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  Volume 6, Issue 3, Article 75
Inequalities for Weighted Power Pseudo Means

    Authors: Vasile Miheşan,  
    Keywords: Weighted power pseudo means, Inequalities.  
    Date Received: 24/11/04  
    Date Accepted: 31/05/05  
    Subject Codes:

26D15, 26E60.

    Editors: Feng Qi,  

In this paper we denote by $ m_n^{[r]}$ the following expression, which is closely connected to the weighted power means of order $ r, M_n^{[r]}$.

Let $ ngeq 2$ be a fixed integer and

begin{displaymath} m_n^{[r]}(mathbf{x;p})= begin{cases} left(frac{P_n}{p_1}... ...i^{p_i/p_1}right., & r=0 end{cases}quad(mathbf{x}in R_r), end{displaymath}
where $ P_n=sum_{i=1}^np_i$ and $ R_r$ denotes the set of the vectors $ mathbf{x}=(x_1,x_2,dots,x_n)$ for which $ x_i>0$ $ (i=1,2,dots,n)$, $ mathbf{p }=(p_1,p_2,dots,p_n),$ $ p_1>0,$ $ p_igeq 0$ $ (i=1,2,dots,n)$ and $ P_nx_1^r>sum_{i=2}^np_ix_i^r$.

Three inequalities are presented for $ m_n^{[r]}$. The first is a comparison theorem. The second and the third is Rado type inequalities. The proofs show that the above inequalities are consequences of some well-known inequalities for weighted power means.

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