is that function must be constant under the following conditions
I'm talking about complex function
$f$ is analytic function on a region $D$ that include the point $z=0$.
for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition $|f(\frac{1}{n})| \leq \dfrac{1}{2^n}$
is that function must be constant ?
Solution 1:
Yes, it must be constant - indeed, that constant must be $0$, since clearly $f(0)=0$ by continuity. Hint for proof: if $f$ were not identically zero, then write $f(z)=z^kg(z)$ where $g$ is analytic and $g(z)\ne0$.