1
$\begingroup$

Let $f$ be an element of the Hardy space $H^2$, i.e. $f$ is an analytic function on the unit disk such that $\sum|a_n|^2<\infty$, where $f(z)=\sum a_n z^n$. Assume also that $f\not\equiv 0$.

Is there a bounded function $g\not\equiv 0$ analytic on the unit disk, such that $|g(z)|\le |f(z)|$, for every $z$?

Improve this question
$\endgroup$
2
  • 3
    $\begingroup$ Yes, because $\log |f(z)|$ is the Poisson integral of a signed measure, and you can define $g$ using its negative part only (or if that is zero, then $|f|$ was bounded below by a positive constant, so everything becomes trivial). $\endgroup$
    – Christian Remling
    Commented Jun 1, 2019 at 3:40
  • 3
    $\begingroup$ And I should have first divided out the zeros by using a Blaschke product. $\endgroup$
    – Christian Remling
    Commented Jun 1, 2019 at 3:48

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .