I have $\xi$ and $\eta$ with following properties: $E\xi = E\eta = 0$, $D\xi = D\eta = 1$. And the correlation coefficient: $\rho = \rho (\xi, \eta)$.
I want to prove the following inequality:
$ E \max (\xi^2, \eta^2) \leq 1 + \sqrt{1 - \rho^2}.$
I don't know how to start as r.v.'s are not independent and since I can't use standard approach:
$ P( \max (\xi^2, \eta^2) \leq x) = P( \xi^2 \leq x, \ \eta^2 \leq x ) \neq P( \xi^2 \leq x)P( \eta^2 \leq x).$