Bohr phenomenon for the special family of analytic functions and harmonic mappings

S. A. Alkhaleefah

Problemy analiza, Tome 9 (2020) no. 3, pp. 3-13.

Résumé

In this paper we obtain the sharp Bohr radius for a family of bounded analytic functions $\mathcal B'$ and for the family of sense-preserving $\mathrm{K}$-quasiconformal harmonic mappings of the form $f = h + \overline g$, where $h\in \mathcal B'$.

Détail

Keywords: Bohr inequality, analytic functions, harmonic mappings, sense-preserving $\mathrm{K}$-quasiconformal mappings.

Bibliographie

[1] Abu Muhanna Y., Ali R. M., Ponnusamy S., “On the Bohr inequality”, Progress in Approximation Theory and Applicable Complex Analysis, Springer Optimization and Its Applications, 117, eds. N. K. Govil et al., 2016, 265–295 | MR

[2] Aizenberg L., “Generalization of results about the Bohr radius for power series”, Stud. Math., 180 (2007), 161–168 | DOI | MR | Zbl

[3] Sib. Math. J., 45:4 (2004), 606–617 | DOI | MR | Zbl

[4] Ali R. M., Barnard R. W., Solynin A. Yu., “A note on the Bohr's phenomenon for power series”, J. Math. Anal. Appl., 449:1 (2017), 154–167 | DOI | MR | Zbl

[5] Alkhaleefah S. A., Kayumov I. R., Ponnusamy S., “On the Bohr inequality with a fixed zero coefficient”, Proc. Amer. Math. Soc., 147:12 (2019), 5263–5274 | DOI | MR | Zbl

[6] Bénéteau C., Dahlner A., Khavinson D., “Remarks on the Bohr phenomenon”, Comput. Methods Funct. Theory, 4:1 (2004), 1–19 | DOI | MR | Zbl

[7] Boas H. P., Khavinson D., “Bohr's power series theorem in several variables”, Proc. Am. Math. Soc., 125:10 (1997), 2975–2979 | DOI | MR | Zbl

[8] Bohr H., “A theorem concerning power series”, Proc. London Math. Soc., s2–13:1 (1914), 1–5 | DOI | MR | Zbl

[9] Djakov P. B., Ramanujan M. S., “A remark on Bohr's theorems and its generalizations”, J. Analysis, 8 (2000), 65–77 | MR | Zbl

[10] Evdoridis S., Ponnusamy S., Rasila A., “Improved Bohr's inequality for locally univalent harmonic mappings”, Indag. Math. (N.S.), 30 (2019), 201–213 | DOI | MR | Zbl

[11] Garcia S. R., Mashreghi J., Ross W. T., Finite Blaschke products and their connections, Springer International Publishing, 2018 | MR | Zbl

[12] Ismagilov A., Ponnusamy S., Kayumov I., “Sharp Bohr type inequality”, J. Math. Anal. and Appl., 486 (2020), 124147, 10 pp. | DOI | MR

[13] Kayumov I. R., Ponnusamy S., “Bohr inequality for odd analytic functions”, Comput. Methods Funct. Theory, 17 (2017), 679–688 | DOI | MR | Zbl

[14] Kayumov I. R., Ponnusamy S., “Bohr's inequalities for the analytic functions with lacunary series and harmonic functions”, J. Math. Anal. and Appl., 465:2 (2018), 857–871 | DOI | MR | Zbl

[15] Kayumov I. R., Ponnusamy S., “On a powered Bohr inequality”, Ann. Acad. Sci. Fenn. Ser. AI Math., 44 (2019), 301–310 | DOI | MR | Zbl

[16] Kayumov I. R., Ponnusamy S., Shakirov N., “Bohr radius for locally univalent harmonic mappings”, Math. Nachr., 291:11–12 (2018), 1757–1768 | DOI | MR | Zbl

[17] Liu Z. H., Ponnusamy S., “Bohr radius for subordination and K-quasiconformal harmonic mappings”, Bull. Malays. Math. Sci. Soc., 42 (2019), 2151–2168 | DOI | MR | Zbl

[18] Paulsen V. I., Popescu G., Singh D., “On Bohr's inequality”, Proc. London Math. Soc., 85:2 (2002), 493–512 | DOI | MR | Zbl

[19] Paulsen V. I., Singh D., “Bohr's inequality for uniform algebras”, Proc. Amer. Math. Soc., 132:12 (2004), 3577–3579 | DOI | MR | Zbl

[20] Ponnusamy S., Vijayakumar R., Wirths K.-J., “New inequalities for the coefficients of unimodular bounded functions”, Results in Mathematics, 75:107 (2020) | DOI | MR | Zbl

[21] Ponnusamy S., Wirths K.-J., “Bohr type inequalities for functions with a multiple zero at the origin”, Comput. Methods Funct. Theory, 2020 | DOI | MR