Let $h(x, y): A \times B \rightarrow (0 , M]$ where $A = [0, a]$ and $B$ are two compact sets in $\mathbb{R}$ and M is a positive constant.
Write
$$f(y) = \int_0^a h(x,y) dx$$
I know that $y_0$ is an interior point (unique) of $B$ and is such that
$$f(y_0) = \int_0^a h(x,y_0) dx > f(y') \quad, \forall y' \neq y_0$$
In other words, $y_0$ minimizes f(y). Now, I noticed that some functions $g(\cdot)$ keeps this minimum such that
$$f_g(y_0) = \int_0^a g(h(x,y_0)) dx > f_g(y')$$
which is true for example when $g(u) = u^2, g(u) = 1/u, g(u) = \Phi^{-1}(1 - \exp(-u))$ but not true for example when $g(u) = \log(u)$. ($\Phi^{-1}$ is the quantile function of a standard normal distribution). I am wondering which conditions should satisfies $g$ such that the minimum is still $y_0$.