A New Upper Bound of the Logarithmic Mean

Volume 4, Issue 4, 2003

Article 80

A NEW UPPER BOUND OF THE LOGARITHMIC MEAN

GAO JIA AND JINDE CAO

DEPARTMENT OF APPLIED MATHEMATICS,
HUNAN CITY UNIVERSITY,
YIYANG 413000, CHINA

DEPARTMENT OF MATHEMATICS,
SOUTHEAST UNIVERSITY,
NANJING 210096, CHINA
E-Mail: jdcao@seu.edu.cn

Received 26 June, 2003; Accepted 05 November, 2003.
Communicated by: J. Sándor

ABSTRACT. Let

and

be positive numbers with $a \ne b$ . The inequalities about the logarithmic-mean

$\displaystyle L(a,b)<H_{p}(a,b)<M_q(a,b)$

are obtained, where $p\ge \frac{1}{2}$ and $q\ge \frac{2}{3}p$ . We would point out that $p=\frac{1}{2}$ and $q=\frac{1}{3}$ are the best constants such that above inequalities hold.

Key words:
Logarithmic mean; Power mean; Heron mean; Best constant.

2000 Mathematics Subject Classification:
26D15, 26D10.

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