# Thread: zeros of an analytic function

1. ## zeros of an analytic function

let f, g be analytic on a domain containing a simple closed curve ɣ and its inside. Show that if |f(z)| > |g(z)| for all z in ɣ, then the two equations f(z)=g(z) and f(z)=0 have an equal number of solutions inside ɣ.

I do not know how to prove this T-T plz help me!!!

and use above to determine how many solutions are there for z^3 = z^2 + z + 6 in |z|<1. I know answer for this is "no solution" but why???

plz!!!

2. If $\displaystyle |h-f|<|f|\;\forall z\in D$ then by Rouche, f and h have the same number of zeros in D counted with multiplicities. Now,if $\displaystyle |g|<|f|$, and we let $\displaystyle h=f-g$, then $\displaystyle |h-f|<|f|$ and so by Rouche, h and f have the same number of zeros in D.

In your case, you wish to know the number of zeros of $\displaystyle z^3-z^2-z-6=0$ in the unit disc. So what happens if we let $\displaystyle h=z^3-z^2-z-6$ and $\displaystyle f=-6$?

3. thank so you much!!!!