Sharp One-Parameter Mean Bounds for Heron Mean
2011-11-23ZONGChengCHUYuming
ZONG Cheng, CHU Yu-ming
(1.College of Science, Hangzhou Normal University, Hangzhou 310036, China; 2.Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China)
Sharp One-Parameter Mean Bounds for Heron Mean
ZONG Cheng, CHU Yu-ming*
(1.College of Science, Hangzhou Normal University, Hangzhou 310036, China; 2.Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China)
one-parameter mean; Heron mean; power mean
1 Introduction
(1)
(2)
respectively.
It is well-known that one-parameter meanJp(a,b) is continuous and strictly increasing with respect top∈Rfor fixeda,b>0 witha≠b.Many mean values are the special case of the one-parameter mean, for example
Recently, the one-parameter and the Heron means have been the subject of intensive research.In particular, many remarkable inequalities for these means can be found in the literature [1-8].
Forr∈Rthe power meanMr(a,b) of orderrof two positive real numbersaandbis defined by
(3)
The main properties of the power mean are given in [9].In [10], Alzer and Janous established a sharp double inequality as follows:
(4)
for alla,b>0 witha≠b.
(5)
2 Main Result
(6)
Let
(7)
then simple computations lead to
f(1)=0
(8)
and
(9)
fort>1.
Next, we prove that
(10)
for alla,b>0 witha≠b.
(11)
We clearly see that
f(t)>0
(12)
for allt>1.
For anyε>0 andx>0, from (1) and (2) we have
(13)
Letx→0, making use of the Taylor expansion one has
(14)
(15)
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Heron均值的一个严格一参数均值界
宗 诚1,褚玉明2
(1.杭州师范大学理学院,浙江 杭州 310036;2.湖州师范学院理学院,浙江 湖州 313000)
一参数均值;Heron平均;幂均值
date:2010-10-25
Supported by the Natural Science Foundation of China (11071069) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (T200924).
Biography:ZONG Cheng(1985—), female, born in Huainan, Anhui Province, pure mathematical major graduate, engaged in complex analysis theory.
*CorrespondingauthorCHU Yu-ming(1966—), male, born in Huzhou, Zhejiang Province, professor, engaged in complex analysis theory.E-mail: chuyuming@hutc.zj.cn
10.3969/j.issn.1674-232X.2011.04.004
O178MSC2010: 26E60ArticlecharacterA
1674-232X(2011)04-0309-03