On the monotonicity of the ratio of some hyperelliptic integrals of order 7

Abstract

In this paper, we study the monotonicity of the ratio of some hyperelliptic integrals of order 7. These integrals are related to the 1-forms ydx,xydx and defined by the hyperelliptic curves of dH=0, with H(x,y)=y2+P7(x), where P7 is a polynomial of degree 7 depending on three parameters α,β and γ. Our results are based on the well-known algebraic criteria.