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An Inequality Between Compositions of Weighted Arithmetic and Geometric Means  
 
  Authors: Finbarr Holland,  
  Keywords: Weighted averages, Carleman's inequality, Convexity, Induction.  
  Date Received: 09/06/06  
  Date Accepted: 08/12/06  
  Subject Codes:

Primary 26D15.

  
  Editors: Grahame Bennett,  
 
  Abstract:

Let $ mathbb{P}$ denote the collection of positive sequences defined on $ mathbb{N}$. Fix $ win mathbb{P}$. Let $ s, t$, respectively, be the sequences of partial sums of the infinite series $ sum w_k$ and $ sum s_k$, respectively. Given $ xin mathbb{P}$, define the sequences $ A(x)$ and $ G(x)$ of weighted arithmetic and geometric means of $ x$ by

$displaystyle A_n(x)=sum_{k=1}^nfrac{w_k}{s_n} x_k, G_n(x)=prod_{k=1}^nx_k^{w_k/s_n}, n=1,2, dotsc $
Under the assumption that $ log t$ is concave, it is proved that $ A(G(x))le G(A(x))$ for all $ xin mathbb{P}$, with equality if and only if $ x$ is a constant sequence.;


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