Rouché's theorem states that, given two functions and analytic in with a simple loop homotopic to
a point in ,
if
for all
on ,
then
and
have the same number of roots inside .
A stronger version has been proved by Estermann (1962). The strong version also has a converse, as shown by Challener and Rubel (1982).
and 
analytic in 
with 
a simple loop homotopic to
a point in 
,
if 
for all 
on 
,
then 
and 
have the same number of roots inside 
.
