Maximal correlation

This is a question about the correlation. Let $X,Y$ be two Gaussion random variable $\mathcal{N}(0,1)$ with correlation $\beta$, then prove that the best constant of Cauchy inequality is 

$$Cov(f(X), g(Y)) \leq \vert \beta \vert \sqrt{Var(f(X)) Var(g(Y))}$$.

In fact, one can define the maximal correlation of random variable by the best constant above and of course it should be bigger than $\beta$. Let us remark how to prove the inequality above quickly. We can use the expansion by Hermit polynomial that we have 

$$\mathbb{E}[H_n(X) H_m(Y)] = \delta_{n,m} \left(\mathbb{E}\frac{1}{n!}[XY]\right)^n.$$

Then a centered $L^2$ functions have projection on $H_0$ zero. Then we have 

$$\mathbb{E}[f(X)g(Y)] = \sum_{n=1}^{\infty}\langle f, H_n \rangle \langle g, H_n \rangle \frac{1}{n!} \beta^n \\ \leq \vert \beta \vert \sqrt{\sum_{n=1}^{\infty}\langle f, H_n \rangle^2 \frac{1}{n!} } \sqrt{\sum_{n=1}^{\infty}\langle g, H_n \rangle^2 \frac{1}{n!} } \\ \leq \beta \vert \sqrt{Var(f(X)) Var(g(Y))}.$$
This concludes the proof.