Limit of nonnegative analytic functions

Asked Jul 19 '16 at 10:24

Active Jul 19 '16 at 10:24

Viewed 55 times

Let $\{g_n\}$ be a sequence of analytic functions on the closed unit disk $\overline{D_1}$ that converges uniformly to $g$ such that $g_n$ is never zero in the open unit disk $D_1$ for all $n$, then $g$ is either always equal to 0 or never zero on $D_1$.

I am not very sure how to do this. I came up with a rough sketch but not sure if it is correct: $g$ is the uniform limit of analytic functions thus analytic. Suppose $g$ has a zero, then by maximum/minimum modulus principles (???), $g$ cannot have a true local max or min thus is always equal to zero. This sounds unconvincing thus I am looking for a better proof. Perhaps Rouche theorem?

Thanks for any help!

asked Jul 19 '16 at 10:24

yoyostein

19,608

1

Yes, that follows from Rouché's theorem. Compare http://math.stackexchange.com/questions/1631841/number-of-roots-of-a-sequence-of-a-uniformly-convergent-holomorphic-functions-im for a similar question. – Martin R Jul 19 '16 at 11:41
1

But you should replace "nonnegative" by "nonzero" ... – Martin R Jul 19 '16 at 11:43
I did some research, Hurwitz theorem looks promising too. – yoyostein Jul 19 '16 at 13:15
... Which is a consequence of Rouche's theorem. – yoyostein Jul 19 '16 at 13:25

Limit of nonnegative analytic functions

0 Answers0