In my studies of various geometric inequalities I reached an inequality which seems true (numerically) but I cannot prove it. Let $p$, $q$, and $r$ be real numbers from the interval $(0,1)$. Let's also define the following function $f({p})=\frac{\sqrt{1-p}}{(2-p)^2}$ Prove (or disprove) that: $ \frac{f(p)+f(q)+f(r)}{\sqrt{p q r}}\leq \frac{f(p)}{p\sqrt{p}}+\frac{f(q)}{q\sqrt{q}}+\frac{f(r)}{r\sqrt{r}} $
I've tried Lagrange multipliers but the resulting equations do not seem tractable.
EDIT: The original question had the condition $p+q+r=2$ which apparently is not necessary, so I dropped it. I can prove that the inequality holds for $p=q$. A possible strategy is to try to establish monotonicity in one of the parameters under certain conditions. Unfortunately I can't manage the calculations.