Proving an inequality involving the 2 first moments of a class of probability distribution Let $X$ be a random variable such that $\langle e^{-X} \rangle = 1$ (which implies $\langle X \rangle \geq 0$). Here I use the notation $\mathbb{E}[X]=\langle X \rangle$.
I would like to show the following (hint from numerical evidence):
$$\frac{\langle X^2 \rangle }{\langle X \rangle }-\sqrt{\langle X^2 \rangle }\leq 2$$
Cauchy-Schwarz inequality gives that the previous expression is positive, but why would it be less than 2?
I found that this would be equivalent to 
$$\langle X\rangle \geq \frac{\langle X^2 \rangle}{2+\sqrt{\langle X^2 \rangle }}  $$
or
$$\langle X^2 \rangle \leq \frac{\langle X \rangle^2}{2}+2\langle X \rangle +\frac{\langle X \rangle }{2}\sqrt{\langle X \rangle^2+8\langle X \rangle }$$
I am pretty sure that it comes from the constraint  $\langle e^{-X} \rangle = 1$ but I don't know how to use it. Any idea? 
 A: A way to solve such problems is to look at the graph $\Gamma=\{(\exp(-x),x,x^2): x\in\mathbb R\}\subset\mathbb R^3\}$ of the map $x\mapsto(\exp(-x),x,x^2)$, at its convex hull $H$, and at the intersection  of $H$ with the set of points of form $(1,a,b)$, namely $M=\{(a,b):(1,a,b)\in H\}\subset\mathbb R^2$.
The points in $M$ are the $(EX,EX^2)=(\langle X\rangle,\langle X^2\rangle)$ that you are interested in.  Look for points in $M$ that are convex combinations of pairs of elements of $\Gamma$.  (The codimension of $M$ in $H$ is 1, so by a theorem of Dubins, extreme points in $M$ are mixtures of pairs of points in $\Gamma$.)
Added 8 hours later.  I don't think the claimed result is true.  If $P(X=-0.340221)=0.711613$ and $P(X=15.555439)=0.288387)$ then $E e^{-X} =1$, $a=E X= 4.243876$, and $b=EX^2= 69.863772$.  But $2+\sqrt b - b/a = -6.103803$, contrary to the OP's conjecture.  (Of course these numbers are how my computer rounded them off.  I told my computer to pick $u<0<v$ at random, to set $P(X=u)=(1-e^{-v})/(e^{-u}-e^{-v})$ and $P(X=v))=(e^{-u}-1)/(e^{-u}-e^{-v})$, so $E\exp^{-X}=1$, and to work out the corresponding $a$ and $b$.) Maybe the OP's numerical experiments were based on the false hypothesis that $X\ge0$ a.s.
