Ordinary algebraic curves with many automorphisms in positive characteristic

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 $\mathrm{\mid }G\mathrm{\mid }\le 34(\mathcal{g}(\mathcal{X})+1{)}^{3\mathrm{/}2}$. 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}$)>cp^2$forsomepositiveconstant$c$.