Find all the entire functions $f:\mathbb{C}\to\mathbb{C}$ that satisfy $$ |f(z)|\leq |z|^5 + \frac{1}{|z|^5} + \frac{1}{|z-1|^3}. $$
I proved, by using the Cauchy estimatives, that $f$ must be a polynomial of degree at most $5$. Also, if we write $$ f(z) = a_0 + a_1z + a_2z^2 + a_3z^3 + a_4z^4 + a_5z^5, $$ it is easy to see that $|a_5|\leq 1$ (just divide both sides of the inequality by $|z|^5$ and let $z\to \infty$ on the Riemann sphere). But I'm still unable to give a complete characterization of all entire functions satisfying the given inequality.
Any help will be appreciated.