Rouché's theorem applies !
Since the norm is never negative, this means that for all x in some region K if
0 < | g(x) | < | f(x) |
for all x such that f(x) =/= 0,
then
f(x) + g(x) has the same amount of zero's as f(x) in the region K.
See :
https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem