stochastic dominance and monotone functions

Question

Suppose $h\geq g$ are two strictly increasing functions over a compact domain $[\underline{x},\bar{x}]$. Furthermore, $h(\underline{x})=g(\underline{x})$ and $h(\bar{x})=g(\bar{x})$. Suppose $F_1$ first order stochastically dominates $F_2$. Then, is it true that $$\int hdF_1-\int hdF_2\geq \int gdF_1-\int gdF_2$$