Skip to main content
SpringerLink
Log in

New Inequalities for the Coefficients of Unimodular Bounded Functions

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to 1, then the sum of absolute values of its terms is less than or equal to 1 for the subdisk \(|z|<1/3\) and 1/3 is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present related inequalities using \(\sum _{n=0}^{\infty }|a_n|^2r^{2n}\) that generalize for example the well known inequality

$$\begin{aligned} \sum _{n=0}^{\infty }|a_n|^2r^{2n} \le 1. \end{aligned}$$

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abu-Muhanna, Y., Ali, R.M., Ponnusamy, S.: On the Bohr inequality progress in approximation theory and applicable complex analysis. Springer Optim. Appl. 117, 265–295 (2016)

    Google Scholar 

  2. Ali, R.M., Barnard, R.W., Solynin, AYu.: A note on the Bohr’s phenomenon for power series. J. Math. Anal. Appl. 449(1), 154–167 (2017)

    Article  MathSciNet  Google Scholar 

  3. Alkhaleefah, S.A., Kayumov, I.R., Ponnusamy, S.: On the Bohr inequality with a fixed zero coefficient. Proc. Am. Math. Soc. 147(12), 5263–5274 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bénéteau, C., Dahlner, A., Khavinson, D.: Remarks on the Bohr phenomenon. Comput. Methods Funct. Theory 4(1), 1–19 (2004)

    Article  MathSciNet  Google Scholar 

  5. Bhowmik, B., Das, N.: Bohr phenomenon for subordinating families of certain univalent functions. J. Math. Anal. Appl. 462(2), 1087–1098 (2018)

    Article  MathSciNet  Google Scholar 

  6. Boas, H.P., Khavinson, D.: Bohr’s power series theorem in several variables. Proc. Am. Math. Soc. 125(10), 2975–2979 (1997)

    Article  MathSciNet  Google Scholar 

  7. Bohr, H.: A theorem concerning power series. Proc. Lond. Math. Soc. 13(2), 1–5 (1914)

    Article  MathSciNet  Google Scholar 

  8. Bombieri, E.: Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze. Boll. Un. Mat. Ital. 17(3), 276–282 (1962)

    MathSciNet  MATH  Google Scholar 

  9. Carlson, F.: Sur les coefficients d’une fonction bornée dans le cercle unité (French) Ark. Mat. Astr. Fys. 27A(1), 8 (1940)

    MATH  Google Scholar 

  10. Defant, A., Prengel, C.: Christopher Harald Bohr meets Stefan Banach. Methods in Banach space theory, 317–339, London Math. Soc. Lecture Note Ser. 337, Cambridge Univ. Press, Cambridge, (2006)

  11. Dixon, P.G.: Banach algebras satisfying the non-unital von Neumann inequality. Bull. Lond. Math. Soc. 27(4), 359–362 (1995)

    Article  MathSciNet  Google Scholar 

  12. Evdoridis, S., Ponnusamy, S., Rasila, A.: Improved Bohr’s inequality for locally univalent harmonic mappings. Indag. Math. (N.S.) 30, 201–213 (2019)

    Article  MathSciNet  Google Scholar 

  13. Garcia, S.R., Mashreghi, J., Ross, W.T.: Finite Blaschke Products and Their Connections. Springer, Cham (2018)

    Book  Google Scholar 

  14. Kayumov, I.R., Ponnusamy, S.: Bohr inequality for odd analytic functions. Comput. Methods Funct. Theory 17, 679–688 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kayumov, I.R., Ponnusamy, S.: Improved version of Bohr’s inequality. C. R. Math. Acad. Sci. Paris 356(3), 272–277 (2018)

    Article  MathSciNet  Google Scholar 

  16. Kayumov, I.R., Ponnusamy, S.: Bohr’s inequalities for the analytic functions with lacunary series and harmonic functions. J. Math. Anal. Appl. 465, 857–871 (2018)

    Article  MathSciNet  Google Scholar 

  17. Kayumov, I.R., Ponnusamy, S., Shakirov, N.: Bohr radius for locally univalent harmonic mappings. Math. Nachr. 291, 1757–1768 (2018)

    Article  MathSciNet  Google Scholar 

  18. Liu, G., Ponnusamy, S.: On Harmonic \(\nu \)-Bloch and \(\nu \)-Bloch-type mappings, Results Math. 73(3)(2018), Art 90, p 21

  19. Liu, G., Liu, Z.H., Ponnusamy, S.: Refined Bohr inequality for bounded analytic functions, arXiv:2006.08930

  20. Liu, M.S., Shang, Y.M., Xu, J.F.: Bohr-type inequalities of analytic functions. J. Inequal. Appl. 345, 13 (2018)

    MathSciNet  Google Scholar 

  21. Paulsen, V.I., Popescu, G., Singh, D.: On Bohr’s inequality. Proc. Lond. Math. Soc. 85(2), 493–512 (2002)

    Article  MathSciNet  Google Scholar 

  22. Paulsen, V.I., Singh, D.: Bohr’s inequality for uniform algebras. Proc. Am. Math. Soc. 132(12), 3577–3579 (2004)

    Article  MathSciNet  Google Scholar 

  23. Paulsen, V.I., Singh, D.: Extensions of Bohr’s inequality. Bull. Lond. Math. Soc. 38(6), 991–999 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of the first author is supported by Mathematical Research Impact Centric Support of DST, India (MTR/2017/000367). The authors want to thank H. Schellwat from Örebro University for his help in getting the article of F. Carlson [9].

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    Saminathan Ponnusamy & Ramakrishnan Vijayakumar

  2. Institut für Analysis und Algebra, TU Braunschweig, 38106, Braunschweig, Germany

    Karl-Joachim Wirths

Authors
  1. Saminathan Ponnusamy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ramakrishnan Vijayakumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karl-Joachim Wirths
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saminathan Ponnusamy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ponnusamy, S., Vijayakumar, R. & Wirths, KJ. New Inequalities for the Coefficients of Unimodular Bounded Functions. Results Math 75, 107 (2020). https://doi.org/10.1007/s00025-020-01240-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-01240-1

Keywords

Mathematics Subject Classification

Search

Navigation