Ordinary algebraic curves with many automorphisms in positive characteristic

Let X be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus g (X) >= 2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K elementwise. For any solvable subgroup G of Aut(X) we prove that vertical bar G vertical bar <= 34(g(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 vertical bar G vertical bar <= 84(g(X) - g(X) and, for solvable groups G, the Gunby-Smith-Yuan bound vertical bar G vertical bar <= 6(g(X)(2) + 12 root 21g(X)(3/2)where g (X) > cp(2) for some positive constant c.