Suppose that $h\in(0,1)$, $l\in(0,1)$, $p\in(0,1)$, and $l<h$. Show that:
What I have done:
I tried expanding the inequality. It gets very messy and I could not find any useful pattern.
I tried taking derivatives with respect to $p$, $h$, and $l$ to try to spot and use any monotonicity (for example, if the LHS function strictly increased in $p$ I could prove the inequality by assuming $p=1-\epsilon$), but I could not find any pattern either.
I used several numerical examples on Mathematica to confirm that the inequality holds. It seems that the assumption that $l<h$ is not even necessary, but it really does not matter whether I can relax it or not.
Any solution/idea on how this can be solved will be highly appreciated!
Thank you in advance.