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i'm looking for some class of functions [tex]\phi(t) [/tex] that satisfy :

[tex]\int_T \ t^n \phi(t) \, dt = \left( \int_T \ t \phi(t) \, dx \right)^n ; n=0,1,2,3 ...[/tex]

from what i understand - if i'm not mistaken - the problem transforms to finding a set of measure spaces whose measure [tex]\ ds =\phi(t)dt[/tex] , and nth norm of [tex] t [/tex]

[tex]\left\|t\right\|_n = \left( \int \ t^n\ ds \right)^\frac{1}{n} [/tex]

satisfy :

1 - [tex]\int ds =1[/tex]

2 - [tex]\left\|t\right\|_n=\left\|t\right\|_1 ; n=2,3,4 ..... [/tex]

obviously this problem is appropriately studied in [tex] L^p [/tex] spaces .

if i'm not mistaken , can you help me please , and if not , would you please advise .

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# A problem in Lp spaces .

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