How to determine values of parameters such that an inequality is satisfied?

Given inequality f[x, y]>0 where f[x, y]=1/16 (-1 + x (2 - x + x^3 (-1 + y)^2 y^2)), how can one find the values x (keeping y fixed) such that the inequality is satisfied. And then repeat the same to find y (keeping x fixed)? The answer appears in equation (107) of this article:

$ $ 2(\sqrt{2}-1) \le x \le 1, \qquad \frac{1}{2}\left( 1- \sqrt{ \frac{x^2 +4x – 4}{x^2} } \right) \le y \le \frac{1}{2}\left( 1+\sqrt{ \frac{x^2 +4x – 4}{x^2} } \right)$ $