Optimal bound on the a-number of curves in positive characteristic

Let $C$ be a curve of genus $g$ over a field of characteristic $p>0$. Write $a(C)$ for the a-number of its Jacobian, and let $r$ be the rank of the Cartier operator on $H^0(C,\Omega_C^1)$. The optimal a-number bound.

$$a(C) \leq \frac{p-1}{p}g+\frac{p-1}{2},$$

equivalently,

$$g \leq pr+\frac{p(p-1)}{2}.$$

This would generalize the known genus bounds for superspecial curves and for curves with $a(C)=g-1$.