Analyse et géométrie complexes
Bohr radius and its asymptotic value for holomorphic functions in higher dimensions
Comptes Rendus. Mathématique, Tome 359 (2021) no. 7, pp. 911-918.

We establish sharp Bohr phenomena for holomorphic functions defined on a bounded balanced domain $G$ in a complex Banach space $X$, which map into a simply connected domain or a convex domain $\Omega$ in the complex plane $ℂ$. Taking $X$ as the $n$-dimensional complex plane and $G$ as the open unit polydisk, we consider a version of the Bohr inequality stronger than the above mentioned one and study the exact asymptotic behaviour of the Bohr radius. Explicit lower bounds on the Bohr radii of this type are also provided. Extending a recent result of Liu and Ponnusamy, we further record a refined form of the Bohr inequality for the particular case $\Omega =𝔻$, i.e. the open unit disk in $ℂ$.

Reçu le :
Accepté le :
Accepté après révision le :
Publié le :
DOI : https://doi.org/10.5802/crmath.237
Classification : 32A05,  32A10,  32A17,  46G20
Bappaditya Bhowmik 1 ; Nilanjan Das 1

1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur - 721302, India.
@article{CRMATH_2021__359_7_911_0,
author = {Bappaditya Bhowmik and Nilanjan Das},
title = {Bohr radius and its asymptotic value for holomorphic functions in higher dimensions},
journal = {Comptes Rendus. Math\'ematique},
pages = {911--918},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {7},
year = {2021},
doi = {10.5802/crmath.237},
zbl = {07398743},
language = {en},
}
TY  - JOUR
AU  - Nilanjan Das
TI  - Bohr radius and its asymptotic value for holomorphic functions in higher dimensions
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 911
EP  - 918
VL  - 359
IS  - 7
PB  - Académie des sciences, Paris
UR  - https://zbmath.org/?q=an%3A07398743
UR  - https://doi.org/10.5802/crmath.237
DO  - 10.5802/crmath.237
LA  - en
ID  - CRMATH_2021__359_7_911_0
ER  - 
%0 Journal Article
%A Nilanjan Das
%T Bohr radius and its asymptotic value for holomorphic functions in higher dimensions
%J Comptes Rendus. Mathématique
%D 2021
%P 911-918
%V 359
%N 7
%U https://doi.org/10.5802/crmath.237
%R 10.5802/crmath.237
%G en
%F CRMATH_2021__359_7_911_0
Bappaditya Bhowmik; Nilanjan Das. Bohr radius and its asymptotic value for holomorphic functions in higher dimensions. Comptes Rendus. Mathématique, Tome 359 (2021) no. 7, pp. 911-918. doi : 10.5802/crmath.237. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.237/

[1] Lev Aizenberg Multidimensional analogues of Bohr’s theorem on power series, Proc. Am. Math. Soc., Volume 128 (2000) no. 4, pp. 1147-1155 | Article | MR 1636918 | Zbl 0948.32001

[2] Lev Aizenberg Generalization of results about the Bohr radius for power series, Stud. Math., Volume 180 (2007) no. 2, pp. 161-168 | Article | MR 2314095 | Zbl 1118.32001

[3] Frédéric Bayart; D. Pellegrino; Juan B. Seoane-Sepúlveda The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{\left(logn\right)/n}$, Adv. Math., Volume 264 (2014), pp. 726-746 | Article | Zbl 1331.46037

[4] Luis Bernal-González; Hernán J. Cabana; Domingo García; Manuel Maestre; Gustavo A. Muñoz-Fernández; Juan B. Seoane-Sepúlveda A new approach towards estimating the $n$-dimensional Bohr radius, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM, Volume 115 (2021) no. 2, 44, 10 pages | MR 4197435 | Zbl 1456.32001

[5] Bappaditya Bhowmik; Nilanjan Das Bohr phenomenon for operator-valued functions, Proc. Edinb. Math. Soc., Volume 64 (2021) no. 1, pp. 72-86 | Article | MR 4249840 | Zbl 07357829

[6] Bappaditya Bhowmik; Nilanjan Das A characterization of Banach spaces with nonzero Bohr radius, Arch. Math., Volume 116 (2021) no. 5, pp. 551-558 | Article | MR 4248548 | Zbl 07343934

[7] Harold P. Boas; Dmitry Khavinson Bohr’s power series theorem in several variables, Proc. Am. Math. Soc., Volume 125 (1997) no. 10, pp. 2975-2979 | Article | MR 1443371 | Zbl 0888.32001

[8] Harald Bohr A theorem concerning power series, Proc. Lond. Math. Soc., Volume 13 (1914), pp. 1-5 | Article | MR 1577494 | Zbl 44.289.01

[9] Louis de Branges A proof of the Bieberbach conjecture, Acta Math., Volume 154 (1985) no. 1-2, pp. 137-152 | Article | MR 772434 | Zbl 0573.30014

[10] Andreas Defant; Leonhard Frerick A logarithmic lower bound for multi-dimensional Bohr radii, Isr. J. Math., Volume 152 (2006), pp. 17-28 | Article | MR 2214450 | Zbl 1124.32003

[11] Andreas Defant; Leonhard Frerick; Joaquim Ortega-Cerdà; Myriam Ounaïes; Kristian Seip The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive, Ann. Math., Volume 174 (2011) no. 1, pp. 485-497 | Article | MR 2811605 | Zbl 1235.32001

[12] P. G. Dixon Banach algebras satisfying the non-unital von Neumann inequality, Bull. Lond. Math. Soc., Volume 27 (1995) no. 4, pp. 359-362 | Article | MR 1335287 | Zbl 0835.47033

[13] Peter L. Duren Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, 1983 | Zbl 0514.30001

[14] Ian Graham; Gabriela Kohr Geometric function theory in one and higher dimensions, Pure and Applied Mathematics, Marcel Dekker, 255, Marcel Dekker, 2003 | MR 2017933 | Zbl 1042.30001

[15] Hidetaka Hamada; Tatsuhiro Honda; Gabriela Kohr Bohr’s theorem for holomorphic mappings with values in homogeneous balls, Isr. J. Math., Volume 173 (2009), pp. 177-187 | Article | MR 2570664 | Zbl 1189.32011

[16] Hidetaka Hamada; Tatsuhiro Honda; Yusuke Mizota Bohr phenomenon on the unit ball of a complex Banach space, Math. Inequal. Appl., Volume 23 (2020) no. 4, pp. 1325-1341 | MR 4168202 | Zbl 1458.32018

[17] Ming-Sheng Liu; Saminathan Ponnusamy Multidimensional analogues of refined Bohr’s inequality, Proc. Am. Math. Soc., Volume 149 (2021) no. 5, pp. 2133-2146 | MR 4232204 | Zbl 1460.32002

[18] Yusuf Abu Muhanna Bohr’s phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ., Volume 55 (2010) no. 11, pp. 1071-1078 | Article | MR 2811955 | Zbl 1215.46018

[19] Vern I. Paulsen; Gelu Popescu; Dinesh Singh On Bohr’s inequality, Proc. Lond. Math. Soc., Volume 85 (2002) no. 2, pp. 493-512 | Article | MR 1912059 | Zbl 1033.47008

[20] Saminathan Ponnusamy; Ramakrishnan Vijayakumar; Karl-Joachim Wirths New inequalities for the coefficients of unimodular bounded functions, Results Math., Volume 75 (2020) no. 3, 107, 11 pages | MR 4119522 | Zbl 1442.30003

[21] Gelu Popescu Bohr inequalities for free holomorphic functions on polyballs, Adv. Math., Volume 347 (2019), pp. 1002-1053 | Article | MR 3922821 | Zbl 07044308

Cité par Sources :