Failure of Korenblum’s maximum principle in Bergman spaces with small exponents
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- by Vladimir Božin and Boban Karapetrović PDF
- Proc. Amer. Math. Soc. 146 (2018), 2577-2584 Request permission
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 $0<p<1$.References
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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
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2577-2584
- MSC (2010): Primary 30H20
- DOI: https://doi.org/10.1090/proc/13946
- MathSciNet review: 3778159