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 Ap states that for 0 < p < ∞ there exists a constant 0 < c < 1 with the following property. If f and g are holomorphic functions in the unit disk D such that |f(z)| ≤ |g(z)| for all c < |z| < 1, then ‖f‖Ap ≤ ‖g‖Ap. Hayman proved Korenblum’s conjecture for p = 2, and Hinkkanen generalized this result by proving the conjecture for all 1 ≤ p < ∞. 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 Ap does not hold when 0 < p < 1.