Refined Bohr inequalities for certain classes of functions: analytic, univalent, and convex. (English) Zbl 1536.30083
Summary: In this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk \(\mathbb{D}\), class of analytic functions \(f\) defined on \(\mathbb{D}\) such that \(\mathrm{Re}\left(f(z)\right) <1\), and class of subordination to a function \(g\) in \(\mathbb{D}\). Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.
MSC:
| 30H05 | Spaces of bounded analytic functions of one complex variable |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 30A10 | Inequalities in the complex plane |
References:
| [1] | Abu-Muhanna, Y., Bohr’s phenomenon in subordination and bounded harmonic classes. Complex Var. Elliptic Equ.55(2010), 1071-1078. · Zbl 1215.46018 |
| [2] | Abu-Muhanna, Y., Ali, R. M., Ng, Z. C., and Hasni, S. F. M, Bohr radius for subordinating families of analytic functions and bounded harmonic mappings. J. Math. Anal. Appl.420(2014), 124-136. · Zbl 1293.30004 |
| [3] | Ahamed, M. B. and Ahammed, S., Bohr type inequalities for the class of self-analytic maps on the unit disk. Comput. Methods Funct. Theory (2023). https://doi.org/10.1007/s40315-023-00482-8 · Zbl 1526.30066 |
| [4] | Ahamed, M. B. and Allu, V., Bohr phenomenon for certain classes of harmonic mappings. Rocky Mountain J. Math.52(2022), no. 4, 1205-1225. · Zbl 1502.31001 |
| [5] | Ahamed, M. B., Allu, V., and Halder, H., Bohr radius for certain classes of close-to-convex harmonic mappings. Anal. Math. Phys.11(2021), 111. · Zbl 1475.31002 |
| [6] | Ahamed, M. B., Allu, V., and Halder, H., Improved Bohr inequalities for certain class of harmonic univalent functions. Complex Var. Elliptic Equ.68(2021), 267-290. · Zbl 1509.31001 |
| [7] | Ahamed, M. B., Allu, V., and Halder, H., The Bohr phenomenon for analytic functions on shifted disks. Ann. Acad. Sci. Fenn. Math.47(2022), 103-120. · Zbl 1482.30024 |
| [8] | Aizenberg, L. and Tarkhanov, N., A Bohr phenomenon for elliptic equations. Proc. London Math. Soc.82(2001), no. 2, 385-401. · Zbl 1019.32001 |
| [9] | Ali, R. M., Abdulhadi, Z., and Ng, Z. C., The Bohr radius for starlike logharmonic mappings. Complex Var. Elliptic Equ.61(2016), 1-14. · Zbl 1332.30022 |
| [10] | Ali, R. M. and Ng, Z. C., The Bohr inequality in the hyperbolic plane. Complex Var. Elliptic Equ. 63(2018), 1539-1557. · Zbl 1395.30010 |
| [11] | Alkhaleefah, S. A., Kayumov, I. R., and Ponnusamy, S., On the Bohr inequality with a fixed zero coefficient. Proc. Amer. Math. Soc.147(2019), 5263-5274. · Zbl 1428.30001 |
| [12] | Allu, V. and Halder, H., Bohr radius for certain classes of starlike and convex univalent functions. J. Math. Anal. Appl.493(2021), 124519. · Zbl 1451.30023 |
| [13] | Allu, V. and Halder, H., Bohr phenomenon for certain subclasses of harmonic mappings. Bull. Sci. Math.173(2021), 103053. · Zbl 1477.30004 |
| [14] | Allu, V. and Halder, H., Operator valued analogues of multidimensional Bohr inequality. Canad. Math. Bull. 65(2022), 1020-1035. · Zbl 1517.32046 |
| [15] | Bénéteau, C., Dahlner, A., and Khavinson, D., Remarks on the Bohr phenomenon. Comput. Methods Funct. Theory4(2004), 1-19. · Zbl 1067.30094 |
| [16] | Bhowmik, B. and Das, N., Bohr phenomenon for subordinating families of certain univalent functions. J. Math. Anal. Appl.462(2018), 1087-1098. · Zbl 1391.30003 |
| [17] | Boas, H. P. and Khavinson, D., Bohr’s power series theorem in several variables. Proc. Amer. Math. Soc.125(1997), 2975-2979. · Zbl 0888.32001 |
| [18] | Bohr, H., A theorem concerning power series. Proc. Lond. Math. Soc. s2-13(1914), 1-5. · JFM 44.0289.01 |
| [19] | Branges, L. D., A proof of the Bieberbach conjecture. Acta Math.154(1985), 137-152. · Zbl 0573.30014 |
| [20] | Das, N., Refinements of the Bohr and Rogosisnki phenomena. J. Math. Anal. Appl.508(2022), no. 1, 125847. · Zbl 1484.30003 |
| [21] | Dixon, P. G., Banach algebras satisfying the non-unital von Neumann inequality. Bull. Lond. Math. Soc.27(1995), no. 4, 359-362. · Zbl 0835.47033 |
| [22] | Duren, P. L., Univalent function, Springer, New York, 1983. · Zbl 0514.30001 |
| [23] | Evdoridis, S., Ponnusamy, S., and Rasila, A., Improved Bohr’s inequality for shifted disks. Results Math.76(2021), 14. · Zbl 1470.30001 |
| [24] | Hamada, H., Bohr phenomenon for analytic functions subordinate to starlike or convex functions. J. Math. Anal. Appl.499(2021), 125019. · Zbl 1462.30030 |
| [25] | Huang, Y., Liu, M.-S., and Ponnusamy, S., The Bohr-type operator on analytic functions and sections. Complex Var. Elliptic Equ.68(2023), 317-332. · Zbl 1509.30002 |
| [26] | Ismagilov, A., Kayumov, A. V., Kayumov, I. R., and Ponnusamy, S., Bohr inequalities in some classes of analytic functions. J. Math. Sci.252(2021), no. 3, 360-373. · Zbl 1468.30008 |
| [27] | Ismagilov, A., Kayumov, I. R., and Ponnusamy, S., Sharp Bohr type inequality. J. Math. Anal. Appl.489(2020), 124147. · Zbl 1441.30077 |
| [28] | Kayumov, I. R., Khammatova, D. M., and Ponnusamy, S., Bohr-Rogosinski phenomenon for analytic functions and Cesáro operators. J. Math. Anal. Appl.496(2021), 124824. · Zbl 1461.30008 |
| [29] | Kayumov, I. R. and Ponnusamy, S., Improved version of Bohr’s inequality. C. R. Acad. Sci. Paris, Ser. I356(2018), 272-277. · Zbl 1432.30002 |
| [30] | Kumar, S., A generalization of the Bohr inequality and its applications. Complex Var. Elliptic Equ.68(2023), 963-973. · Zbl 1520.30003 |
| [31] | Lata, S. and Singh, D., Bohr’s inequality for non-commutative Hardy spaces. Proc. Amer. Math. Soc.150(2022), no. 1, 201-211. · Zbl 1487.46070 |
| [32] | Liu, G., Liu, Z., and Ponnusamy, S., Refined Bohr inequality for bounded analytic functions. Bull. Sci. Math.173(2021), 103054. · Zbl 1477.30047 |
| [33] | Liu, M.-S. and Ponnusamy, S., Multidimensional analogues of refined Bohr’s inequality. Porc. Amer. Math. Soc.149(2021), no. 5, 2133-2146. · Zbl 1460.32002 |
| [34] | Liu, M.-S., Ponnusamy, S., and Wang, J., Bohr’s phenomenon for the classes of quasi-subordination and K-quasiregular harmonic mappings. RACSAM114(2020), 115. · Zbl 1439.30001 |
| [35] | Liu, M.-S., Shang, Y-M., and Xu, J-F., Bohr-type inequalities of analytic functions. J. Inequal. Appl.345(2018), 2018. · Zbl 1498.30001 |
| [36] | Paulsen, V. I., Popescu, G., and Singh, D., On Bohr’s inequality. Proc. Lond. Math. Soc.85(2002), no. 2, 493-512. · Zbl 1033.47008 |
| [37] | Ponnusamy, S., Vijayakumar, R., and Wirths, K.-J., Improved Bohr’s phenomenon in quasi-subordination classes. J. Math. Anal. Appl.506(2022), no. 1, 125645. · Zbl 1475.30005 |
| [38] | Rogosinski, W., Über Bildschranken bei Potenzreihen und ihren Abschnitten. Math. Z.17(1923), 260-276. · JFM 49.0231.03 |
| [39] | Rogosinski, W., On the coefficients of subordination functions. Proc. Lond. Math. Soc.48(1943), no. 2, 48-82. · Zbl 0028.35502 |
| [40] | Ruscheweyh, S., Two remarks on bounded analytic functions. Serdica11(1985), 200-202. · Zbl 0581.30009 |
| [41] | Sidon, S., Uber einen satz von Hernn Bohr. Math. Zeit.26(1927), 731-732. · JFM 53.0281.04 |
| [42] | Tomic, M., Sur un Theoreme de H. Bohr. Math. Scand.11(1962), 103-106. · Zbl 0109.30202 |
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