Hello, thank you for reading.

f(z) is a holomorfic (<=>analytic) function in a disk D (D = {z: |z-z0|<R}, for some R) (notice that it's an open disk), and f is not identically zero.

I need to prove/disprove the next two claims.

My trouble is that I can in no way understand the difference between them! So what I'd actually like is an explenation of the difference of the two claims...

Thank you very much for reading/responding.

Here are the claims:

1) in every sub-Disk that is contained in D (not sure how to say that in English, but "totally contained", meaning it doesn't equal D) there is a finite number of "zeroes" of f(z). (z is zero <=> f(z)=0).

2) every circle that is contained in D (circle: D'={z: |z-z1|=R for some R), circles a finite number of zeroes of f(z).

Simply cannot see the difference between the claims. D is open - that's the problem...

Thanks in advance!

Tomer.