Let $\xi$ and $\eta$ be random variables and $\mathbb{E}\xi=\mathbb{E}\eta=0$ also $\mathbb{D}\xi=\mathbb{D}\eta=1$. Here $\rho=\rho(\xi,\eta)$ is a correlation coefficient. Need to show $$\mathbb{E}\max\{\xi^2,\eta^2\}\leq 1 + \sqrt{1-\rho^2}.$$
So, I know that $\rho=\rho(\xi,\eta)=\frac{\mathbb{E}(\xi-\mathbb{E}\xi)(\eta-\mathbb{E}\eta)} {\sqrt{\mathbb{D}\xi \mathbb{D}\eta}}$.
But how to start and what qualities to use?