A best-possible double inequality between Seiffert and harmonic means

Chu Yu-Ming, Wang Miao-Kun, Wang, Wang Zi-Kui

Description

In this paper, we establish a new double inequality between the Seiffert and harmonic means. The achieved results is inspired by the papers of Sándor (Arch. Math., 76, 34-40, 2001) and Hästö (Math. Inequal. Appl., 7, 47-53, 2004), and the methods from Wang et al. (J. Math. Inequal., 4, 581-586, 2010). The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal. 2010 Mathematics Subject Classification: 26E60.

Sujets

• Harmonic mean
• Inequality

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 6
• Langue: English

Extrait

Chuet al.Journal of Inequalities and Applications2011,2011:94 http://www.journalofinequalitiesandapplications.com/content/2011/1/94

R E S E A R C HOpen Access A bestpossible double inequality between Seiffert and harmonic means 1* 12 YuMing Chu, MiaoKun Wangand ZiKui Wang

* Correspondence: [e-mail] 1 Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China Full list of author information is available at the end of the article

Abstract In this paper, we establish a new double inequality between the Seiffert and harmonic means. The achieved results is inspired by the papers of Sándor (Arch. Math., 76, 3440, 2001) and Hästö (Math. Inequal. Appl., 7, 4753, 2004), and the methods from Wang et al. (J. Math. Inequal., 4, 581586, 2010). The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal. 2010 Mathematics Subject Classification:26E60. Keywords:harmonic mean, Seiffert mean, inequality

1 Introduction Fora,b >0 witha≠b, the Seiffert meanP(a,b) was introduced by Seiffert [1] as fol lows: a−b P(a,b) = (1:1) 4 arctana/b−π Recently, the bivariate mean values have been the subject of intensive research. In particular, many remarkable inequalities for the Seiffert mean can be found in the lit erature [19]. LetH(a,b) = 2ab/(a+b), G a,b=ab ,L(a,b) = (ab)/(loga logb),I(a,b) = 1/e b a1/(ba2) 2p p1/p (b/a) ,A(a,b) = (a+b)/2,C(a,b) = (a+b)/(a+b), andM p (a,b) = ((a+b)/2) (p≠0) andbe the harmonic, geometric, logarithmic, identric, arith M 0 a,b=ab metic, contraharmonic, andpth power means of two different positive numbersaand b, respectively. Then, it is well known that min{a,b}<H(a,b) =M −1 (a,b)<G(a,b) =M 0 (a,b)<L(a,b) <I a,b<A a,b=M 1 a,b<C a,b<max{a,b} For alla,b >0 witha≠b, Seiffert [1] established thatL(a,b)< P(a,b)< I(a,b); Jagers [4] proved thatM 1/2 (a,b)< P(a,b)< M 2/3 (a,b) andM 2/3 (a,b) is the best possible upper power mean bound for the Seiffert meanP(a,b); Seiffert [7] estab lished thatP(a,b)> A(a,b)G(a,b)/L(a,b) andP(a,b)>2A(a,b)/π; Sándor [6] pre sented that (A(a,b) +G(a,b))/2<P(a,b)<A(a,b)(A(a,b) +G(a,b))/2 and 3 2 A(a,b)G(a,b)<P(a,b)<(G(a,b) + 2A(a,b))/ ; Hästö [3] proved thatP(a,b)>

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