In this paper, we study the monotonicity of the ratio of the Abelian integrals, ${\oint }_{{\gamma }_i(h)}ydx$ and ${\oint }_{{\gamma }_i(h)}xydx$, in an interval, where $i=1,2$, and ${\gamma }_i(h)$ is a compact component of some hyperelliptic curves with genus 2 as $h\in {\gamma }_i(h)$. We give positive answers to the two conjectures proposed by Wang et al. (2014) [18].

中文翻译：

---

某些Abelian积分之比的单调性

在本文中，我们研究了Abelian积分之比的单调性， ${\oint }_{{\gamma }_{\mathrm{一世}}（H）}ÿdX$ 和 ${\oint }_{{\gamma }_{\mathrm{一世}}（H）}XÿdX$在一定间隔内 $\mathrm{一世}=\mathrm{1个}，2$和 ${\gamma }_{\mathrm{一世}}（H）$ 是某些超椭圆曲线的紧成分，其中属2为 $H\in {\gamma }_{\mathrm{一世}}（H）$。我们对Wang等人提出的两个猜想给出了肯定的答案。（2014）[18]。