Given Gain function, how to design a causal, stable and minimum phase IIR filter?
If $H(\omega)=e^{\alpha(\omega)+j\phi(\omega)}$ is a minimum phase frequency response, then the attenuation $\alpha(\omega)$ and the phase $\phi(\omega)$ are related by the following Hilbert transform relationship:

$$\phi(\omega)=-\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\alpha(\Omega)}{\omega-\Omega}d\Omega$$

So $\phi(\omega)$ is uniquely determined by $\alpha(\omega)$. This is the corresponding wikipedia entry.