Failure of Korenblum’s maximum principle in Bergman spaces with small exponents

Abstract:

The well-known conjecture due to B. Korenblum about the maximum principle in Bergman space $A^p$ states that for $0<p<\infty$ there exists a constant $0<c<1$ with the following property. If $f$ and $g$ are holomorphic functions in the unit disk $\mathbb {D}$ such that $|f(z)|\leq |g(z)|$ for all $c<|z|<1$, then $\|f\|_{A^p}\leq \|g\|_{A^p}$. Hayman proved Korenblum’s conjecture for $p=2$, and Hinkkanen generalized this result by proving the conjecture for all $1\leq p<\infty$. The case $0<p<1$ of the conjecture has so far remained open. In this paper we resolve this remaining case of the conjecture by proving that Korenblum’s maximum principle in Bergman space $A^p$ does not hold when $0<p<1$.