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Transactions of the American Mathematical Society

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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
MathSciNet review: 4068249
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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: \begin{equation*} \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 , \end{equation*} 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
MR Author ID: 915441
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
MR Author ID: 822942
Email: muro@cifasis-conicet.gov.ar

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