New criteria for the monotonicity of the ratio of two Abelian integrals

New criteria to determine the monotonicity of the ratio of two Abelian integrals are given. When two Abelian integrals have the forms ∫Γhf1(x)ydx and ∫Γhf2(x)ydx or the forms ∫Γhf1(x)ydx and ∫Γhf2(x)ydx and Γh are ovals belonging to the level set {(x,y)|H(x,y)=h}, where H(x,y) has the form y2/2+Ψ(x) or ϕ(x)y2/2+Ψ(x), we give new criteria, which are defined directly by the functions which appear in the above Abelian integrals, and prove that the monotonicity of the criteria implies the monotonicity of the ratios of the Abelian integrals. The new criteria are applicable in a large class of problems, some of which simplify the existing proofs and some of which generalize known results.