Can i find some constants $a$ and $b$ in $\mathbb{R}_+^d$ such that for all $\mathbf{x} \in \mathbb{R}_{+}^{d}$, the inverse beta function :

$$IB(\mathbf x) = \frac{\Gamma\left(\lvert \mathbf x\rvert\right)}{\prod\limits_{i=1}^d \Gamma(x_i)}$$

satisfies $$\mathbf a^{\mathbf x} \le IB(\mathbf x) \le \mathbf b^{\mathbf x}\text{ ?}$$

Or is there some other known bounds on this function that have another shape ? this is the shape i think of, but maybe another shape can be usefull too.

If it is usefull, you can take the restiction that all $x_i$ except the first are integers.

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Can i find some constants $\mathbf a$ and $\mathbf b$ in $\mathbb{R}_+^d$ such that for all $\mathbf{x} \in \mathbb{R}_{+}^{d}$, the inverse beta function :

$$IB(\mathbf x) = \frac{\Gamma\left(\lvert \mathbf x\rvert\right)}{\prod\limits_{i=1}^d \Gamma(x_i)}$$

satisfies $$\mathbf a^{\mathbf x} \le IB(\mathbf x) \le \mathbf b^{\mathbf x}\text{ ?}$$

Or is there some other known bounds on this function that have another shape ? this is the shape i think of, but maybe another shape can be usefull too.

If it is usefull, you can take the restiction that all $x_i$ except the first are integers.

[Please re-tag the question if you think these are not the right tags]