Sharp bounds for Seiffert mean in terms of root mean square

Chu Yu-Ming, Hou Shou-Wei, Shen, Shen Zhong-Hua

Description

We find the greatest value α and least value β in (1 / 2,1) such that the double inequality S ( α a + ( 1 - α ) b , α b + ( 1 - α ) a ) < T ( a , b ) < S ( β a + ( 1 - β ) b , β b + ( 1 - β ) a ) holds for all a,b > 0 with a ≠ b . Here, T ( a, b ) = ( a - b )/[2 arctan(( a - b )/( a + b ))] and S ( a, b ) = [( a 2 + b 2 )/2] 1 / 2 are the Seiffert mean and root mean square of a and b , respectively. 2010 Mathematics Subject Classification : 26E60.

Sujets

• Root mean square
• Generalized mean
• Inequality

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 5
• Langue: English

Extrait

Chuet al.Journal of Inequalities and Applications2012,2012:11 http://www.journalofinequalitiesandapplications.com/content/2012/1/11

R E S E A R C HOpen Access Sharp bounds for Seiffert mean in terms of root mean square 1* 22 YuMing Chu, ShouWei Houand ZhongHua Shen

* Correspondence: [e-mail] 1 Department of Mathematics and Computing Science, Hunan City University, Yiyang 413000, China Full list of author information is available at the end of the article

Abstract We find the greatest valueaand least valuebin (1/2,1) such that the double inequality Sαa+ 1−αb,αb+ 1−αa<T a,b<Sβa+ 1−βb,βb+ 1−βa

holds for alla,b> 0 witha≠b. Here,T(a, b) = (ab)/[2 arctan((ab)/(a+b))] andS(a, 2 21/2 b) = [(a+bthe Seiffert mean and root mean square of)/2] areaandb, respectively. 2010 Mathematics Subject Classification: 26E60. Keywords:Seiffert mean, root mean square, power mean, inequality

1 Introduction Fora,b> 0 witha≠bthe Seiffert meanT(a, b) and root mean squareS(a, b) are defined by a−b T(a,b) = a−b (1:1) 2 arctan a+b and 2 2 a+b (1:2) S(a,b) = 2 respectively. Recently, both mean values have been the subject of intensive research. In particular, many remarkable inequalities and properties forTandScan be found in the literature [114]. √ pp 1/p Let A a,b=a+b/2,G a,b=ab , andM p (a, b) = ((a+b)/2) (p≠0) and M 0 a,b=ab be the arithmetic, geometric, andpth power means of two positive numbersaandb, respectively. Then it is well known that G a,b=M 0 a,b<A a,b=M 1 a,b<T a,b<S a,b=M 2 a,b for alla, b> 0 witha≠b. Seiffert [1] proved that inequalities A a,b<T a,b<S a,b

© 2012 Chu et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.