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*}