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)

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)

[1] Alzer H.On Stolarsky’s mean value family [J].Internat J Math Ed Sci Tech,1987,20(1):186-189.

[2] Alzer H.über eine einparametrige Familie Von Mittelwerten[J].Bayer Akad Wiss Math-Natur Kl Sitzungsber,1988,1987:1-9.

[3] Alzer H.über eine einparametrige Familie Von Mittelwerten Ⅱ[J].Bayer Akad Wiss Math-Natur Kl Sitzungsber,1989,1988:23-29.

[4] Qi Feng.The extended mean values: definition, properties, monotonicities, comparision, convexities, generalizations, and applications[J].Cubo Math Educ,2003,5(3):63-90.

[5] Cheung W S, Qi feng.Logarithmic convexity of the one-parameter mean values[J].Taiwanese J Math,2007,11(1):231-237.

[6] Qi Feng, Cerone P, Dragomir S S,etal.Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values[J].Appl Math Comput,2009,208(1):129-133.

[7] Zheng Ningguo, Zhang Zhihua, Zhang Xiaoming.Schur-convexity of two types of one-parameter mean values innvariables[J].J Inequal Appl,2007,Article ID 78175.

[8] 毛其吉.兩正數(shù)的冪平均、對數(shù)平均與對偶海倫平均[J].蘇州教育學(xué)院學(xué)報,1999,16(1/2):82-85.

[9] Bullen P S, Mitrinovic D S, Vasic P M.Means and their inequalities[M].Dordrecht: D Reidel Publishing Co,1998.

[10] Alzer H, Janous W.Solution of problem 8*[J].Crux Math,1987,13:173-178.

Heron均值的一個嚴格一參數(shù)均值界

宗 誠1，褚玉明2

(1.杭州師范大學(xué)理學(xué)院，浙江 杭州 310036；2.湖州師范學(xué)院理學(xué)院，浙江 湖州 313000)

一參數(shù)均值；Heron平均；冪均值

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