Means and Their Inequalities

Bullen, P. S.; Mitrinović, D. S.; Vasić, P. M.

doi:10.1007/978-94-017-2226-1

Cited by 323 publications

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“…As pointed out in [1], the theory of inequalities plays an important role in all the fields of mathematics. And the power mean is the most important one in all the means.…”

Section: Definition 12mentioning

confidence: 99%

“…And the power mean is the most important one in all the means. Many mathematicians wrote a great number of papers, and established the inequalities involving the power means and the related problems (see, e.g., [1,4,5,8,9,13,14,[17][18][19]21]). Recently, the authors studied the optimal real number λ such that the following J. Wen and W.-L. Wang 3 inequality:…”

Section: Definition 12mentioning

confidence: 99%

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The optimization for the inequalities of power means

Wen

Wang

2006

Journal of Inequalities and Applications

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Let M [t] n (a) be the tth power mean of a sequence a of positive real numbers, where a = (a 1 ,a 2 ,...,a n ),n ≥ 2, and α,λ ∈ R m ++ ,m ≥ 2, m j=1 λ j = 1,min{α} ≤ θ ≤ max {α}. In this paper, we will state the important background and meaning of the inequalityn (a); a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.Copyright © 2006 J. Wen and W.-L. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Symbols and introductionWe will use some symbols in the well-known monographs [1,5,13]:is a finite set, and it is not empty.Recall that the definitions of the tth power mean and Hardy mean of order r for a sequence a = (a 1 ,...,a n ) (n ≥ 2) are, respectively,, if 0< |t| < +∞,n (a) = n √ a 1 a 2 ··· a n , if t = 0, Hindawi Publishing Corporation

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“…As pointed out in [1], the theory of inequalities plays an important role in all the fields of mathematics. And the power mean is the most important one in all the means.…”

Section: Definition 12mentioning

confidence: 99%

Section: Definition 12mentioning

confidence: 99%

The optimization for the inequalities of power means

Wen

Wang

2006

Journal of Inequalities and Applications

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“…The most important properties of these and other mean values are given in the monograph [5]. Using the notation of power means, we can write (1.1) as…”

Section: H Alzermentioning

confidence: 99%

On a Gamma Function Inequality of Gautschi

Alzer¹

2002

Proceedings of the Edinburgh Mathematical Society

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“…Recently, C. E. M. Pearce and J. E. Pecaric have published the following extensions. If J2"=] u t = 1> t n e n S!|'(x, u) is the weighted power mean of order t. And, if «, = 1 (i = 1,..., n), then we have the sum of order t. Many remarkable properties of these power sums can be found in the monograph [3].…”

Section: Introductionmentioning

confidence: 99%