3.II.13G

(a) Let $f$ and $g$ be two analytic functions on a domain $D$ and let $\gamma \subset D$ be a simple closed curve homotopic in $D$ to a point. If $|g(z)|<|f(z)|$ for every $z$ in $\gamma$, prove that $\gamma$ encloses the same number of zeros of $f$ as of $f+g$.

(b) Let $g$ be an analytic function on the disk $|z|<1+\epsilon$, for some $\epsilon>0$. Suppose that $g$ maps the closed unit disk into the open unit disk (both centred at 0 ). Prove that $g$ has exactly one fixed point in the open unit disk.

(c) Prove that, if $|a|<1$, then

$z^{m}\left(\frac{z-a}{1-\bar{a} z}\right)^{n}-a$

has $m+n$ zeros in $|z|<1$.

*Typos? Please submit corrections to this page on GitHub.*