Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: Vladimir Božin and Boban Karapetrović
Journal: Proc. Amer. Math. Soc. 146 (2018), 2577-2584
MSC (2010): Primary 30H20
DOI: https://doi.org/10.1090/proc/13946
Published electronically: January 26, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ \vert f(z)\vert\leq \vert g(z)\vert$ for all $ c<\vert z\vert<1$, then $ \Vert f\Vert _{A^p}\leq \Vert g\Vert _{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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30H20

Retrieve articles in all journals with MSC (2010): 30H20


Additional Information

Vladimir Božin
Affiliation: Faculty of Mathematics University of Belgrade Studentski trg 16 Serbia
Email: bozinv@mi.sanu.ac.rs

Boban Karapetrović
Affiliation: Faculty of Mathematics University of Belgrade Studentski trg 16 Serbia
Email: bkarapetrovic@matf.bg.ac.rs

DOI: https://doi.org/10.1090/proc/13946
Keywords: Bergman spaces, Korenblum's maximum principle
Received by editor(s): May 31, 2017
Received by editor(s) in revised form: September 3, 2017
Published electronically: January 26, 2018
Additional Notes: The authors were supported by NTR Serbia, Project ON174032
Communicated by: Stephan Ramm Garcia
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society