Let $q$ be a prime power and let $F$ be a global function field, i.e. a finite extension of $\mathbb F_q(t)$. Suppose that $\mathbb F_q$ is its full constant field. Suppose in addition that $F$ has degree $n$ and it is Galois over $\mathbb F_q(t)$. I am interested in an estimate for the genus of $F$ in this general case.
The problem becomes easy if one restricts to the case in which the characteristic of the base field does not divide $n$, as one can simply replace the different exponents with the ramification exponent -1 in the Hurwitz formula.
An option is using Castelnuovo inequality recursively, but this leads to a bad estimate I believe.
Any ideas for a good estimate?