I would like to learn pointers to how to prove "a function $f(x)$ always greater than function $g(x)$ in a interval $(a,b)$". In other words show that $f(x) > g(x)$ in $(a,b)$.
Their end point values are same.
$f(a) = g(a)$ $f(b) = g(b)$
$f(x)$ and $g(x)$ are functions used in Decision tree induction.
$f(x)$ is information gain.
$ f(x) = - \sum p(x)\log_2(p(x)) $
g(x) is GINI index function.
$ g(x) = 1 - \sum p(x)^2 $
$p(x)$ functions are probabilities therefore $ \sum{p(x)} = 1$.
I tried to show that $f(x) - g(x) > 0$ is always true. But I think I need formal name of this approach or other approaches.
I am engineer by profession therefore I would like to learn full names of approaches to search in the internet.
I added functions and more information about this problem.
A good explanation of my problem is given here. Example is about SPAM classification. http://people.cs.vt.edu/~ramakris/Courses/CS6604/lectures/lec9/lec9.pdf
$f(x)$ and $g(x)$ functions are discrete, probabilities are computed using given data. Above link has a figure which shows that between $(0,1)$, $f(x)$ is always greater that $g(x)$.
I thought that generic answers is possible but it seems that I am mistaken. Every problem is somewhat unique.