As the title suggests, I would like to determine all holomorphic functions $f$ from the open unit disk $D=\{z:|z|<1\}$ to $\mathbb{C}$ satisfying $f''(1/n)+f(1/n)=0$ for all $n=2,3,4...$.
I have two useful theorems which I may use to solve this problem:
Theorem 1: If $f(z)=\sum_{n=0}^{\infty}a_{n}(z-c)^{n}$ converges on $|z-c|<R$, then $f'(z)=\sum_{n=0}^{\infty}na_{n}(z-c)^{n-1}$ converges on $|z-c|<R$ as well. Consequently, power series are infinitely differentiable on $|z-c|<R$.
Theorem 2 (Identity Theorem for Power Series): If $f(z)=\sum_{n=0}^{\infty}a_{n}(z-c)^{n}=0$ for $z=z_{k} \neq c, (k=1,2,3,...)$ and $z_{n}$ converges to $c$, then $a_{n}=0$ $\forall n$.
Other than these theorems, is there any other more straightforward and computational method to determine the desired result? Any help will be greatly appreciated.
take g(z)=f"+f , g is holomorphic on D since f is holomorphic and f" is also holomorphic because f is smooth on D. Then apply the uniqueness theorem. Therefore f(z)= c cos z + d sin z c, d are constant.