Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal to $\left \lfloor{\frac{g+3}{2}}\right \rfloor$ (where $\mathbb{S}^2$ is the unit sphere). Can this bound be improved?

In a similar vein, if we now assume that $\Sigma$ has $r$ boundary components, Gabard showed that there exists a branched holomorphic covering $G : \Sigma \to \mathbb{D}$ satisfying $\deg G \leq g + r$, improving a theorem of Ahlfors (where $\mathbb{D}$ is the unit disk). Do you believe this bound is optimal?

More generally, do you know similar results?