How to show that the square of a concave function satisfies this upper bound? 
Suppose $F(x)∈[0,1]$ and $x∈[0,X]$. Suppose $F(0)=0$ and $F(X)=1$. Suppose that $F(x)$ is continuous and strictly concave for $x∈[F(x^∗)x^∗,X]$. Suppose that $\frac{x^* F(x^*)}{X} \leq F(x^* F(x^*))$.
Show that:
$F(x^∗)^2<F(x^*F(x^∗))$

By using concavity I have shown that if $F(x)$ is strictly concave for $x\in[0,X]$, then the inequality is satisfied. This can be done applying concavity with $x_1=x^*$, $x_2=0$ and $\lambda=F(x^*)$. Given that result, I believe that the above one can be proven as well, but not sure how to do it. We cannot longer use the "trick" of setting $x_2=0$. Any nice alternative?
Please, answer if you can solve it or if you can give me any insight on how to attack the problem. It will be of enormous help!
 A: It seems bad for me. Look at the picture below:

Now I'm going to explain what is illustrated here. First, points $O$ and $P$ on a coordinate plane stand for $(0, 0)$ and $(X, F(X))$, respectively. Then, $B = (x^*, 1)$, and $A = \left(x^*, F\left(x^*\right)\right)$. We assume that $A$ is located above $OP$ as on the pic.
Now $C$ is defined as the intersection of $OB$ and a horizontal line through $A$. One can see that $C$ is $\left(x^*F\left(x^*\right), F\left(x^*\right)\right)$. The condition that
$$\frac{x^*F\left(x^*\right)}{X}\leqslant F\left(x^*F\left(x^*\right)\right)$$
means that point $\left(x^*F\left(x^*\right), F\left(x^*F\left(x^*\right)\right)\right)$ is located somewhere above $OP$.
Let this point lie below $OA$. Then we can connect points $\left(x^*F\left(x^*\right), F\left(x^*F\left(x^*\right)\right)\right)$, $A$ and $P$ with a concave curve and continue it onto $\left[0, x^F\left(x^\right)\right) properly.
One can see that the intersection of $OA$ and the vertical line through $C$ has $y$-coordinate equal to
$$F\left(x^*\right)\cdot\frac{x^*F\left(x^*\right)}{x^*} = F\left(x^*\right)^2,$$
and we've just constructed a function where it exceeds $F\left(x^*F\left(x^*\right)\right)$.
Maybe I misunderstood the statement?
