I want to show that $$\arg\max_{x}\left(\int^b_a f(x,p) dp\right)>\arg\max_{x}\left(\int^b_a g(p)h(f(x,p)) dp\right)$$ where the parameters have the following properties $x,p,a,b>0$,
and the functions are of the following form $f(x,p)>0$,
$f'_{p}>0$, $f''_{p}\leq0$, $\lim\limits_{x \to 0} f'_{x}>0$, $\lim\limits_{x \to X} f'_{x}<0$, $f''_{x}<0$
$g(p)>0, \quad g'_{p}<0,\quad g''_{p}>0$
$h(.)$ is an increasing and concave function.
I can show the above numerically, but I am at a loss how to do it analytically. Can I somehow use Jensen's inequality (or any theorem for that matter) to show that the $x$ which maximizes the first expression has to be strictly larger than the one that maximizes the second expression? I hope that the question is not too trivial for this forum, but I have been stuck with it for quite a while now, and would appreciate some help.
