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  Volume 3, Issue 4, Article 48
 
On Weighted Inequalities with Geometric Mean Operator Generated by the Hardy-type Integral Transform
    Authors: Maria Nassyrova, Lars-Erik Persson, Vladimir Stepanov,  
    Keywords: Integral inequalities, Weights, Geometric mean operator, Kernels, Riemann-Liouville operators.  
    Date Received: 27/11/01  
    Date Accepted: 29/04/02  
    Subject Codes:

26D15,26D10.

  
    Editors: Bohumir Opic,  
 
    Abstract:

The generalized geometric mean operator

$displaystyle G_{K}f(x)=exp dfrac{1}{K(x)}int_{0}^{x}k(x,y)log f(y)dy,$

with $ K(x):=int_{0}^{x}k(x,y)dy$ is considered. A characterization of the weights $ u(x)$ and $ v(x)$ so that the inequality

$displaystyle left( int_{0}^{infty }left( G_{K}f(x)ight) ^{q}uleft( xight) dxight) ^{1/q} qquadqquadqquad $

$displaystyle qquadqquadqquadleq Cleft( int_{0}^{infty }f(x)^{p}v(x)dxight) ^{1/p},quad fgeq 0, $

holds is given for all

         
       
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