Large finite products of small fractions

Abstract

Fix positive reals $a,b,c,d$, and let $h(x)$ be a real function behaving sort of like $\sin x$ near 0. Then, provided $m$ grows linearly with $n$. there exists a positive constant $C$ such that$$ \prod_{j=0}^m\frac{h\left((cj+a)\frac{d}{n}\right)}{h\left((cj+b)\frac{d}{n}\right)}\sim C n^{\frac{a-b}c}. $$