Bounds for Toader Mean in Terms of Arithmetic and Second Seiffert Means

Abstract

In the article, we prove that the double inequalities \begin{align*} &\alpha_{1}T(a,b)+(1-\alpha_{1})A(a,b) 0\) with \(a\neq b\) 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.