Suppose $f(x),g(x),h(x)$ are continuous, and $h(x)>g(x)>0$. Does it always hold true that $$\left|\int f(x)g(x)\,\mathrm dx\right|<\left|\int f(x)h(x)\,\mathrm dx\right|?$$
Does it play a role if $f(x)$ changes sign on the domain of integration? If not, what would be the sufficient condition for the estimation?
Consider $$ \int_{-1}^1 1\cdot x\,\mathrm{d}x=0 $$ and $$ \int_{-1}^1 \frac{2+x}4\cdot x\,\mathrm{d}x=\frac16 $$
Suppose that $f$ doesn't change sign and does not vanish, and that $|X|\gt0$. Then $f(x)=c\,|f(x)|$, where $c=1$ over all of $X$ or $c=-1$ over all of $X$. $$ \begin{align} \left|\,\int_Xf(x)h(x)\,\mathrm{d}x\,\right|-\left|\,\int_Xf(x)g(x)\,\mathrm{d}x\,\right| &=|c|\left|\,\int_X|f(x)|h(x)\,\mathrm{d}x\,\right|-|c|\left|\,\int_X|f(x)|g(x)\,\mathrm{d}x\,\right|\\ &=\int_X|f(x)|(h(x)-g(x))\,\mathrm{d}x\\[6pt] &\gt0 \end{align} $$