Condition for expectation decreasing as upper bound increases

Question

Suppose I have two continuous functions $f(y_I)$ and $g(y_I,m)$, and I am interested in calculating the conditional expectation that $y_I \in [0,m]$: $$\int_{0}^m f(y_I) g(y_I,m)dy_I.$$ What conditions imply that the conditional expectation is decreasing in $m$? I know that $f(y_I)$ is strictly decreasing in $y_I$ and that $g(y_I, m)$ is strictly decreasing in $m$ and strictly decreasing in $y_I$.