I want to show that , with , has exactly one root in the open disc . I'm trying to do this with Rouche's theorem, that is:
finding a suitable function with exactly one zero in , that satisfies for all
I believe it's not all too hard to find such function, like however showing the desired inequality is for me problematic. Any help/suggestions?
I tried, with . Then with Rouche trying to show that f(z) has only 1 zero in D(0,1)
however it seems harder then I thought finding a suitable function with exactly one zero in D(0,1) and such that for all in D(0,1)
It must be a pretty clever choice for g since the most obvious choices like is not going to work.
Rouché's theorem says that if |f(z)| < |g(z)| on the boundary of some domain G, then g and f + g have the same number of zeros in G. The result you have quoted is the symmetric version of Rouché's theorem, though even in that version the inequality only needs to hold on the boundary of G, not in the whole of G. It is the usual version, not the symmetric version, that is useful here.
The inequality holds whenever z is on the boundary of D(0,1) (that is, when |z| = 1), so Rouché's theorem applies there.