TITLE: Analytic functions close to $\bar{z}$
QUESTION [6 upvotes]: Is there an analytic function $f\colon\Bbb{C}\longrightarrow \Bbb{C}$ such that for any $z$ on the unit circle $\lvert f(z) - \overline{z}\rvert < 1 $?

REPLY [7 votes]: Let me give an alternative proof. Suppose $f$ is the desired function. Then for $z\bar{z} = 1$:
$$ |z f(z) - 1| = |z| |f(z) - \bar{z}| < 1 $$
so $g(z) = zf(z) - 1$ is an analytic function satisfying $|g(z)| < 1$ whenever $|z| = 1$. By the maximum modulus principle 
$$ 1 =  |0 \cdot f(0) - 1| = |g(0)| < 1$$
which gives a contradiction.