A Criterion for Determining the Monotonicity of the Ratio of Two Abelian Integrals

A new method to determine the monotonicity of the ratio of two Abelian integrals is given. We find two criterion functions defined directly by the functions which appear in the Abelian integrals. The monotonicities of the criterion functions imply the monotonicity of the ratio of the Abelian integrals. By using this criterion, several known results can be deduced easier, and some new results are obtained.