KöMaL 2014 A609

Let $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ be complex numbers satisfying $\mathop{\rm Im} a_j\ge 1$ and $\mathop{\rm Im} b_j\le -1$ ($j=1,2,\dots,n$ ), and let $f(z) = \frac{(z-a_1)(z-a_2)\dots (z-a_n)}{(z-b_1)(z-b_2)\dots (z-b_n)}$. Prove that the function $f'(z)$ has no root in the set $|\mathop{\rm Im}z|<1$.