The monotonicity of the ratio of two hyperelliptic Abelian integrals for a class of symmetric potential systems of degree eight

This paper studies the monotonicity of the ratio of two hyperelliptic Abelian integrals I0(h)=∮Γhydx and I1(h)=∮Γhxydx, where Γh is a compact component of the level set {(x,y)∈R2:H(x,y)=h,h∈(h1,h2)}, with H(x,y)=y2+A(x), where A(x) is an even polynomial of degree 8 depending on two parameters α and β.