Suppose that $X$ is a positive continuous random variable with infinitely differentiable pdf $f_\theta (x)$ and suppose that its expectation is increasing in $\theta$. That is, the function $$ g_{1}(\theta) = \mathbb{E} [X] = \int_{\mathbb{R}_+} x \, f_\theta(x) \, \mathrm{d} x $$ is increasing in $\theta$.
Now suppose that $h$ is infinitely differentiable, positive, and increasing ($h$ doesn't depend on $\theta$, either). Is it true that the function $$ g_2 (\theta) = \int_{\mathbb{R}_+} h(x) \, f_\theta(x) \, \mathrm{d} x $$ is increasing in $\theta$? (or are there any extra restrictions, such as convexity or concavity, which might ensure that the statement is correct?).
I've tried some examples, like $X \sim $ Normal$(\mu,1)$ along with some increasing transformations, and it seems to work. I've started taking derivatives inside the integral, but I haven't succeeded in proving it. Any help would be much appreciated, thanks!