THE MONOTONICITY OF THE RATIO OF TWO ABELIAN INTEGRALS

In this paper, we study the monotonicity of the ratio of two Abelian integrals I-0(h) = integral(Gamma h) ydx and I-1(h) = integral(Gamma h) xy dx, where Gamma(h) is a compact component of the level set {(x, y) : y(2) + Psi(x) = h, h is an element of J}; here J is an open interval. We first give a new criterion for determining the monotonicity of the ratio of the above two Abelian integrals. Then using this new criterion, we obtain some new Hamiltonian functions H(x, y) so that the ratio of the associated two Abelian integrals is monotone. Especially when H(x, y) has the form y(2) + P-5(x), we obtain the sufficient and necessary conditions that the ratio of two Abelian integrals is monotone, where P-5(x) is a polynomial of x with degree five.