Is this inequality true? (Inequality involving probability distribution and products)

Question

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

$$\prod_{z \in S}{g(z)^{f(z)}}+\prod_{z \in S}{h(z)^{f(z)}} \leq \prod_{z \in S}{[g(z)+h(z)]^{f(z)}}?$$

My first impulse was to use Jensen's Inequality, but to no avail. This inequality is part of a much larger theorem I am trying to prove. Thanks for your help!

Thanks. It's actually an old question that no one ever answered. — Mike 105, Sep 27 '13 at 23:40

score 2 · Accepted Answer · edited Apr 13 '17 at 12:20

2

This is just the superadditivity of the (generalized) geometric mean. My answer here shows how to prove it when $f$ is a uniform distribution on $n$ points. The general case follows in the same way, using the generalized AM-GM inequality instead of the usual AM-GM inequality.

edited Apr 13 '17 at 12:20

Community

1

answered Sep 27 '13 at 23:55

Thank you so much! You're answer allowed me to get over my mental block. – Mike 105 Sep 29 '13 at 16:55
Glad to help out! – Sep 29 '13 at 16:59

Is this inequality true? (Inequality involving probability distribution and products)

1 Answers1