REFINED BOHR INEQUALITIES AND A REFINED BOHR-ROGOSINSKI INEQUALITY ON COMPLEX BANACH SPACES

  • Molla Basir AHAMED1 ,
  • Sabir AHAMMED1 ,
  • Hidetaka HAMADA2 ,
  • *
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  • 1. Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India;
    2. Faculty of Science and Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome Higashi-Ku, Fukuoka 813-8503, Japan
Molla Basir AHAMED,E-mail: E-mail: mbahamed.math@jadavpuruniversity.in; Sabir AHAMMED, E-mail: sabira.math.rs@jadavpuruniversity.in

Received date: 2024-07-25

  Revised date: 2024-10-29

  Online published: 2026-05-22

Supported by

The first author was supported by the SERB, SUR/2022/002244, Govt. India and the second author was supported by the UGC-JRF (NTA Ref. No.: 201610135853), New Delhi, India, and the third author was partially supported by the JSPS KAKENHI (JP22K03363).

Abstract

In this paper, we first establish refined versions of the Bohr inequalities for the class of holomorphic functions from the unit ball $B_X$ of a complex Banach space $X$ into $\mathbb{C}$. As applications, we will establish refined Bohr inequalities of functional type or of norm type for holomorphic mappings with lacunary series on the unit ball $B_X$ with values in higher dimensional spaces. Next, we obtain the Bohr-Rogosinski inequality for the class of holomorphic functions on $B_X.$ In addition, we establish an improved version of the Bohr inequality for holomorphic functions on $B_X$. All the results are proved to be sharp.

Cite this article

Molla Basir AHAMED1 , Sabir AHAMMED1 , Hidetaka HAMADA2 , * . REFINED BOHR INEQUALITIES AND A REFINED BOHR-ROGOSINSKI INEQUALITY ON COMPLEX BANACH SPACES[J]. Acta mathematica scientia, Series B, 2026 , 46(1) : 19 -38 . DOI: 10.1007/s10473-026-0102-9

References

[1] Ahamed M B, Allu V, Halder H. The Bohr phenomenon for analytic functions on shifted disks. Ann Acad Sci Fenn Math, 2022, $\textbf{47}$: 103-120
[2] Aizenberg L. Multidimensional analogues of Bohr's theorem on power series. Proc Amer Math Soc, 2000, ${\bf 128}$: 1147-1155
[3] Ali R M, Barnard R W, Solynin A Y. A note on Bohr's phenomenon for power series. J Math Anal Appl, 2017, ${\bf 449}$: 154-167
[4] Allu V, Arora V. Bohr-Rogosinski type inequalities for concave univalent functions. J Math Anal Appl, 2023, $\textbf{520}$: 126845
[5] Arora V. Bohr's phenomenon for holomorphic and harmonic functions with lacunary series in complex Banach spaces. Complex Var Elliptic Equ, 2024, $\textbf{69}$(3): 492-503
[6] Boas H P, Khavinson D. Bohr's power series theorem in several variables. Proc Amer Math Soc, 1997, $\textbf{125}$(10): 2975-2979
[7] Bohr H. A theorem concerning power series. Proc Lond Math Soc, 1914, $\textbf{S2-13}$(1): 1-5
[8] Chen K, Liu M S, Ponnusamy S. Bohr-type inequalities for unimodular bounded analytic functions. Results Math, 2023, $\textbf{78}$: Article 183
[9] Evdoridis S, Ponnusamy S, Rasila A. Improved Bohr's inequality for locally univalent harmonic mappings. Indag Math, 2019, $\textbf{30}$: 201-213
[10] Evdoridis S, Ponnusamy S, Rasila A. Improved Bohr's inequality for shifted disks. Results Math, 2021, $\textbf{76}$: Article 14
[11] Fournier R, Ruscheweyh S. On the Bohr radius for simply connected plane domains. CRM Proc Lect Notes, 2010, $\textbf{51}$: 165-171
[12] Graham I, Kohr G.Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics. New York: Marcel Dekker, 2003
[13] Hamada H. Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces. Math Nachr, 2023, ${\bf 296}$: 2795-2808
[14] Hamada H, Honda T. Bohr radius for pluriharmonic mappings in separable complex Hilbert spaces. Bull Malays Math Sci Soc, 2024, $\textbf{ 47}$: Article 47
[15] Hamada H, Honda T. Bohr phenomena for holomorphic mappings with values in several complex variables. Results Math, 2024, $\textbf{79}$: Article 239
[16] Hamada H, Honda T, Kohr G. Bohr's theorem for holomorphic mappings with values in homogeneous balls. Israel J Math, 2009, $\textbf{173}$: 177-187
[17] Hamada H, Honda T, Mizota Y. Bohr phenomenon on the unit ball of a complex Banach space. Math Inequal Appl, 2020, $\textbf{23}$(4): 1325-1341
[18] Ismagilov A, Kayumov I R, Ponnusamy S. Sharp Bohr type inequality. J Math Anal Appl, 2020, ${\bf 489}$: Article 124147
[19] Kayumov I R, Khammatova D M, Ponnusamy S. Bohr-Rogosinski phenomenon for analytic functions and Cesáro operators. J Math Anal Appl, 2021, $\textbf{496}$: Article 124824
[20] Kayumov I R, Ponnusamy S. Bohr inequality for odd analytic functions. Comput Methods Funct Theory, 2017, $\textbf{17}$: 679-688
[21] Kayumov I R, Ponnusamy S. Improved version of Bohr's inequality. C R Acad Sci Paris Ser I, 2018, $\textbf{356}$: 272-277
[22] Kayumov I R, Ponnusamy S. Bohr's inequalities for the analytic functions with lacunary series and harmonic functions. J Math Anal Appl, 2018, ${\bf 465}$: 857-871
[23] Kumar S.On the multidimensional Bohr radius. Proc Amer Math Soc, 2023, ${\bf 151}$: 2001-2009
[24] Kumar S. A generalization of the Bohr inequality and its applications. Complex Var Elliptic Equ, 2023, ${\bf 68}$: 963-973
[25] Kumar S, Ponnusamy S, Williams G B. The Bohr-type inequalities for holomorphic functions with lacunary series in complex Banach space. New York J Math, 2025, ${\bf 31}$: 259-281
[26] Lin R, Liu M, Ponnusamy S. The Bohr-type inequalities for holomorphic mappings with a lacunary series in several complex variables. Acta Math Sci, 2023, $\textbf{43}$: 63-79
[27] Liu G, Liu Z, Ponnusamy S. Refined Bohr inequality for bounded analytic functions. Bull Sci Math, 2021, $\textbf{173}$: Article 103054
[28] Liu M S, Ponnusamy S. Multidimensional analogues of refined Bohr's inequality. Porc Amer Math Soc, 2021, $\textbf{149}$(5): 2133-2146
[29] Liu M S, Ponnusamy S, Wang J. Bohr's phenomenon for the classes of quasi-subordination and $K$-quasiregular harmonic mappings. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 2020, $\textbf{114}$: Article 115
[30] Liu X, Liu T S. The Bohr inequality for holomorphic mappings with lacunary series in several complex variables. J Math Anal Appl, 2020, $\textbf{485}$: Article 123844
[31] Paulsen V I, Popescu G, Singh D. On Bohr's inequality. Proc Lond Math Soc, 2002, $\textbf{85}$(2): 493-512
[32] Ponnusamy S, Vijayakumar R.Note on improved Bohr inequality for harmonic mappings. arXiv: 2104.06717
[33] Ponnusamy S, Vijayakumar R, Wirths K J. Improved Bohr's phenomenon in quasi-subordination classes. J Math Anal Appl, 2022, $\textbf{506}$(1): Article 125645
[34] Ponnusamy S, Wirths K J. Bohr type inequalities for functions with a multiple zero at the origin. Comput Methods Funct Theory, 2020, $\textbf{20}$: 559-570
[35] Rogosinski W. Über Bildschranken bei Potenzreihen und ihren Abschnitten. Math Z, 1923, $\textbf{17}$: 260-276
[36] Sidon S. Über einen satz von Hernn Bohr. Math Z, 1927, ${\bf 26}$: 731-732
[37] Tomić M. Sur un théorème de H Bohr. Math Scand, 1962, ${\bf 11}$: 103-106
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