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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)?

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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: $$H(p_1,\ldots,p_n) = -\sum_i p_i \log p_i.$$

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