FAILURE OF KORENBLUM’S MAXIMUM PRINCIPLE IN BERGMAN SPACES WITH SMALL EXPONENTS.

The well-known conjecture due to B. Korenblum about the maximum principle in Bergman space A^p 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 llfllA^p ≤ llgllA^p . 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 pA^per 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. [ABSTRACT FROM AUTHOR]