Carleman's Inequality

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/CarlemansInequality.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CarlemansInequality.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

The Inequality

(∑_(i=1)^n)((((Subscript[a, 1]) (Subscript[a, 2]) (Subscript[⋯a, i])))^(1/i))≤E (∑_(i=1)^n)(Subscript[a, i])

Sum[((Subscript[a, 1])(Subscript[a, 2])(Subscript[⋯a, i]))^(1/i),{i,n}] ≤ E Sum[(Subscript[a, i]),{i,n}]

Numerical Testing

Carleman[l_List]:=Module[{i,n=Length[l]},
Sum[Product[l[[i]],{i,r}]^(1/r),{r,n}]<=E Plus@@l]

PowerExpand[Carleman[Table[a,{5}]]]

Union@Table[Carleman[Table[Random[Real,{0,100}],{10}]],{1000}]