Comparing f(sigma(x)) and sigma(f(x))

[Perisona]

Hello,

The function in question has the following properties:

1. Is monotonically increasing
2. Is concave downwards
3. f(0) =0

I was trying to find a relation between sum(f(x_i)) and f(sum(x_i)) i=1 to i=n and all x_i positive.

I tried a few things, including this:

nx>=x
f'(nx)<=f'(x)
integrating 0 to x
(1/n)f(nx)<=f(x)
taking sigma
sum((1/n)f(nx))<=sum(f(x))

i tried using this along with jensen's inequality but couldn't draw any conclusions. Any help would be greatly appreciated.

 

[Dick]

Think about the linear function L(x)=x*f(sum(x_i))/sum(x_i). L(0)=0, L(sum(x_i))=f(sum(x_i)). But for 0<=x<=sum(x_i), L(x)<=f(x) because of your convexity condition. Can you fill in the rest? You don't need the monotone increasing condition.

 

[Perisona]

The function is concave downards f''(x)<0 so shouldn't the L(x) and f(x) relation reverse sign?

Also, how do I introduce a sigma(f(x_i)) into the inequalities?

 

[Dick]

Nooo. The chord to a concave downward function is below the function. Like f(x)=(-x^2). sigma(f(x_i))>=sigma(L(x_i))=L(sigma(x_i)).