Is $\log|h|$ BMO when $h$ is analytic?

Question

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. A function $f$ is called a bounded mean oscillation (BMO for short) function on $\mathbb{D}$ if $$\sup_{disc\,\subset\, \mathbb{D}}\dfrac{1}{|disc|}\int_{disc} |f - f_{disc}| <\infty.$$ Here $disc$ is an open disc in $\mathbb{C}$ and $$f_{disc} = \dfrac{1}{|disc|}\int_{disc}f.$$ A well-known BMO function is of course $f=\log|z|$. In fact this is a BMO function on the whole complex plane.

My question: Assume that $h$ is analytic on $\mathbb{D}$. Is it true that $\log|h|$ is BMO on $\mathbb{D}$?

Answer 1

No, the function $f(z)=\exp(1/(1-z))$ is a counterexample.