Authors: Božin, Vladimir 
Karapetrović, Boban
Title: Failure of korenblum’s maximum principle in bergman spaces with small exponents
Journal: Proceedings of the American Mathematical Society
Volume: 146
Issue: 6
First page: 2577
Last page: 2584
Issue Date: 1-Jan-2018
Rank: M22
ISSN: 0002-9939
DOI: 10.1090/proc/13946
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.
Keywords: Bergman spaces | Korenblum’s maximum principle
Publisher: Amerian Mathematial Soiety
Project: Analysis and algebra with applications 

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