Totally monotone functions with applications to the Bergman space

Abstract:

Using a theorem of S. Bernstein [1] we prove a special case of the following maximum principle for the Bergman space conjectured by B. Korenblum [3]: There exists a number $\delta \in (0,1)$ such that if $f$ and $g$ are analytic functions on the open unit disk ${\mathbf {D}}$ with $|f(z)| \leq |g(z)|$ on $\delta \leq |z| < 1$ then ${\left \| f \right \|_2} \leq {\left \| g \right \|_2}$, where ${\left \| {} \right \|_2}$ is the ${L^2}$ norm with respect to area measure on ${\mathbf {D}}$. We prove the above conjecture when either $f$ or $g$ is a monomial; in this case we show that the optimal constant $\delta$ is greater than or equal to $1/\sqrt 3$.