suppose that $f$ is an entire function such that $f'(0)=0$ and $f''(1+1/n)=7-3/n$ for all $n$ natural number. we have to find all $f$ that satisfies these properties.
What I have done is: define $g(z)=f''(z)-10+3z$, so $g(1+1/n)=0$ so uniqueness theorem implies that $f''(z)=10-3z$ and therefore $f'(z)=10z-3z^2/2 +a$, As $f'(0)=0$ so $f(z)=5z^2-z^3/2+b(constant)$ is my solution is correct?
