On Bohr phenomenon for a class of bounded analytic and harmonic functions

Year 2025, Volume: 1 Issue: 1, 22 - 33, 28.11.2025

Ismaila Amusa , Adesanmi Mogbademu

Abstract

In this paper, we consider a class of bounded analytic functions f(z)= \sum\limits_{n=0}^{\infty}a_nz^n in the open unit disk, subject to the coefficient bound |a_n| \le 1- |a_0|, n \ge 1. We study functions formed through compositions with Schwarz functions and derive the corresponding Bohr-type bounds. Our investigation also includes sense-preserving harmonic mappings f(z) = h(z) + \overline{g(z)}, where h(z) belongs to the considered analytic class and g'(0)=0. These findings contribute to the improvements and generalizations of some existing results in the literature.

Keywords

Bohr inequality , Schwarz function , sense-preserving , harmonic mapping

References

• [1] H.A. Bohr (1914), A theorem concerning power series, Proc. London Math. Soc., 13(1), 1–5. https://doi.org/10.1112/plms/s2-13.1.1
• [2] M.S. Liu, Y.M. Shang, J.F. Xu (2018), Bohr-type inequalities of analytic functions, J. Ineq. Appl., 345, 1–13. https://doi.org/10.1186/s13660-018-1937-y
• [3] Y. Abu-Muhanna, R.M. Ali, S. Ponnusamy (2016), On the Bohr inequality, Progress in Approximation Theory and Applicable Complex Analysis, Springer Optimization and Its Applications, 117, 265–295.
• [4] E. Bombieri, J. Bourgain (2004), A remark on Bohr’s inequality, Int. Math. Res. Not., 80, 4307–4330. https://doi.org/10.1155/S1073792804143444
• [5] D. Singh, V.I. Paulsen, G. Popescu (2002), On Bohr’s inequality, Proc. London Math. Soc., 85(2), 493–512. https://doi.org/10.1112/S0024611502013692
• [6] I.R. Kayumov, S. Ponnusamy (2018), Improved version of Bohr’s inequality, C. R. Acad. Sci. Paris Ser., 356, 272–277. https://doi.org/10.1016/j.crma.2018.01.010
• [7] M.B. Ahamed (2022), The sharp refined Bohr-Rogosinski’s inequalities for certain classes of harmonic mappings, Complex Var. Ellipt. Equ., 69(4), 586–606. https://doi.org/10.1080/17476933.2022.2155636
• [8] M.B. Ahamed, S. Ahammed (2023), Bohr-Rogosinski-type inequalities for certain classes of functions: Analytic, univalent, and convex, Results Math., 78, 171. https://doi.org/10.1007/s00025-023-01953-z
• [9] S. Ahammed, M.B. Ahamed (2024), Refined Bohr inequalities for certain classes of functions: analytic, univalent, and convex, Canad. Math. Bull., 67(1), 9–25. https://doi.org/10.4153/S0008439523000474
• [10] M.B. Ahamed (2022), The Bohr-Rogosinski radius for a certain class close-to-convex harmonic mappings, Comput. Methods Funct. Theory, 1–19. https://doi.org/10.1007/s40315-022-00444-6
• [11] S. Ahammed, M.B. Ahamed (2025), Operator valued Bohr-type inequalities for certain integral transforms, Mediterranean J. Math., 22(34). https://doi.org/10.1007/s00009-025-02803-8
• [12] M.B. Ahamed, V. Allu, H. Halder (2022), The Bohr phenomenon for analytic functions on shifted disks, Ann. Acad., Sci. Fenn. Math., 27, 103–120. https://doi.org/10.54330/afm.112561
• [13] S. Ahammed, M.B. Ahamed (2023), Refined Bohr inequality for functions in Cn and in complex Banach spaces, Complex Var. Ellipt. Equ., 69(10), 1700–1722. https://doi.org/10.1080/17476933.2023.2244438
• [14] A. Ismagilov, I. Kayumov, S. Ponnusami (2020), Sharp Bohr-type inequality, J. Math. Anal. Appl., 489(2), 124147. https://doi.org/10.1016/j.jmaa.2020.124147
• [15] D. Nilanjan, B. Bappaditya (2018), Bohr phenomenon for subordinating families of certain univalent function, J. Math. Anal. Appl., 462(2), 1087–1098. https://doi.org/10.1016/j.jmaa.2018.01.035
• [16] M.A. Rosihan,W.B. Roger, Y.S. Alexender (2017), A note on Bohr’s Phenomenon for power series, J. Math. Anal. Appl., 449, 154–167.
• [17] D. Singh, V.I. Paulsen (2004), Bohr’s inequality for uniform algebras, Proc. Amer. Math. Soc., 132(12), 493–512.
• [18] I.S. Amusa, A.A. Mogbademu (2025), Some Bohr-type inequalities for sense-preserving Harmonic mappings, Probl. Anal. Issues Anal., 14(32), 3–21. https://doi.org/10.15393/j3.art.2025.16770
• [19] S.A. Alkhaleefah (2020), Bohr Phenomenon for the special family of analytic functions and Harmonic mappings, Probl. Anal. Issues Anal., 9(3), 3–13. https://doi.org/10.15393/j3.art.2020.7990
• [20] I.S. Amusa, A.A. Mogbademu (2025), Some Bohr-type inequalities for certain class of Holomorphic functions, J. Nig. Math. Soc., 44(1), 85–97.
• [21] H. Xiaojun, W. Qihan, L. Boyong (2021), Bohr-type inequalities for bounded analytic functions of Schwarz functions, AIMS Math., 6(12), 3608–3621. https://doi.org/10.3934/math.2021791
• [22] D. Nilanjan, B. Bappaditya (2019), Bohr phenomenon for locally univalent functions and logarithmic power series, Comput. Methods Funct. Theory, 19, 729–745. https://doi.org/10.1007/s40315-019-00291-y