TITLE: A problem related to Rouche's Theorem.
QUESTION [6 upvotes]: In Rouche's theorem, If we replace analytic property of functions $f(z)$ and $h(z)$ with meromorphic then this theorem will not be valid anymore. 
I want to illustrate this fact by producing some $f(z)$ and $h(z)$ which are meromorphic on some bounded domain D (where D have piecewise smooth boundary $\partial D$). Such that $f(z)$ and $h(z)$ have no poles on $\partial D$ and $|h(z)|<|f(z)|$, $\forall z\in\partial D$ but $f+h$ and $h$ have different number of zeros in D.

REPLY [3 votes]: (I assume you mean $f+h$ and $f$.)  Take $D$ the closed unit disc, $h(z) = z^{-1}$ and $f(z) = 2$.