On the multidimensional Bohr radius
HTML articles powered by AMS MathViewer
- by Shankey Kumar
- Proc. Amer. Math. Soc. 151 (2023), 2001-2009
- DOI: https://doi.org/10.1090/proc/16280
- Published electronically: February 28, 2023
- HTML | PDF | Request permission
Abstract:
This article provides a relation between the classical Bohr radius, the arithmetic Bohr radius, and the multidimensional Bohr radius. Through this relation, we give answers to some questions, one is raised by Defant et al. and other one is an open problem, both are related to the multidimensional Bohr radius. Moreover, we provide better new estimates of Bohr radii for holomorphic functions defined on Banach spaces.References
- Yusuf Abu Muhanna, Rosihan M. Ali, and Saminathan Ponnusamy, On the Bohr inequality, Progress in approximation theory and applicable complex analysis, Springer Optim. Appl., vol. 117, Springer, Cham, 2017, pp. 269–300. MR 3644748
- Lev Aizenberg, Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1147–1155. MR 1636918, DOI 10.1090/S0002-9939-99-05084-4
- Lev Aizenberg, Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series, Selected topics in complex analysis, Oper. Theory Adv. Appl., vol. 158, Birkhäuser, Basel, 2005, pp. 87–94. MR 2147589, DOI 10.1007/3-7643-7340-7_{6}
- Lev Aizenberg, Generalization of results about the Bohr radius for power series, Studia Math. 180 (2007), no. 2, 161–168. MR 2314095, DOI 10.4064/sm180-2-5
- L. Aizenberg, A. Aytuna, and P. Djakov, An abstract approach to Bohr’s phenomenon, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2611–2619. MR 1657738, DOI 10.1090/S0002-9939-00-05270-9
- Lev Aizenberg, Aydin Aytuna, and Plamen Djakov, Generalization of a theorem of Bohr for bases in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl. 258 (2001), no. 2, 429–447. MR 1835551, DOI 10.1006/jmaa.2000.7355
- Luis Bernal-González, Hernán J. Cabana, Domingo García, Manuel Maestre, Gustavo A. Muñoz-Fernández, and 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 115 (2021), no. 2, Paper No. 44, 10. MR 4197435, DOI 10.1007/s13398-020-00986-1
- Bappaditya Bhowmik and Nilanjan Das, Bohr radius and its asymptotic value for holomorphic functions in higher dimensions, C. R. Math. Acad. Sci. Paris 359 (2021), 911–918. MR 4322990, DOI 10.5802/crmath.237
- Harold P. Boas, Majorant series, J. Korean Math. Soc. 37 (2000), no. 2, 321–337. Several complex variables (Seoul, 1998). MR 1775963
- Harold P. Boas and Dmitry Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2975–2979. MR 1443371, DOI 10.1090/S0002-9939-97-04270-6
- Harald Bohr, A Theorem Concerning Power Series, Proc. London Math. Soc. (2) 13 (1914), 1–5. MR 1577494, DOI 10.1112/plms/s2-13.1.1
- Enrico Bombieri, Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze, Boll. Un. Mat. Ital. (3) 17 (1962), 276–282 (Italian). MR 162918
- E. Bombieri and J. Bourgain, A remark on Bohr’s inequality, Int. Math. Res. Not. 80 (2004), 4307–4330. MR 2126627, DOI 10.1155/S1073792804143444
- Andreas Defant and Leonhard Frerick, A logarithmic lower bound for multi-dimensional Bohr radii, Israel J. Math. 152 (2006), 17–28. MR 2214450, DOI 10.1007/BF02771973
- Andreas Defant and Leonhard Frerick, The Bohr radius of the unit ball of $\ell ^n_p$, J. Reine Angew. Math. 660 (2011), 131–147. MR 2855822, DOI 10.1515/crelle.2011.080
- Andreas Defant, Manuel Maestre, and Christopher Prengel, The arithmetic Bohr radius, Q. J. Math. 59 (2008), no. 2, 189–205. MR 2428075, DOI 10.1093/qmath/ham028
- Andreas Defant, Manuel Maestre, and Christopher Prengel, Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables, J. Reine Angew. Math. 634 (2009), 13–49. MR 2560405, DOI 10.1515/CRELLE.2009.068
- P. G. Dixon, Banach algebras satisfying the non-unital von Neumann inequality, Bull. London Math. Soc. 27 (1995), no. 4, 359–362. MR 1335287, DOI 10.1112/blms/27.4.359
- Hidetaka Hamada, Tatsuhiro Honda, and Gabriela Kohr, Bohr’s theorem for holomorphic mappings with values in homogeneous balls, Israel J. Math. 173 (2009), 177–187. MR 2570664, DOI 10.1007/s11856-009-0087-9
- Ilgiz R. Kayumov and Saminathan Ponnusamy, On a powered Bohr inequality, Ann. Acad. Sci. Fenn. Math. 44 (2019), no. 1, 301–310. MR 3919139, DOI 10.5186/aasfm.2019.4416
- Ming-Sheng Liu and Saminathan Ponnusamy, Multidimensional analogues of refined Bohr’s inequality, Proc. Amer. Math. Soc. 149 (2021), no. 5, 2133–2146. MR 4232204, DOI 10.1090/proc/15371
Bibliographic Information
- Shankey Kumar
- Affiliation: School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni 752050, India
- MR Author ID: 1406679
- Email: shankeygarg93@gmail.com
- Received by editor(s): June 9, 2022
- Received by editor(s) in revised form: August 16, 2022
- Published electronically: February 28, 2023
- Additional Notes: The work of the author was supported by the Institute Post Doctoral Fellowship of NISER Bhubaneswar, India.
- Communicated by: Javad Mashreghi
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2001-2009
- MSC (2020): Primary 32A05, 32A10; Secondary 46B45
- DOI: https://doi.org/10.1090/proc/16280
- MathSciNet review: 4556195