Consider an analytic function $f : U \longrightarrow \mathbb{C}$ where $U$ is an open subset of the complex numbers which contains the closed unit disk. I have $|f(x)| \geq 1$ for any $ x \in [-1, 1]$. Is the following true ? 
$$ |\int\_0^{2\pi} f(e^{it})\overline{f(e^{-it})}dt| \geq 4$$



A New Analytic Inequality 
Yes, indeed 
$$\frac{1}{2\pi}\int\_0^{2\pi} f(e^{it})\overline{f(e^{-it})}dt= |f(0)|^2,$$
as you can check applying the Cauchy formula to the holomorphic function $f(z)\overline{f(\bar z)}$.


