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Sharp coefficient bounds for starlike functions associated with the Bell numbers

  • Virendra Kumar , Nak Eun Cho EMAIL logo , V. Ravichandran and H. M. Srivastava
From the journal Mathematica Slovaca

Abstract

Let SB be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class SB several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class SB are investigated.


The second-named author was supported by the Basic Science Research Program through the National Re-search Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).




  1. (Communicated by Stanisława Kanas)

References

[1] Ali, R. M.: Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. 26 (2003), 63–71.Search in Google Scholar

[2] Bell, E. T.: The iterated exponential integers, Ann. of Math. (2) 39 (1938), 539–557.10.2307/1968633Search in Google Scholar

[3] Bell, E. T.: Exponential polynomials, Ann. of Math. Ann. of Math. (2) 35 (1934), 258–277.10.2307/1968431Search in Google Scholar

[4] Cho, N. E.—Kumar, S.—Kumar, V.—Ravichandran, V.: Differential subordination and radius estimates for starlike functions associated with the Booth lemniscate, Turkish J. Math. 42 (2018), 1380–1399.Search in Google Scholar

[5] Cho, N. E.—Kumar, V.—Ravichandran, V.: Sharp bounds on the higher-order Schwarzian derivatives for Janowski classes, Symmetry 10(348) (2018); https://doi.org/10.3390/sym10080348.10.3390/sym10080348Search in Google Scholar

[6] Dorff, M.—Szynal, J.: Higher-order Schwarzian derivatives for convex univalent functions, Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 15 (2009), 7–11.Search in Google Scholar

[7] Dziok, J.—Raina, R. K.—Sokół, J.: Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers, Comput. Math. Appl. 61 (2011), 2605–2613.10.1016/j.camwa.2011.03.006Search in Google Scholar

[8] Dziok, J.—Raina, R. K.—Sokół, J.: On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Math. Comput. Model. 57 (2013), 1203–1211.10.1016/j.mcm.2012.10.023Search in Google Scholar

[9] Harmelin, R.: Aharonov invariants and univalent functions, Israel J. Math. 43 (1982), 244–254.10.1007/BF02761945Search in Google Scholar

[10] Hayman, W. K.: On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc. (3) 18 (1968), 77–94.10.1112/plms/s3-18.1.77Search in Google Scholar

[11] Keogh, F. R.—Merkes, E. P.: A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12.10.1090/S0002-9939-1969-0232926-9Search in Google Scholar

[12] Koepf, W.: Close-to-convex functions and linear-invariant families, Ann. Acad. Sci. Fenn. Ser. A I Math 8 (1983), 349–355.10.5186/aasfm.1983.0815Search in Google Scholar

[13] Leverenz, C. R.: Hermitian forms in function theory, Trans. Amer. Math. Soc. 286 (1984), 675–688.10.1090/S0002-9947-1984-0760980-7Search in Google Scholar

[14] Libera, R. J.—Zlotkiewicz, E. J.: Coefficient bounds for the inverse of a function with derivatives in 𝓟, Proc. Amer. Math. Soc. 87 (1983), 251–257.10.2307/2043698Search in Google Scholar

[15] Ma, W. C.—Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis (Z. Li, F. Ren, L. Yang, S. Zhang, eds.), Tianjin, China, 1992, Conference Proceedings and Lecture Notes in Analysis, Vol. I, International Press, Cambridge, Massachusetts, 1994, pp. 157–169.Search in Google Scholar

[16] Ohno, R.—Sugawa, T.: Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions, Kyoto J. Math. 58(2) (2018), 227–241.10.1215/21562261-2017-0015Search in Google Scholar

[17] Noonan, J. W.—Thomas, D. K.: On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 223 (1976), 337–346.10.2307/1997533Search in Google Scholar

[18] Pommerenke, C.: On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. (1) 41 (1966), 111–122.10.1112/jlms/s1-41.1.111Search in Google Scholar

[19] Ravichandran, V.—Verma, S.: Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris Sér. I 353 (2015), 505–510.10.1016/j.crma.2015.03.003Search in Google Scholar

[20] Schippers, E.: Distortion theorems for higher-order Schwarzian derivatives of univalent functions, Proc. Amer. Math. Soc. 128 (2000), 3241–3249.10.1090/S0002-9939-00-05623-9Search in Google Scholar

[21] Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14(1) (2017), Art. ID 19, 1–10.10.1007/s00009-016-0829-ySearch in Google Scholar

Received: 2018-09-12
Accepted: 2019-01-08
Published Online: 2019-10-05
Published in Print: 2019-10-25

© 2019 Mathematical Institute Slovak Academy of Sciences

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