  Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible,
nonsingular) algebraic curve of genus $\mathcal{g}(\mathcal{X}) \ge 2$ defined
over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let
$Aut(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix
$\mathbb{K}$ element-wise. For any solvable subgroup $G$ of $Aut(\mathcal{X})$
we prove that $|G|\leq 34 (\mathcal{g}(\mathcal{X})+1)^{3/2}$. There are known
curves attaining this bound up to the constant $34$. For $p$ odd, our result
improves the classical Nakajima bound $|G|\leq
84(\mathcal{g}(\mathcal{X})-1)\mathcal{g}(\mathcal{X})$, and, for solvable
groups $G$, the Gunby-Smith-Yuan bound $|G|\leq
6(\mathcal{g}(\mathcal{X})^2+12\sqrt{21}\mathcal{g}(\mathcal{X})^{3/2})$ where
$\mathcal{g}(\mathcal{X})>cp^2$ for some positive constant $c$.
