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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Mixed Bohr radius in several variables


Authors: Daniel Galicer, Martín Mansilla and Santiago Muro
Journal: Trans. Amer. Math. Soc. 373 (2020), 777-796
MSC (2010): Primary 32A05, 32A22; Secondary 46B15, 46B20, 46G25, 46E50
DOI: https://doi.org/10.1090/tran/7870
Published electronically: November 5, 2019
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Abstract: Let $ K(B_{\ell _p^n},B_{\ell _q^n}) $ be the $ n$-dimensional $ (p,q)$-Bohr radius for holomorphic functions on $ \mathbb{C}^n$. That is, $ K(B_{\ell _p^n},B_{\ell _q^n}) $ denotes the greatest number $ r\geq 0$ such that for every entire function $ f(z)=\sum _{\alpha } a_{\alpha } z^{\alpha }$ in $ n$-complex variables, we have the following (mixed) Bohr-type inequality:

$\displaystyle \sup _{z \in r \cdot B_{\ell _q^n}} \sum _{\alpha } \vert a_{\alpha } z^{\alpha } \vert \leq \sup _{z \in B_{\ell _p^n}} \vert f(z) \vert ,$    

where $ B_{\ell _r^n}$ denotes the closed unit ball of the $ n$-dimensional sequence space $ \ell _r^n$.

For every $ 1 \leq p, q \leq \infty $, we exhibit the exact asymptotic growth of the $ (p,q)$-Bohr radius as $ n$ (the number of variables) goes to infinity.


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Additional Information

Daniel Galicer
Affiliation: Departamento de Matemática - Pab I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina; and IMAS-CONICET
Email: dgalicer@dm.uba.ar

Martín Mansilla
Affiliation: Departamento de Matemática - Pab I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina; and IMAS-CONICET
Email: mmansilla@dm.uba.ar

Santiago Muro
Affiliation: Departamento de Matemática - Pab I, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina; and CIFASIS-CONICET, Rosario, Argentina
Email: muro@cifasis-conicet.gov.ar

DOI: https://doi.org/10.1090/tran/7870
Keywords: Bohr radius, power series, homogeneous polynomials, domains of convergence for monomial expansions, unconditional bases
Received by editor(s): December 21, 2017
Received by editor(s) in revised form: May 30, 2018
Published electronically: November 5, 2019
Additional Notes: This work was partially supported by projects CONICET PIP 11220130100329CO, ANPCyT PICT 2015-2224, ANPCyT PICT 2015-2299, ANPCyT PICT 2015-3085, UBACyT 20020130100474BA, UBACyT 20020130300052BA, UBACyT 20020130300057BA
The second author was supported by a CONICET doctoral fellowship
Article copyright: © Copyright 2019 American Mathematical Society