Sharp Bohr type inequality. (English) Zbl 1441.30077
Summary: This article is devoted to the sharp improvement of the classical Bohr inequality for bounded analytic functions defined on the unit disk. We also prove two other sharp versions of the Bohr inequality by replacing the constant term by the absolute of the function and the square of the absolute of the function, respectively.
MSC:
| 30H05 | Spaces of bounded analytic functions of one complex variable |
| 30B10 | Power series (including lacunary series) in one complex variable |
| 30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |
References:
| [1] | Abu-Muhanna, Y., Bohr’s phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ., 55, 11, 1071-1078 (2010) · Zbl 1215.46018 |
| [2] | Abu-Muhanna, Y.; Ali, R. M., Bohr’s phenomenon for analytic functions into the exterior of a compact convex body, J. Math. Anal. Appl., 379, 2, 512-517 (2011) · Zbl 1214.30004 |
| [3] | Aizenberg, L., Multidimensional analogues of Bohr’s theorem on power series, Proc. Am. Math. Soc., 128, 4, 1147-1155 (1999) · Zbl 0948.32001 |
| [4] | Aizenberg, L., Generalization of Carathéodory’s inequality and the Bohr radius for multidimensional power series, (Selected Topics in Complex Analysis. Selected Topics in Complex Analysis, Oper. Theory Adv. Appl., vol. 158 (2005), Birkhäuser: Birkhäuser Basel), 87-94 · Zbl 1086.32003 |
| [5] | Aizenberg, L., Generalization of results about the Bohr radius for power series, Stud. Math., 180, 161-168 (2007) · Zbl 1118.32001 |
| [6] | Aizenberg, L.; Aytuna, A.; Djakov, P., Generalization of a theorem of Bohr for basis in spaces of holomorphic functions of several complex variables, J. Math. Anal. Appl., 258, 2, 429-447 (2001) · Zbl 0988.32005 |
| [7] | Ali, R. M.; Abu-Muhanna, Y.; Ponnusamy, S., On the Bohr inequality, (Govil, N. K.; etal., Progress in Approximation Theory and Applicable Complex Analysis. Progress in Approximation Theory and Applicable Complex Analysis, Springer Optimization and Its Applications, vol. 117 (2016)), 265-295 |
| [8] | 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) · Zbl 1428.30001 |
| [9] | Bénéteau, C.; Dahlner, A.; Khavinson, D., Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory, 4, 1, 1-19 (2004) · Zbl 1067.30094 |
| [10] | Bhowmik, B.; Das, N., Bohr phenomenon for subordinating families of certain univalent functions, J. Math. Anal. Appl., 462, 2, 1087-1098 (2018) · Zbl 1391.30003 |
| [11] | Boas, H. P.; Khavinson, D., Bohr’s power series theorem in several variables, Proc. Am. Math. Soc., 125, 10, 2975-2979 (1997) · Zbl 0888.32001 |
| [12] | Bohr, H., A theorem concerning power series, Proc. Lond. Math. Soc., 13, 2, 1-5 (1914) · JFM 44.0289.01 |
| [13] | Bombieri, E., Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze, Boll. Unione Mat. Ital., 17, 3, 276-282 (1962) · Zbl 0109.04801 |
| [14] | Bombieri, E.; Bourgain, J., A remark on Bohr’s inequality, Int. Math. Res. Not., 80, 4307-4330 (2004) · Zbl 1069.30001 |
| [15] | Defant, A.; Frerick, L., A logarithmic lower bound for multi-dimensional Bohr radii, Isr. J. Math., 152, 1, 17-28 (2006) · Zbl 1124.32003 |
| [16] | Defant, A.; Frerick, L.; Ortega-Cerdà, J.; Ounaïes, M.; Seip, K., The Bohnenblust-Hille inequality for homogenous polynomials is hypercontractive, Ann. Math. (2), 174, 512-517 (2011) |
| [17] | Defant, A.; García, D.; Maestre, M., Bohr’s power series theorem and local Banach space theory, J. Reine Angew. Math., 557, 173-197 (2003) · Zbl 1031.46014 |
| [18] | Defant, A.; Garcia, S. R.; Maestre, M., Asymptotic estimates for the first and second Bohr radii of Reinhardt domains, J. Approx. Theory, 128, 53-68 (2004) · Zbl 1049.32001 |
| [19] | Defant, A.; Maestre, M.; Schwarting, U., Bohr radii of vector valued holomorphic functions, Adv. Math., 231, 5, 2837-2857 (2012) · Zbl 1252.32001 |
| [20] | Defant, A.; Prengel, C., Christopher Harald Bohr meets Stefan Banach, (Methods in Banach Space Theory. Methods in Banach Space Theory, London Math. Soc. Lecture Note Ser., vol. 337 (2006), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 317-339 · Zbl 1130.46028 |
| [21] | Djakov, P. B.; Ramanujan, M. S., A remark on Bohr’s theorems and its generalizations, J. Anal., 8, 65-77 (2000) · Zbl 0969.30001 |
| [22] | Evdoridis, S.; Ponnusamy, S.; Rasila, A., Improved Bohr’s inequality for locally univalent harmonic mappings, Indag. Math. (N.S.), 30, 201-213 (2019) · Zbl 1404.31002 |
| [23] | Garcia, S. R.; Mashreghi, J.; Ross, W. T., Finite Blaschke Products and Their Connections (2018), Springer: Springer Cham · Zbl 1398.30002 |
| [24] | Ismagilov, A.; Kayumova, A.; Kayumov, I. R.; Ponnusamy, S., Bohr type inequalities in some classes of analytic functions, (Complex Analysis. Complex Analysis, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., vol. 153 (2018), VINITI: VINITI Moscow), 69-83, (in Russian) |
| [25] | Kayumov, I. R.; Ponnusamy, S., Bohr inequality for odd analytic functions, Comput. Methods Funct. Theory, 17, 679-688 (2017) · Zbl 1385.30003 |
| [26] | Kayumov, I. R.; Ponnusamy, S., Bohr’s inequalities for the analytic functions with lacunary series and harmonic functions, J. Math. Anal. Appl., 465, 2, 857-871 (2018) · Zbl 1394.31001 |
| [27] | Kayumov, I. R.; Ponnusamy, S., Improved version of Bohr’s inequality, C. R. Math. Acad. Sci. Paris, 356, 3, 272-277 (2018) · Zbl 1432.30002 |
| [28] | Kayumov, I. R.; Ponnusamy, S., On a powered Bohr inequality, Ann. Acad. Sci. Fenn., Ser. A 1 Math., 44, 301-310 (2019) · Zbl 1423.30008 |
| [29] | Kayumov, I. R.; Ponnusamy, S., Bohr-Rogosinski radius for analytic functions, preprint, see · Zbl 1456.30001 |
| [30] | Kayumov, I. R.; Ponnusamy, S.; Shakirov, N., Bohr radius for locally univalent harmonic mappings, Math. Nachr., 291, 11-12, 1757-1768 (2018) · Zbl 1398.30003 |
| [31] | Liu, M. S.; Shang, Y. M.; Xu, J. F., Bohr-type inequalities of analytic functions, J. Inequal. Appl., Article 345 pp. (2018) · Zbl 1498.30001 |
| [32] | Paulsen, V. I.; Popescu, G.; Singh, D., On Bohr’s inequality, Proc. Lond. Math. Soc., 85, 2, 493-512 (2002) · Zbl 1033.47008 |
| [33] | Paulsen, V. I.; Singh, D., Bohr’s inequality for uniform algebras, Proc. Am. Math. Soc., 132, 12, 3577-3579 (2004) · Zbl 1062.46041 |
| [34] | Ponnusamy, S.; Wirths, K.-J., Bohr type inequalities for functions with a multiple zero at the origin, Comput. Methods Funct. Theory (2020), in press · Zbl 1461.30004 |
| [35] | Popescu, G., Multivariable Bohr inequalities, Trans. Am. Math. Soc., 359, 11, 5283-5317 (2007) · Zbl 1128.47009 |
| [36] | Ricci, G., Complementi a un teorema di H. Bohr riguardante le serie di potenze, Rev. Unión Mat. Argent., 17, 185-195 (1955/1956) · Zbl 0072.07301 |
| [37] | Sidon, S., Über einen Satz von Herrn Bohr, Math. Z., 26, 1, 731-732 (1927) · JFM 53.0281.04 |
| [38] | Tomić, M., Sur un théorème de H. Bohr, Math. Scand., 11, 103-106 (1962) · Zbl 0109.30202 |
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