JIPAM

Certain Bounds for the Differences of Means  
 
  Authors: Peng Gao,  
  Keywords: Ky Fan's inequality, Levinson's inequality, Generalized weighted power means, Mean value theorem.  
  Date Received: 17/08/02  
  Date Accepted: 03/06/03  
  Subject Codes:

26D15,26D20

  
  Editors: Kenneth B. Stolarsky,  
 
  Abstract:

Let $ P_{n,r}(mathbf{x})$ be the generalized weighted power means. We consider bounds for the differences of means in the following form:

 
$displaystyle max left{frac {C_{u, v,beta}}{x^{betaalpha_1},frac {C_{u, v,beta}}{x^{2beta-alpha}_n} right}sigma_{n, w^{prime},beta}$ $displaystyle geq frac { P^{alpha}_{n,u} - P^{alpha}_{n,v}}{alpha $
$displaystyle geq min left{frac {C_{u, v,beta} }{x^{2beta-alpha_1},frac {C_{u, v,beta}}{x^{2beta-alpha}_n} right}sigma_{n,w,beta}.$

Here $ beta neq 0,sigma_{n,t,beta}(mathbf{ x})=sum_{i=1}^{n}omega_i[x^{beta}_i-P^{beta}_{n,t}(mathbf{x})]^2$ and $ C_{u, v,beta}= frac {u-v}{2beta^2}$ . Some similar inequalities are also considered. The results are applied to inequalities of Ky Fan's type.

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