IJPAM: Volume 72, No. 1 (2011)

THE SHARP COMBINATION BOUNDS OF ARITHMETIC
AND LOGARITHMIC MEANS FOR SEIFFERT'S MEAN

Yefang Qiu$^1$, Miaokun Wang$^2$, Yuming Chu$^3$
$^{1,2}$Department of Mathematics
Zhejiang Sci-Tech University
Zhejiang, Hangzhou, 310018, P.R. CHINA
$^2$Department of Mathematics
Huzhou Teachers College
Zhejiang, Huzhou, 313000, P.R. CHINA


Abstract. For $a,b>0$, the arithmetic mean $A(a,b)$, logarithmic mean $L(a,b)$ and Seiffert's mean $P(a,b)$ are defined by

\begin{displaymath}
A(a,b)=\frac{a+b}{2},\quad L(a,b)=\left\{\begin{array}{ll}
...
...c{b-a}{\log b-\log a},&b\neq a,\\
a,& b=a
\end{array}\right.\end{displaymath}

and

\begin{displaymath}P(a,b)=\left\{\begin{array}{ll}
\frac{a-b}{4\arctan\sqrt{\frac{a}{b}}-\pi},&b\neq a,\\
a,&b=a,\end{array}\right.
\end{displaymath}

respectively.

In this paper we find the greatest value $\alpha$ and least value $\beta$ such that inequality $A^{\alpha}(a,b)L^{1-\alpha}(a,b)<P(a,b)<A^{\beta}(a,b)L^{1-\beta}(a,b)$ holds for all $a,b>0$ with $a\neq b$.

Received: January 7, 2010

AMS Subject Classification: 26E60

Key Words and Phrases: arithmetic mean, logarithmic mean, Seiffert's mean

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 72
Issue: 1