Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-08-18T13:52:19.756Z Has data issue: false hasContentIssue false
  • English
  • Français

Refined Bohr inequalities for certain classes of functions: analytic, univalent, and convex

Part of: General properties Geometric function theory Spaces and algebras of analytic functions

Published online by Cambridge University Press:  09 June 2023

Sabir Ahammed Open the ORCID record for Sabir Ahammed [Opens in a new window]  and
Molla Basir Ahamed Open the ORCID record for Molla Basir Ahamed [Opens in a new window]
Sabir Ahammed
Affiliation:
Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India e-mail: sabira.math.rs@jadavpuruniversity.in
Molla Basir Ahamed*
Affiliation:
Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India e-mail: sabira.math.rs@jadavpuruniversity.in
*
e-mail: mbahamed.math@jadavpuruniversity.in
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk $ \mathbb {D} $, class of analytic functions $ f $ defined on $ \mathbb {D} $ such that $\mathrm {Re}\left (f(z)\right )<1 $, and class of subordination to a function g in $ \mathbb {D} $. Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.

Keywords

Bounded analytic functionsBohr inequalityBohr–Rogosinski inequalitySchwarz–Pick lemma

MSC classification

Primary: 30A10: Inequalities in the complex domain 30H05: Bounded analytic functions 30C35: General theory of conformal mappings
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 9 - 25
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abu-Muhanna, Y., Bohr’s phenomenon in subordination and bounded harmonic classes . Complex Var. Elliptic Equ. 55(2010), 1071–1078.CrossRefGoogle Scholar
Abu-Muhanna, Y., Ali, R. M., Ng, Z. C., and Hasni, S. F. M, Bohr radius for subordinating families of analytic functions and bounded harmonic mappings . J. Math. Anal. Appl. 420(2014), 124–136.CrossRefGoogle Scholar
Ahamed, M. B. and Ahammed, S., Bohr type inequalities for the class of self-analytic maps on the unit disk . Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00482-8CrossRefGoogle Scholar
Ahamed, M. B. and Allu, V., Bohr phenomenon for certain classes of harmonic mappings . Rocky Mountain J. Math. 52(2022), no. 4, 1205–1225.CrossRefGoogle Scholar
Ahamed, M. B., Allu, V., and Halder, H., Bohr radius for certain classes of close-to-convex harmonic mappings . Anal. Math. Phys. 11(2021), 111.CrossRefGoogle Scholar
Ahamed, M. B., Allu, V., and Halder, H., Improved Bohr inequalities for certain class of harmonic univalent functions . Complex Var. Elliptic Equ. 68(2021), 267–290.CrossRefGoogle Scholar
Ahamed, M. B., Allu, V., and Halder, H., The Bohr phenomenon for analytic functions on shifted disks . Ann. Acad. Sci. Fenn. Math. 47(2022), 103–120.CrossRefGoogle Scholar
Aizenberg, L. and Tarkhanov, N., A Bohr phenomenon for elliptic equations . Proc. London Math. Soc. 82(2001), no. 2, 385–401.CrossRefGoogle Scholar
Ali, R. M., Abdulhadi, Z., and Ng, Z. C., The Bohr radius for starlike logharmonic mappings . Complex Var. Elliptic Equ. 61(2016), 1–14.CrossRefGoogle Scholar
Ali, R. M. and Ng, Z. C., The Bohr inequality in the hyperbolic plane . Complex Var. Elliptic Equ. 63(2018), 1539–1557.CrossRefGoogle Scholar
Alkhaleefah, S. A., Kayumov, I. R., and Ponnusamy, S., On the Bohr inequality with a fixed zero coefficient . Proc. Amer. Math. Soc. 147(2019), 5263–5274.CrossRefGoogle Scholar
Allu, V. and Halder, H., Bohr radius for certain classes of starlike and convex univalent functions . J. Math. Anal. Appl. 493(2021), 124519.CrossRefGoogle Scholar
Allu, V. and Halder, H., Bohr phenomenon for certain subclasses of harmonic mappings . Bull. Sci. Math. 173(2021), 103053.CrossRefGoogle Scholar
Allu, V. and Halder, H., Operator valued analogues of multidimensional Bohr inequality . Canad. Math. Bull. 65(2022), 1020–1035.CrossRefGoogle Scholar
Bénéteau, C., Dahlner, A., and Khavinson, D., Remarks on the Bohr phenomenon . Comput. Methods Funct. Theory 4(2004), 1–19.CrossRefGoogle Scholar
Bhowmik, B. and Das, N., Bohr phenomenon for subordinating families of certain univalent functions . J. Math. Anal. Appl. 462(2018), 1087–1098.CrossRefGoogle Scholar
Boas, H. P. and Khavinson, D., Bohr’s power series theorem in several variables . Proc. Amer. Math. Soc. 125(1997), 2975–2979.CrossRefGoogle Scholar
Bohr, H., A theorem concerning power series . Proc. Lond. Math. Soc. s2-13(1914), 1–5.CrossRefGoogle Scholar
Branges, L. D., A proof of the Bieberbach conjecture . Acta Math. 154(1985), 137–152.CrossRefGoogle Scholar
Das, N., Refinements of the Bohr and Rogosisnki phenomena . J. Math. Anal. Appl. 508(2022), no. 1, 125847.CrossRefGoogle Scholar
Dixon, P. G., Banach algebras satisfying the non-unital von Neumann inequality . Bull. Lond. Math. Soc. 27(1995), no. 4, 359–362.CrossRefGoogle Scholar
Duren, P. L., Univalent function , Springer, New York, 1983.Google Scholar
Evdoridis, S., Ponnusamy, S., and Rasila, A., Improved Bohr’s inequality for shifted disks . Results Math. 76(2021), 14.CrossRefGoogle Scholar
Hamada, H., Bohr phenomenon for analytic functions subordinate to starlike or convex functions . J. Math. Anal. Appl. 499(2021), 125019.CrossRefGoogle Scholar
Huang, Y., Liu, M.-S., and Ponnusamy, S., The Bohr-type operator on analytic functions and sections . Complex Var. Elliptic Equ. 68(2023), 317–332.CrossRefGoogle Scholar
Ismagilov, A., Kayumov, A. V., Kayumov, I. R., and Ponnusamy, S., Bohr inequalities in some classes of analytic functions . J. Math. Sci. 252(2021), no. 3, 360–373.CrossRefGoogle Scholar
Ismagilov, A., Kayumov, I. R., and Ponnusamy, S., Sharp Bohr type inequality . J. Math. Anal. Appl. 489(2020), 124147.CrossRefGoogle Scholar
Kayumov, I. R., Khammatova, D. M., and Ponnusamy, S., Bohr-Rogosinski phenomenon for analytic functions and Cesáro operators . J. Math. Anal. Appl. 496(2021), 124824.CrossRefGoogle Scholar
Kayumov, I. R. and Ponnusamy, S., Improved version of Bohr’s inequality . C. R. Acad. Sci. Paris, Ser. I 356(2018), 272–277.CrossRefGoogle Scholar
Kumar, S., A generalization of the Bohr inequality and its applications . Complex Var. Elliptic Equ. 68(2023), 963–973.CrossRefGoogle Scholar
Lata, S. and Singh, D., Bohr’s inequality for non-commutative Hardy spaces . Proc. Amer. Math. Soc. 150(2022), no. 1, 201–211.CrossRefGoogle Scholar
Liu, G., Liu, Z., and Ponnusamy, S., Refined Bohr inequality for bounded analytic functions . Bull. Sci. Math. 173(2021), 103054.CrossRefGoogle Scholar
Liu, M.-S. and Ponnusamy, S., Multidimensional analogues of refined Bohr’s inequality . Porc. Amer. Math. Soc. 149(2021), no. 5, 2133–2146.CrossRefGoogle Scholar
Liu, M.-S., Ponnusamy, S., and Wang, J., Bohr’s phenomenon for the classes of quasi-subordination and K-quasiregular harmonic mappings . RACSAM 114(2020), 115.CrossRefGoogle Scholar
Liu, M.-S., Shang, Y-M., and Xu, J-F., Bohr-type inequalities of analytic functions . J. Inequal. Appl. 345(2018), 2018.Google Scholar
Paulsen, V. I., Popescu, G., and Singh, D., On Bohr’s inequality . Proc. Lond. Math. Soc. 85(2002), no. 2, 493–512.CrossRefGoogle Scholar
Ponnusamy, S., Vijayakumar, R., and Wirths, K.-J., Improved Bohr’s phenomenon in quasi-subordination classes . J. Math. Anal. Appl. 506(2022), no. 1, 125645.CrossRefGoogle Scholar
Rogosinski, W., Über Bildschranken bei Potenzreihen und ihren Abschnitten . Math. Z. 17(1923), 260–276.CrossRefGoogle Scholar
Rogosinski, W., On the coefficients of subordination functions . Proc. Lond. Math. Soc. 48(1943), no. 2, 48–82.Google Scholar
Ruscheweyh, S., Two remarks on bounded analytic functions . Serdica 11(1985), 200–202.Google Scholar
Sidon, S., Uber einen satz von Hernn Bohr . Math. Zeit. 26(1927), 731–732.CrossRefGoogle Scholar
Tomic, M., Sur un Theoreme de H. Bohr . Math. Scand. 11(1962), 103–106.CrossRefGoogle Scholar