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Bohr–Rogosinski-Type Inequalities for Certain Classes of Functions: Analytic, Univalent, and Convex

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Abstract

In this article, we prove the sharp improvement of the classical Rogosinski inequality for the class of analytic self maps on the unit disk. Moreover, we prove two sharp Bohr-type inequalities by replacing the initial coefficient with the absolute value of the function in the majorant series. In addition, we obtain sharp refined Bohr–Rogosinski-type inequalities for the subordination class of univalent and convex function. Furthermore, considering integral power of the absolute value of the initial coefficient, we prove a sharp version of refined Bohr inequality for the class of functions with real part less than one.

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References

  1. Abu-Muhanna, Y.: Bohr’s phenomenon in subordination and bounded harmonic classes. Complex Var. Elliptic Equ. 55, 1071–1078 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Abu Muhanna, Y., Ali, R.M., Ponnusamy, S.: On the Bohr inequality. In: Progress in approximation theory and applicable complex analysis. In: Springer Optimization and Its Applications, vol. 117, pp. 269–300. Springer, Cham (2017)

  3. Ahamed, M.B., Allu, V.: Bohr phenomenon for certain classes of harmonic mappings. Rocky Mt. J. Math. 52(4), 1205–1225 (2021)

    MathSciNet  MATH  Google Scholar 

  4. Ahamed, M.B., Allu, V., Halder, H.: Bohr radius for certain classes of close-to-convex harmonic mappings. Anal. Math. Phys. 11, 111 (2021)

    MathSciNet  MATH  Google Scholar 

  5. Ahamed, M.B., Allu, V., Halder, H.: Improved Bohr inequalities for certain class of harmonic univalent functions. Complex Var. Elliptic Equ. 68(2), 67–290 (2023)

    MathSciNet  MATH  Google Scholar 

  6. Ahamed, M.B., Allu, V., Halder, H.: The Bohr phenomenon for analytic functions on shifted disks. Ann. Acad. Sci. Fenn. Math. 47, 103–120 (2022)

    MathSciNet  MATH  Google Scholar 

  7. Aizenberg, L.: Multidimensional analogues of Bohr’s theorem on power series. Proc. Am. Math. Soc. 128, 1147–1155 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Aizenberg, L.: Remarks on the Bohr and Rogosinski phenomenon for power series. Anal. Math. Phys. 2, 69–78 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Aizenberg, L., Elin, M., Shoikhet, D.: On the Rogosinski radius for holomorphic mappings and some of its applications. Stud. Math. 168(2), 147–158 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Aizenberg, L., Gotliv, V., Vidras, A.: Bohr and Rogosinski abscissas for ordinary Dirichlet series. Comput. Methods Funct. Theory 9(1), 65–74 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Aizenberg, L., Tarkhanov, N.: A Bohr phenomenon for elliptic equations. Proc. Lond. Math. Soc. 82(2), 385–401 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Ali, R.M., Ng, Z.C.: The Bohr inequality in the hyperbolic plane. Complex Var. Elliptic Equ. 63(11), 1539–1557 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Ali, R.M., Jain, N.K., Ravichandran, V.: Bohr radius for classes of analytic functions. Results Math. 74, 179 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Allu, V., Arora, V.: Bohr–Rogosinski type inequalities for concave univalent functions, 520(1)(2023), 126845

  15. Allu, V., Halder, H.: Bohr phenomenon for certain subclasses of harmonic mappings. Bull. Sci. Math. 173, 103053 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Allu, V., Halder, H.: Operator valued analogue of multidimensional Bohr inequality. Canad. Math. Bull. (2022, to appear)

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

    MathSciNet  MATH  Google Scholar 

  18. Aytuna, A., Djakov, P.: Bohr property of bases in the space of entire functions and its generalizations. Bull. Lond. Math. Soc. 45(2), 411–420 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. Bhowmik, B., Das, N.: Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series. Comput. Methods Funct. Theory 19, 729–745 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Bohr, H.: A theorem concerning power series. Proc. Lond. Math. Soc. s2-13, 1–5 (1914)

  23. Das, N.: Refinements of the Bohr and Rogosisnki phenomena. J. Math. Anal. Appl. 508(1), 125847 (2022)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Duren, P.L.: Univalent Functions. Springer, New York (1983)

    MATH  Google Scholar 

  26. Evdoridis, S., Ponnusamy, S., Rasila, A.: Improved Bohr’s inequality for shifted disks. Results Math. 76, 14 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Goluzin, G.M.: On subordinate univalent functions (Russian). Trudy. Mat. Inst. Steklov. 38, 68–71 (1951)

    MathSciNet  Google Scholar 

  28. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics, 255. Marcel Dekker Inc, New York (2003)

    Google Scholar 

  29. Huang, Y., Liu, M.-S., Ponnusamy, S.: Refined Bohr-type inequalities with area measure for bounded analytic functions. Anal. Math. Phys. 10(4), 50 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Huang, Y., Liu, M.-S., Ponnusamy, S.: The Bohr-type operator on analytic functions and sections. Complex Var. Elliptic Equ 68(2), 317–332 (2023)

    MathSciNet  MATH  Google Scholar 

  31. Ismagilov, A., Kayumov, A.V., Kayumov, I.R., Ponnusamy, S.: Bohr inequalities in some classes of analytic functions. J. Math. sci. 252, 360–373 (2021)

    MATH  Google Scholar 

  32. Ismagilov, A., Kayumov, I.R., Ponnusamy, S.: Sharp Bohr type inequality. J. Math. Anal. Appl. 489, 124147 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Landau, E., Gaier, D.: Darstellung und Begrüundung Einiger Neuerer Ergebnisse der Funktionentheorie. Springer, Berlin (1986)

    MATH  Google Scholar 

  34. Kayumov, I.R., Khammatova, D.M., Ponnusamy, S.: Bohr–Rogosinski phenomenon for analytic functions and Cesáro operators. J. Math. Anal. Appl. 496(2), 124824 (2021)

    MATH  Google Scholar 

  35. Kayumov, I.R., Ponnusamy, S.: Improved Version of Bohr’s Inequality. C. R. Acad. Sci. Paris Ser. I(356), 272–277 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Kayumov, I.R., Ponnusamy, S.: Bohr-Rogosinski radius for analytic functions. Preprint, see arXiv:1708.05585

  37. Kayumov, I.R., Ponnusamy, S.: On a powered Bohr inequality. Ann. Acad. Sci. Fenn. Ser. A 44, 301–310 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Kumar, S., Sahoo, S.: Bohr inequalities for certain integral operators. Mediterr. J. Math. 18, 268 (2021)

    MathSciNet  MATH  Google Scholar 

  39. Lata, S., Singh, D.: Bohr’s inequality for non-commutative Hardy spaces. Proc. Am. Math. Soc. 150(1), 201–211 (2022)

    MathSciNet  MATH  Google Scholar 

  40. Liu, G.: Bohr-type inequality via proper combination. J. Math. Anal. Appl. 503(1), 125308 (2021)

    MathSciNet  MATH  Google Scholar 

  41. Liu, G., Liu, Z., Ponnusamy, S.: Refined Bohr inequality for bounded analytic functions. Bull. Sci. Math. 173, 103054 (2021)

    MathSciNet  MATH  Google Scholar 

  42. Liu, M.-S., Ponnusamy, S.: Multidimensional analogues of refined Bohr’s inequality. Porc. Am. Math. Soc. 149(5), 2133–2146 (2021)

    MathSciNet  MATH  Google Scholar 

  43. Liu, M.-S., Ponnusamy, S., Wang, J.: Bohr’s phenomenon for the classes of Quasi-subordination and \(K\)-quasiregular harmonic mappings. RACSAM 114, 115 (2020)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Ponnusamy, S., Vijayakumar, R.: Note on improved Bohr inequality for harmonic mappings. In: Lecko, A., Thomas, D.K. (eds.) Current Research in Mathematical and Computer Sciences III. Publisher UWM, Olsztyn (2022) pp. 353360. See also arXiv:2104.06717

  48. Ponnusamy, S., Vijayakumar, R., Wirths, K.-J.: New inequalities for the coefficients of unimodular bounded functions. Results Math. 75, 107 (2020)

    MathSciNet  MATH  Google Scholar 

  49. Ponnusamy, S., Wirths, K.-J.: Bohr type inequalities for functions with a multiple zero at the origin. Comput. Methods Funct. Theory 20, 559–570 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Ponnusamy, S., Vijayakumar, R., Wirths, K.-J.: Improved Bohr’s phenomenon in quasi-subordination classes. J. Math. Anal. Appl. 506(1), 125645 (2022)

    MathSciNet  MATH  Google Scholar 

  51. Ruscheweyh, S.: Two remarks on bounded analytic functions. Serdica 11, 200–202 (1985)

    MathSciNet  MATH  Google Scholar 

  52. Rogosinski, W.: Über Bildschranken bei Potenzreihen und ihren Abschnitten. Math. Z. 17, 260–276 (1923)

    MathSciNet  MATH  Google Scholar 

  53. Schur, I., Szegö, G.: Über die Abschnitte einer im Einheitskreise beschrankten Potenzreihe. Sitz.-Ber. Preuss. Acad. Wiss. Berlin Phys.-Math. Kl. 545–560 (1925)

  54. Sidon, S.: Uber einen satz von Hernn Bohr. Math. Zeit. 26, 731–732 (1927)

    MATH  Google Scholar 

  55. Tomic, M.: Sur un Theoreme de H. Bohr. Math. Scand. 11, 103–106 (1962)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are greatly indebted to the anonymous referees for their elaborate comments and valuable suggestions which improve significantly the presentation of the paper. The second author is supported by UGC-JRF (NTA Ref. No.: 201610135853), New Delhi, India.

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  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, West Bengal, India

    Molla Basir Ahamed & Sabir Ahammed

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  1. Molla Basir Ahamed
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Correspondence to Molla Basir Ahamed.

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Ahamed, M.B., Ahammed, S. Bohr–Rogosinski-Type Inequalities for Certain Classes of Functions: Analytic, Univalent, and Convex. Results Math 78, 171 (2023). https://doi.org/10.1007/s00025-023-01953-z

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