Question 5
Let $p$ be a positive integer and let $q,z$ be real numbers. Suppose $q^{p+1}\le z\le1$ and $0\le q\le1$. Prove that
$$\prod_{k=1}^p\left|\frac{z-q^k}{z+q^k}\right|\le\prod_{k=1}^p\frac{1-q^k}{1+q^k}.$$
$$\prod_{k=1}^p\left|\frac{z-q^k}{z+q^k}\right|\le\prod_{k=1}^p\frac{1-q^k}{1+q^k}.$$