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

Zai-Yin He, Yue-Ping Jiang, Yu-Ming Chu

Abstract


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 \(a\neq b\) 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.


Keywords


Toader mean; Second Seiffert mean; Arithmetic mean

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References


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DOI: http://dx.doi.org/10.26713%2Fcma.v10i3.1200

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