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).

  • $\begingroup$ $\Gamma$ is log-convex on $\mathbb{R}^+$, so given the sum, $B$ is minimized when all $a_i$s are equal. $B$ diverges as $a_i\to0$, since $\Gamma(x)\sim x^{-1}$. Beyond that, what sort of bounds are you looking for? $\endgroup$ – Omer Mar 6 '11 at 5:12
  • $\begingroup$ Also, you might be able to play off of approximations to $\binom{n}{m} \lt (2^n)/\sqrt(n)$ or something like that. If you knew relations among the $a_i$, tighter bounds could be formed for you. Can you tell us more? Gerhard "Ask Me About Upper Bounds" Paseman, 2011.03.05 $\endgroup$ – Gerhard Paseman Mar 6 '11 at 6:19

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