Suppose $f(z)$ is a discrete probability distribution with space $S$. Suppose $g(z),h(z)>0$ for all $z \in S$. Is it true that
$$\prod_{z \in S}{g(z)^{f(z)}}+\prod_{z \in S}{h(z)^{f(z)}} \leq \prod_{z \in S}{[g(z)+h(z)]^{f(z)}}?$$
I'm writing a thesis and this inequality is a lemma to a much broader theorem regarding concavity that I am currently trying to prove. Thanks for your help!
This is an extension of Mahler's inequality. See http://en.wikipedia.org/wiki/Mahler%27s_inequality
We know from the general version of the AM-GM ineqaulity that $\sum f(z) g(z) \geq \Pi g(z)^{f(z)}$ for a discrete distribution $f(z)$ and $g(z) \geq 0$.
In our case, we apply this twice:
To $\frac{g(z)}{g(z) + h(z)}$ (Since $g(z), h(z) > 0$, we can do this): $\sum f(z)\frac{g(z)}{g(z) + h(z)} \geq \Pi \left(\frac{g(z)}{g(z)+ h(z)}\right)^{f(z)}$
To $\frac{h(z)}{g(z) + h(z)}$ (Since $g(z), h(z) > 0$, we can do this again): $\sum f(z)\frac{h(z)}{g(z) + h(z)} \geq \Pi \left(\frac{h(z)}{g(z)+ h(z)}\right)^{f(z)}$
Adding these two equations gives us: $$ \sum f(z)\frac{g(z)}{g(z) + h(z)} + \sum f(z)\frac{h(z)}{g(z) + h(z)} \geq \Pi \left(\frac{g(z)}{g(z)+ h(z)}\right)^{f(z)} + \Pi \left(\frac{h(z)}{g(z)+ h(z)}\right)^{f(z)} $$ Rearranging, we get: $$ \left(\sum f(z)\frac{g(z) + h(z)}{g(z) + h(z)}\right)\Pi\left(g(z)+ h(z)\right)^{f(z)} \geq \Pi g(z)^{f(z)} + \Pi h(z)^{f(z)} $$ Since the first term on the left hand side is 1, we get the desired inequality: $$ \Pi\left(g(z)+ h(z)\right)^{f(z)} \geq \Pi g(z)^{f(z)} + \Pi h(z)^{f(z)} $$
Notation shorthand: $\sum$ means $\sum_{z \in S}$ and $\Pi$ means $\Pi_{z \in S}$.
