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Coefficient bounds for convex functions associated with cosine function

  • Umar Raza , Mohsan Raza EMAIL logo und PaweƂ Zaprawa
Veröffentlicht/Copyright: 9. Juni 2025
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Abstract

In this paper, we study the class 𝒞cos of normalized analytic functions f satisfying 1 + z f''(z) / f'(z) â‰ș cos(z). We obtain the sharp coefficient bounds and Hankel determinants of second and third order for functions for 𝒞cos. We also present the similar results for inverse and logarithm coefficients. These results improve the results recently obtained in [Marimuthu et al.: Coefficient estimates for starlike and convex functions associated with cosine function, Hacet. J. Math. Stat. 52 (2023), 596–618. Furthermore, our results provide examples of invariance of the coefficient bounds among the subclass of convex functions.

2020 Mathematics Subject Classification: Primary 30C45; 30C50

The work here is supported by HEC grant: 20-16367/NRPU/R&D/HEC/2021-2020.


  1. (Communicated by StanisƂawa Kanas)

References

[1] Babalola, K. O.: On H3(1) Hankel determinant for some classes of univalent functions. In: Inequality Theory and Applications (Y. J. Cho, ed.), Vol. 6, Nova Science Publishers, New York, 2010, pp. 1–7.Suche in Google Scholar

[2] Bano, K.—Raza, M.: Starlike functions associated with cosine functions, Bull. Iranian Math. Soc. 47 (2021), 1513–1532.10.1007/s41980-020-00456-9Suche in Google Scholar

[3] Janteng, A.—Halim, S. A.—Darus, M.: Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7(2) (2006), Art. No. 50.Suche in Google Scholar

[4] Janteng, A.—Halim, S. A.—Darus, M.: Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1(13) (2007), 619–625.Suche in Google Scholar

[5] Lecko, A.—Sim, Y. J.—ƚmiarowska, B.: The sharp bound of the hankel determinant of the third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory 13(5) (2019), 2231–2238.10.1007/s11785-018-0819-0Suche in Google Scholar

[6] Libera, R. J.—ZƂotkiewicz, E. J.: Coefficient bounds for the inverse of a function with derivative in đ’«, Proc. Amer. Math. Soc. 87(2) (1983), 251–257.10.1090/S0002-9939-1983-0681830-8Suche in Google Scholar

[7] Libera, R. J.—ZƂotkiewicz, E. J.: Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225–230.10.1090/S0002-9939-1982-0652447-5Suche in Google Scholar

[8] Ma, W. C.—Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157–169, Conf. Proc. Lecture Notes Anal., Int. Press, Cambridge, MA, 1994.Suche in Google Scholar

[9] Marimuthu, K.—Uma, J.—Bulboaca, T.: Coefficient estimates for starlike and convex functions associated with cosine function, Hacet. J. Math. Stat. 52 (2023), 596–618.Suche in Google Scholar

[10] Obradović M.—Tuneski, N.: Hankel determinants of second and third order for the class 𝒼 of univalent functions, Math. Slovaca 71(3) (2021), 649–654.10.1515/ms-2021-0010Suche in Google Scholar

[11] Obradović, M.—Tuneski, N.: Two types of the second Hankel determinant for the class 𝒰 and the general class 𝒼, Acta Comment. Univ. Tartu. Math. 27 (2023), 59–67.10.12697/ACUTM.2023.27.05Suche in Google Scholar

[12] Pommerenke, C.: On the coefficients and hankel determinants of univalent functions, J. Lond. Math. Soc. 1 (1966), 111–122.10.1112/jlms/s1-41.1.111Suche in Google Scholar

[13] Pommerenke, C.: On the hankel determinants of univalent functions, Mathematika 14(1) (1967), 108–112.10.1112/S002557930000807XSuche in Google Scholar

[14] Raza, M.—Thomas, D. K.—Riaz, A.: Coefficient estimates for starlike and convex functions related to sigmoid functions, Ukrainian Math. J. 75 (2023), 782–799.10.1007/s11253-023-02228-0Suche in Google Scholar

[15] Riaz, A.—Raza, M.: The third Hankel determinant for starlike and convex functions associated with lune, Bull. Sci. Math. 187 (2023), Art. ID 103289.10.1016/j.bulsci.2023.103289Suche in Google Scholar

[16] Sim, Y. J.—Thomas, D. K.—Zaprawa, P.: The second Hankel determinant for starlike and convex functions of order alpha, Complex Var. Elliptic Equ. 67(10) (2022), 2423–2443.10.1080/17476933.2021.1931149Suche in Google Scholar

[17] Thomas, D. K.—Verma, S.: Invariance of the coefficients of strongly convex functions, Bull. Aust. Math. Soc. 95(3), (2017), 436–445.10.1017/S0004972716000976Suche in Google Scholar

[18] Zaprawa, P.: On Hankel determinant H2(3) for univalent functions, Results Math. 73 (2018), Art. No. 89.10.1007/s00025-018-0854-1Suche in Google Scholar

[19] Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (2017), Art. No. 19.10.1007/s00009-016-0829-ySuche in Google Scholar

Received: 2024-07-20
Accepted: 2024-12-04
Published Online: 2025-06-09
Published in Print: 2025-06-26

© 2025 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2025-0041/html
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