About the roots of a complex function

Question

Let $D=\{z\in\mathbb{C}:|z|<1\}$. Let $f_1$ and $f_2$ be two analytic functions on $D$. Consider $p:D\longrightarrow \mathbb{C}$ defined as $p(z)=f_1(z)-f_2(z)+k$ and $q(z)=f_1(z)-f_2(z)+k+k’$ where $k,0\neq k’\in\mathbb{C}$. Suppose $p$ has a root in $D$. Then can I choose $k’$ such that $q$ has a root in $D$?

Answer 1

Rouche's theorem tells you, among other things, that as long as $|k'|$ is smaller than the minimum of $|p|$ on the boundary of $D$, then $p$ and $q$ have the same number of zeroes (counted with multiplicity) on $D$.

If $p$ has a root on the unit circle (so that the minimum mentioned above is $0$ and therefore non-viable), then shrink $D$ slightly so that the root on the inside is still inside, but there is no root on the boundary, then do the same argument.

Finally, if $p(z)=0$, then no, there is no such $k'$.