# $f\left(\sum X_i\right) \leq \sum f(X_i)$, where $X_i\gt 0$; for what functions is this true?

In a previous post, the following inequality has been proven $${\left( {\sum\limits_{i = 1}^n {{W_i}} } \right)^a} \le \sum\limits_{i = 1}^n {{W_i}^a}$$ where $W_i\gt 0$, $0\lt a\lt 1$. I guess it is more correct to say that this is always greater, and it is valid for $a\gt 0$ not just $0\lt a \lt 1$.

I am trying to see if one can generalize it to something like $$f\left( \sum\limits_{i = 1}^n W_i \right) \le \sum\limits_{i = 1}^n {f({W_i})}$$ where ${W_i}\gt 0$ ?

Under what circumstances and functions can this be true ?

It has been proven for the power function $f(x)=kx^a$ where $a,k\gt 0$.

Are there any other cases? Do you think there is any basic inequality to prove this?

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The condition $$f\left( {\sum\limits_{i = 1}^n {{W_i}} } \right) \le \sum\limits_{i = 1}^n {f({W_i})}$$ where ${W_i}$>0 is equivalent to $$f\left( {\sum\limits_{i = 1}^2 {{W_i}} } \right) \le \sum\limits_{i = 1}^2 {f({W_i})}$$ where ${W_i}$>0. These are subadditive functions. See Subadditivity

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I know von Neumann entropy is subadditive, but that's a function of a different kind, defined on a different kind of space. But I think the Shannon entropy has the same property. –  Raskolnikov Nov 26 '10 at 12:07