Euler function is defined, for $|x|\le 1$, as follows: $$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$

Upper bounds for $\phi$ can be simply derived from ending the product early, e.g. $$\phi(x)<\prod_{i=1}^2(1-x^i)=1-x-x^2+x^3$$

What lower bounds are known for $\phi(x)$?

I'm especially interested in bounds which can be computed efficiently as this is needed for a computer application.

`mpmath`

can compute it exactly, does this help? $\endgroup$ – joro Nov 16 '15 at 16:34