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?