Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $a_1,\ldots,a_n$ be a set of $n$ positive numbers.

Are there known lower and upper bounds on:

$\displaystyle\frac{\prod_{i} \Gamma(a_i)}{\Gamma(\sum_i a_i)}$

where $\Gamma$ is the Gamma function (a generalization of the factorial distribution)?

share|improve this question

1 Answer 1

You can use Stirling's approximation. Considering the multinomial coefficient in $a_i$ instead of what you wrote, and putting $p_i = a_i/\sum_i a_i$, you should get that the logarithm of the multinomial coefficient is $$(1+o(1))(\sum_i a_i) H(p_1,\ldots,p_n),$$ where $H(\cdots)$ is the entropy function: $$H(p_1,\ldots,p_n) = -\sum_i p_i \log p_i.$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.