I couldn't find any bounds on the Beta function, defined for $a_1,\ldots,a_n$ positive:

$B(a_1,\ldots,a_n) = \prod_i \Gamma(a_i) / \Gamma(\sum_i a_i)$.

where $\Gamma(x)$ is the gamma function.

Are there any useful lower *and* upper bounds for this function, where the bounds depend on $\sum a_i$ and the $a_i$ themselves perhaps? I am not talking about asymptotic bounds (but if you are aware of one that does not use Striling's approximation, that could also be useful perhaps).