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See the Theorem 2.2 for a proof . Generalized means are a family of functions for aggregating sets of numbers, that include as special cases the arithmetic, geometric, and harmonic means. Project supported by National Natural Science Foundation of China (Grant No. Generalized means are a family of functions for aggregating sets of numbers, that include as special cases the arithmetic, geometric, and harmonic means. x n n, if r = 0. is an increasing continuous function of r ∈ R. Moreover, M is strictly increasing unless all x i are equal. 1=r: As r!0 the rth weighted power mean tends to: M0 w (a 1;:::;a n) = Yn i=1 aw i i : which we call 0th weighted power mean. Mathematics subject classification(1991): 26D15, 26A48.
A separation of weighted power mean inequality was derived in this paper. We de ne the rth weighted power mean of non-negative numbers a 1;:::;a n as follows: Mr w (a 1;:::;a n) = Xn i=1 w ia r i!
Tamavas established mixed weighted power mean inequality in 1999. Let w 1;:::;w n be positive real numbers such that w 1 + +w n= 1. In the article, a new proof of the weighted power mean inequalities is given using Cauchy-Schwarz-Buniakowski’s inequality, and another two simple and short proofs of mono-tonicity for the generalized weighted mean values with two parameters are showed. As its applications, some separations of other inequalities were given.Prof., Department of Mathematics, Hexi University, 734000, Zhangye, P.R. As its applications, some separations of other inequalities …
The generalized mean is also known as power mean. The power mean inequality is a generalization of AM-GM which places the arithemetic and geometric means on a continuum of different means.
Weighted Power Means Inequality.
The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the final average, some data points contribute more than others. For a sequence of positive weights wi with sum= 1 we define the weighted power mean.You must activate Javascript to use this site. See the theorem 2.3 for that .
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We will prove the weighted power means inequality, which states that for any two real numbers r < s, the weighted power means of orders r and s of n positive real numbers x 1, x 2, …, x n satisfy the inequality.
A separation of weighted power mean inequality was derived in this paper. The purpose of this paper is to establish weighted power mean inequalities gen-eralizing Jocic ’s inequality (1) and to investigate the equality conditions of some of´ these inequalities for the Schatten p-norms.
For nonnegative real numbers , demonstrate that if then .