An entire function $f(z)$ has the properties that $f(0)=10,f'(0)=0$, and $f''(1+ \frac{1}{n})=2- \frac{3}{n}$ for each $n=1,2,3,...$ Express $f(z)$ by an exact formula and prove that it is the unique function with these conditions.
My work: Since $f(z)$ is entire, define $g(z)= f ''(z)-5+3z$, and so, we have $g(1+\frac{1}{n})=0$. Thus by uniqueness theorem, we obtain $f''(z)=5-3z$. Therefore, $f'(z)=5z-\frac{3z^2}{2}+a$. since $f'(0)=0$, so we get $f(z)=\frac{5z^2}{2}-\frac{z^3}{2}+b$, Since $f(0)=10$, we thus obtain, $f(z)=\frac{5z^2}{2}-\frac{z^3}{2}+10$. Did i miss anything? How to prove that it is unique functions with these conditions? Your kind help will appreciated. Thank you so much!
What you did is fine. In order to prove that it is unique, suppose that $g$ satisfies the same conditions. Then, by the identity theorem, $g''=f''$. Therefore $g'-f'$ is constant. But $g'(0)=f'(0)$ and therefore $g'-f'=0$. So, $g-f$ is constant. But $g(0)-f(0)=0$ and therefore $g-f=0$, that is, $g=f$.