Let $f$ and $g$ be two analytic functions with the same finite number of zeroes. Then there exists an analytic function $h:\mathbb{C}\to\mathbb{C}\setminus\{0\}$ and a closed path $\gamma$ with interior containing all zeroes of $f$ and $g$ such that $|f(z)-h(z)g(z)|<|f(z)|$ on $\gamma$.
This really looks like some inverse statement to Rouché's theorem, but how can I prove it?