Inequality concerning the pairwise correlation coefficients of three random variables I was asked to prove:
The correlation coefficients, $\rho_{12}$, $\rho_{23}$, $\rho_{13}$ between three random variables $X_1$, $X_2$, $X_3$ obey
$$(1+\rho_{12})(1+\rho_{13})(1+\rho_{23})\ge\frac{1}{2}(1+\rho_{12}+\rho_{23}+\rho_{13})^2$$
I'm able to prove (using triangle inequality) a weaker result:
$$(1+\rho_{12})(1+\rho_{13})(1+\rho_{23})\ge\frac{1}{8}(1+\rho_{12}+\rho_{23}+\rho_{13})^3$$
but have no clue in attacking the original one. Anyone have an idea? Thanks in advance!

For ease of notation, I'll let $a = \rho_{12}$, $b = \rho_{13}$ and $c = \rho_{23}$. The matrix
$$
\begin{pmatrix}
1 & a & b \\
a & 1 & c \\
b & c & 1
\end{pmatrix}
$$
is positive semi-definite, so its determinant is non-negative:
$$
1 + 2abc \ge a^2 + b^2 + c^2.
$$
Now consider $(1 + a + b + c)^2$:
\begin{align*}
(1 + a + b + c)^2 & = 1 + 2(a + b + c) + (a^2 + b^2 + c^2) + 2(ab + ac + bc) \\
& \le 1 + 2(a + b + c) + 1 + 2abc + 2(ab + ac + bc) \\
& = 2(1 + a + b + c + ab + ac + bc + abc) \\
& = 2(1 + a)(1 + b)(1 + c).
\end{align*}
