
In the article, we prove that the double inequalities \begin{align*} &\alpha_{1}T(a,b)+(1-\alpha_{1})A(a,b)<TD(a,b)<\beta_{1}T(a,b)+(1-\beta_{1})A(a,b),\\ &T^{\alpha_{2}}(a,b)A^{1-\alpha_{2}}(a,b)<TD(a,b)<T^{\beta_{2}}(a,b)A^{1-\beta_{2}}(a,b) \end{align*} hold for all \(a,b>0\) with \(a eq b\) if and only if \(\alpha_{1}\leq 3/4\), \(\beta_{1}\geq1\), \(\alpha_{2}\leq 3/4\) and \(\beta_{2}\geq 1\), where \(A(a,b)\), \(TD(a,b)\) and \(T(a,b)\) are the arithmetic, Toader and second Seiffert means of \(a\) and \(b\), respectively.