THE BOHR'S PHENOMENON FOR THE CLASS OF K-QUASICONFORMAL HARMONIC MAPPINGS

  • Raju BISWAS ,
  • Rajib MANDAL
Expand
  • Department of Mathematics, Raiganj University, Raiganj, West Bengal-733134, India
Rajib MANDAL, E-mail: rajibmathresearch@gmail.com

Received date: 2024-10-16

  Online published: 2026-05-22

Supported by

This research was supported by University Grants Commission (IN) fellowship (F.44-1/2018 (SA-III)).

Abstract

The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\bar{g}$ in the unit disk $\mathbb{D}: = \{z\in\mathbb{C}: |z| < 1\}$. In order to achieve these objectives, we employ the non-negative quantity $S_\rho(h)$ and the concept of replacing the initial coefficients of the majorant series by the absolute values of the analytic function and its derivative, as well as other various settings. Moreover, we obtain the sharp Bohr-Rogosinski radius for harmonic mappings in the unit disk by replacing the bounding condition on the analytic function $h$ with the half-plane condition.

Cite this article

Raju BISWAS , Rajib MANDAL . THE BOHR'S PHENOMENON FOR THE CLASS OF K-QUASICONFORMAL HARMONIC MAPPINGS[J]. Acta mathematica scientia, Series B, 2026 , 46(2) : 790 -811 . DOI: 10.1007/s10473-026-0215-1

References

[1] Abu-Muhanna Y, Ali R M, Ponnusamy S. On the Bohr inequality//Govil N K, Mohapatra R, Qazi M, Schmeisser G. Progress in Approximation Theory and Applicable Complex Analysis. Berlin: Springer, 2017, 117: 269-300
[2] Aizenberg L. Multidimensional analogues of Bohr's theorem on power series. Proc Amer Math Soc, 2000, 128: 1147-1155
[3] Aizenberg L, Aytuna A, Djakov P. Generalization of theorem on Bohr for bases in spaces of holomorphic functions of several complex variables. J Math Anal Appl, 2001, 258: 429-447
[4] Alkhaleefah S A, Kayumov I R, Ponnusamy S. On the Bohr inequality with a fixed zero coefficient. Proc Am Math Soc, 2019, 147(12): 5263-5274
[5] Alkhaleefah S A, Kayumov I R, Ponnusamy S.Bohr-Rogosinski inequalities for bounded analytic functions. Lobachevskii J Math, 2020, 41: 2110-2119
[6] Ahamed M B, Allu V, Halder H. The Bohr phenomenon for analytic functions on a shifted disk. Ann Fenn Math, 2022, 47: 103-120
[7] Ahamed M B, Allu V, Halder H. Improved Bohr inequalities for certain class of harmonic univalent functions. Complex Var Elliptic Equ, 2023, 68: 267-290
[8] Allu V, Arora V, Shaji A. On the second Hankel determinant of logarithmic coefficients for certain univalent functions. Mediterr J Math, 2023, 20: Art 81
[9] Allu V, Halder H. Bohr radius for certain classes of starlike and convex univalent functions. J Math Anal Appl, 2021, 493(1): 124519
[10] Allu V, Halder H. Bohr phenomenon for certain subclasses of harmonic mappings. Bull Sci Math, 2021, 173: 103053
[11] Allu V, Halder H. Bohr phenomenon for certain close-to-convex analytic functions. Comput Methods Funct Theory, 2022, 22: 491-517
[12] Allu V, Halder H. Bohr inequality for certain harmonic mappings. Indag Math, 2022, 33(3): 581-597
[13] Allu V, Arora V. Bohr-Rogosinski type inequalities for concave univalent functions. J Math Anal Appl, 2023, 520: 126845
[14] Biswas R. Second Hankel determinant of logarithmic coefficients for $\mathcal G(\alpha)$ and $\mathcal P(M)$. J Anal, 2024, 32: 3019-3037
[15] Bénéteau C, Dahlner A, Khavinson D. Remarks on the Bohr phenomenon. Comput Methods Funct Theory,2004, 4(1): 1-19
[16] Boas H P, Khavinson D. Bohr's power series theorem in several variables. Proc Amer Math Soc, 1997, 125(10): 2975-2979
[17] Bohr H. A theorem concerning power series. Proc London Math Soc, 1914, 13(2): 1-5
[18] Bombieri E. Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze. Bolletino dell Unione Mat Ital, 1962, 17(3): 276282
[19] Bombieri E, Bourgain J. A remark on Bohr's inequality. Int Math Res Not, 2004, 80: 4307-4330
[20] Dai S Y, Pan Y F. Note on Schwarz-Pick estimates for bounded and positive real part analytic functions. Proc Amer Math Soc, 2008, 136: 635-640
[21] Defant A, Frerick L, Ortega-Cerdà J,et al. The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. Ann Math,2011, 174(2): 512-517
[22] Duren P.Harmonic Mapping in the Plane. Cambridge: Cambridge University Press, 2004
[23] Evdoridis S, Ponnusamy S, Rasila A. Improved Bohr's inequality for locally univalent harmonic mappings. Indag Math (NS), 2019, 30: 201-213
[24] Evdoridis S, Ponnusamy S, Rasila A. Improved Bohr's inequality for shifted disks. Results Math, 2021, 76: Art 14
[25] Fournier R, Ruscheweyh S. On the Bohr radius for simply connected plane domains. CRM Proc Lect Notes, 2010, 51: 165-171
[26] Gamelin T W. Complex Analysis.New York: Springer-Verlag, 2000
[27] Garcia S R, Mashreghi J, Ross W T.Finite Blaschke Products and Their Connections. Cham: Springer, 2018
[28] Huang Y, Liu M S, Ponnusamy S. Refined Bohr-type inequalities with area measure for bounded analytic functions. Anal Math Phys, 2020, 10: Art 50
[29] Huang Y, Liu M S, Ponnusamy S. Bohr-type inequalities for harmonic mappings with a multiple zero at the origin. Mediterr J Math, 2021, 18: Art 75
[30] Ismagilov A, Kayumov I R, Ponnusamy S. Sharp Bohr type inequality. J Math Anal Appl, 2020, 489: 124147
[31] Kalaj D. Quasiconformal harmonic mapping between Jordan domains. Math Z, 2008, 260(2): 237-252
[32] Kayumov I R, Khammatova D M, Ponnusamy S. Bohr-Rogosinski phenomenon for analytic functions and Ces\'aro operators. J Math Anal Appl, 2021, 493(2): 124824
[33] Kayumov I R, Ponnusamy S.Bohr-Rogosinski radius for analytic functions. arXiv.1708.05585
[34] Kayumov I R, Ponnusamy S. Bohr inequality for odd analytic functions. Comput Methods Funct Theory, 2017, 17: 679-688
[35] Kayumov I R, Ponnusamy S. Improved version of Bohr's inequality. C R Math Acad Sci Paris, 2018, 356(3): 272-277
[36] Kayumov I R, Ponnusamy S. Bohr's inequalities for the analytic functions with Lacunary series and harmonic functions. J Math Anal Appl, 2018, 465: 857-871
[37] Kayumov I R, Ponnusamy S, Shakirov N. Bohr radius for locally univalent harmonic mappings. Math Nachr, 2018, 291: 1757-1768
[38] Krantz S G. Geometric Function Theory.Explorations in Complex Analysis. Boston: Birkhäuser, 2006
[39] Lewy H. On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull Amer Math Soc, 1936, 42: 689-692
[40] Lin R Y, Liu M S, Ponnusamy S. The Bohr-type inequalities for holomorphic mappings with lacunary series in several complex variables. Acta Math Sci, 2023, 43: 63-79
[41] Liu G, Liu Z H, Ponnusamy S. Refined Bohr inequality for bounded analytic functions. Bull Sci Math, 2021, 173: 103054
[42] Liu G, Ponnusamy S. Improved Bohr inequality for harmonic mappings. Math Nachr, 2023, 296: 716-731
[43] Liu M S, Ponnusamy S, Wang J. Bohr's phenomenon for the classes of quasi-subordination and $K$-quasiregular harmonic mappings. RACSAM, 2020, 114(3): 115
[44] Liu Z H, Ponnusamy S.Bohr radius for subordination and $K$-quasiconformal harmonic mappings. Bull Malays Math Sci Soc, 2019, 42: 2151-2168
[45] Liu M S, Ponnusamy S.Multidimensional analogues of refined Bohr's inequality. Proc Amer Math Soc, 2021, 149: 2133-2146
[46] Liu M S, Shang Y M, Xu J F. Bohr-type inequalities of analytic functions. J Inequal Appl, 2018, 2018: Art 345
[47] Martio O. On harmonic quasiconformal mappings. Ann Acad Sci Fenn A I, 1968, 425: 3-10
[48] Mandal R, Biswas R, Guin S K. Geometric studies and the Bohr radius for certain normalized harmonic mappings. Bull Malays Math Sci Soc, 2024, 47: Art 131
[49] Ponnusamy S, Vijayakumar R, Wirths K J. New inequalities for the coefficients of unimodular bounded functions. Results Math, 2020, 75: Art 107
[50] Ponnusamy S, Wirths K J. Bohr type inequalities for functions with a multiple zero at the origin. Comput Methods Funct Theory, 2020, 20: 559-570
[51] Rogosinski W. Über Bildschranken bei Potenzreihen und ihren Abschnitten. Math Z,1923, 17: 260-276
[52] ST. Ruscheweyh, Two remarks on bounded analytic functions. Serdica1985, 11(2): 200-202
Options
Outlines

/