Asymptotic behaviour of Bohr’s radii for polynomials on simply connected domains

Molla Basir Ahamed; Vasudevarao Allu; Himadri Halder

doi:10.5802/crmath.625

Research article - Complex analysis and geometry

Asymptotic behaviour of Bohr’s radii for polynomials on simply connected domains

Molla Basir Ahamed ¹; Vasudevarao Allu ²; Himadri Halder ²

¹ Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India
² School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1047-1058

Abstract (Original language)
Résumé

Let $\mathcal{H}^{\infty }$ be the Banach space of bounded analytic functions in the unit disk $\mathbb{D}=\bigl \lbrace z\in \mathbb{C} : \vert z \vert <1\bigr \rbrace $ endowed with the norm $\Vert f \Vert _{\infty }=\sup _{z\in \mathbb{D}} \left|f(z) \right|$. The Bohr radius $R$ for $\mathcal{H}^{\infty }$ is defined by

\[ R=\sup _{0<r<1}\Biggl \lbrace r : \sum _{n=0}^{\infty } \left|a_n(f) \right|r^n\le \Vert f \Vert _{\infty } \ \text{for all $f(z)=\sum _{n=0}^{\infty }a_nz^n\in \mathcal{H}^{\infty }$}\Biggr \rbrace \]

and it is well-known that $R=1/3$, the Bohr radius, is best possible. The class $\mathcal{P}_n$ of complex polynomials of degree at most $n$ which are bounded by $1$ and the Bohr radius for this class is defined by

\[ R_n \mathrel {:=} \sup _{0<r<1}\Biggl \lbrace r : \sum _{n=0}^{n} \left|a_n(p) \right|r^n\le \Vert p \Vert _{\infty } \ \text{for all $p(z)=\sum _{n=0}^{n}a_nz^n\in \mathcal{P}_n$}\Biggr \rbrace \]

and the estimate

\[ \frac{c_1}{3^{n/2}}<R_n-\frac{1}{3}<c_2\frac{\log n}{n} \]

is valid for large values of $n$ for absolute positive constants $c_1$ and $c_2$. In this paper, we prove the above estimate for simply connected domains.

Soit $\mathcal{H}^{\infty }$ l’espace de Banach des fonctions analytiques bornées dans le disque unité $\mathbb{D}=\bigl \lbrace z\in \mathbb{C} : \vert z \vert <1\bigr \rbrace $, muni de la norme $\Vert f \Vert _{\infty }=\sup _{z\in \mathbb{D}} \left|f(z) \right|$. Le rayon de Bohr $R$ pour $\mathcal{H}^{\infty }$ est défini par

\[ R=\sup _{0<r<1}\Biggl \lbrace r : \sum _{n=0}^{\infty } \left|a_n(f) \right|r^n\le \Vert f \Vert _{\infty } \ \text{pour tout $f(z)=\sum _{n=0}^{\infty }a_nz^n\in \mathcal{H}^{\infty }$}\Biggr \rbrace \]

et il est bien connu que $R=1/3$, le rayon de Bohr, est optimal. La classe $\mathcal{P}_n$ des polynômes complexes de degré au plus $n$, qui sont bornés par $1$, et le rayon de Bohr pour cette classe sont définis par

\[ R_n \mathrel {:=} \sup _{0<r<1}\Biggl \lbrace r : \sum _{n=0}^{n} \left|a_n(p) \right|r^n\le \Vert p \Vert _{\infty } \ \text{pour tout $p(z)=\sum _{n=0}^{n}a_nz^n\in \mathcal{P}_n$}\Biggr \rbrace \]

et l’estimation

\[ \frac{c_1}{3^{n/2}}<R_n-\frac{1}{3}<c_2\frac{\log n}{n} \]

est valable pour de grandes valeurs de $n$, pour des constantes absolues positives $c_1$ et $c_2$. Dans cet article, nous démontrons cette estimation pour les domaines simplement connexes.

Received: 2022-10-23
Revised: 2023-01-25
Accepted: 2024-02-12
Published online: 2025-09-29

DOI: 10.5802/crmath.625

Classification: 30C45, 30C50, 30C80
Keywords: Bohr inequality, Bohr radius, Bohr phenomenon, analytic functions, majorant series, Dirichlet series, harmonic mappings, holomorphic functions, Reinhardt domains, multidimensional Bohr radius
Mots-clés : Inégalité de Bohr, rayon de Bohr, phénomène de Bohr, fonctions analytiques, série majorante, séries de Dirichlet, applications harmoniques, fonctions holomorphes, domaines de Reinhardt, rayon de Bohr multidimensionnel

Author's affiliations:

Molla Basir Ahamed ¹; Vasudevarao Allu ²; Himadri Halder ²

¹ Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India
² School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Asymptotic behaviour of {Bohr{\textquoteright}s} radii for polynomials on simply connected domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1047--1058},
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Molla Basir Ahamed; Vasudevarao Allu; Himadri Halder. Asymptotic behaviour of Bohr’s radii for polynomials on simply connected domains. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1047-1058. doi: 10.5802/crmath.625

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