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

  • Thread starter Perisona
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In summary, the conversation discusses the properties of a function, including its monotonicity and concavity, as well as using Jensen's inequality to find a relation between sum(f(x_i)) and f(sum(x_i)) when all x_i are positive. The speaker suggests using the linear function L(x)=x*f(sum(x_i))/sum(x_i) and the concavity condition to prove that L(x)<=f(x) for 0<=x<=sum(x_i). The use of sigma(f(x_i)) in the inequalities is also mentioned.
  • #1
Perisona
2
0
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.
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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)).
 

What is the difference between f(sigma(x)) and sigma(f(x))?

The main difference between these two expressions is the order in which the functions are applied to the input variable x. In f(sigma(x)), the function sigma is applied first and then the function f is applied to the result. In sigma(f(x)), the function f is applied to x first and then the output is passed to the function sigma.

Which expression is evaluated first, f(sigma(x)) or sigma(f(x))?

The expression that is evaluated first depends on the order in which the functions are written. If f(sigma(x)) is written first, then it will be evaluated first. If sigma(f(x)) is written first, then it will be evaluated first.

Do f(sigma(x)) and sigma(f(x)) always give the same result?

No, the results of these two expressions may not always be the same. This is because the functions f and sigma may have different properties and may not always commute, meaning the order of their application can affect the result.

How do I know whether to use f(sigma(x)) or sigma(f(x)) in my calculations?

The choice between these two expressions depends on the specific problem and the properties of the functions f and sigma. It may be helpful to simplify the expressions or to use specific values for x to compare the results and determine which expression is more appropriate.

Are there any specific cases where f(sigma(x)) and sigma(f(x)) will always give the same result?

Yes, if the functions f and sigma commute, meaning the order of their application does not affect the result, then f(sigma(x)) and sigma(f(x)) will always give the same result. In this case, the expressions are said to be equivalent.

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