Abstract
In this paper, we study the class ðcos of normalized analytic functions f satisfying 1 + z f''(z) / f'(z) ⺠cos(z). We obtain the sharp coefficient bounds and Hankel determinants of second and third order for functions for ðcos. 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), 596â618. Furthermore, our results provide examples of invariance of the coefficient bounds among the subclass of convex functions.
-
(Communicated by StanisÅawa 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. 1â7.Search in Google Scholar
[2] Bano, K.âRaza, M.: Starlike functions associated with cosine functions, Bull. Iranian Math. Soc. 47 (2021), 1513â1532.10.1007/s41980-020-00456-9Search in Google Scholar
[3] Janteng, A.âHalim, S. A.âDarus, M.: Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7(2) (2006), Art. No. 50.Search in Google Scholar
[4] Janteng, A.âHalim, S. A.âDarus, M.: Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1(13) (2007), 619â625.Search in Google Scholar
[5] Lecko, A.âSim, Y. J.âÅmiarowska, 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), 2231â2238.10.1007/s11785-018-0819-0Search in Google Scholar
[6] Libera, R. J.âZÅotkiewicz, E. J.: Coefficient bounds for the inverse of a function with derivative in ð«, Proc. Amer. Math. Soc. 87(2) (1983), 251â257.10.1090/S0002-9939-1983-0681830-8Search in Google Scholar
[7] Libera, R. J.âZÅotkiewicz, E. J.: Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225â230.10.1090/S0002-9939-1982-0652447-5Search in Google Scholar
[8] Ma, W. C.âMinda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157â169, Conf. Proc. Lecture Notes Anal., Int. Press, Cambridge, MA, 1994.Search in Google Scholar
[9] Marimuthu, K.âUma, J.âBulboaca, T.: Coefficient estimates for starlike and convex functions associated with cosine function, Hacet. J. Math. Stat. 52 (2023), 596â618.Search in Google Scholar
[10] ObradoviÄ M.âTuneski, N.: Hankel determinants of second and third order for the class ð® of univalent functions, Math. Slovaca 71(3) (2021), 649â654.10.1515/ms-2021-0010Search in Google Scholar
[11] ObradoviÄ, M.âTuneski, N.: Two types of the second Hankel determinant for the class ð° and the general class ð®, Acta Comment. Univ. Tartu. Math. 27 (2023), 59â67.10.12697/ACUTM.2023.27.05Search in Google Scholar
[12] Pommerenke, C.: On the coefficients and hankel determinants of univalent functions, J. Lond. Math. Soc. 1 (1966), 111â122.10.1112/jlms/s1-41.1.111Search in Google Scholar
[13] Pommerenke, C.: On the hankel determinants of univalent functions, Mathematika 14(1) (1967), 108â112.10.1112/S002557930000807XSearch in Google Scholar
[14] Raza, M.âThomas, D. K.âRiaz, A.: Coefficient estimates for starlike and convex functions related to sigmoid functions, Ukrainian Math. J. 75 (2023), 782â799.10.1007/s11253-023-02228-0Search in Google Scholar
[15] Riaz, A.âRaza, 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.2023.103289Search in Google Scholar
[16] Sim, Y. J.âThomas, D. K.âZaprawa, P.: The second Hankel determinant for starlike and convex functions of order alpha, Complex Var. Elliptic Equ. 67(10) (2022), 2423â2443.10.1080/17476933.2021.1931149Search in Google Scholar
[17] Thomas, D. K.âVerma, S.: Invariance of the coefficients of strongly convex functions, Bull. Aust. Math. Soc. 95(3), (2017), 436â445.10.1017/S0004972716000976Search in Google Scholar
[18] Zaprawa, P.: On Hankel determinant H2(3) for univalent functions, Results Math. 73 (2018), Art. No. 89.10.1007/s00025-018-0854-1Search 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-ySearch in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Joins of normal matrices, their spectrum, and applications
- On isbellâs density theorem for bitopological pointfree spaces II
- Every positive integer is a sum of at most n + 2 centered n-gonal numbers
- A generalisation of q-additive functions
- Generalised class groups in dihedral and fake â€p-extensions
- Singular value bounds with applications to norm and numerical radius inequalities
- Coefficient bounds for convex functions associated with cosine function
- On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
- Numerical implementation of solving a boundary value problem including both delay and parameter
- Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
- Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
- On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
- A note on derivations into annihilators of the ideals of banach algebras
- Characterization of nonlinear mixed skew lie and jordan n-Type derivation on â-Algebras
- Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
- A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
- Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function
Articles in the same Issue
- Joins of normal matrices, their spectrum, and applications
- On isbellâs density theorem for bitopological pointfree spaces II
- Every positive integer is a sum of at most n + 2 centered n-gonal numbers
- A generalisation of q-additive functions
- Generalised class groups in dihedral and fake â€p-extensions
- Singular value bounds with applications to norm and numerical radius inequalities
- Coefficient bounds for convex functions associated with cosine function
- On solutions to q-difference equations for q-appell functions in the spirit of Olsson and Exton
- Numerical implementation of solving a boundary value problem including both delay and parameter
- Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions
- Global attractivity of nonlinear delay dynamic equations on time scales via Lyapunov functional method
- On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems
- A note on derivations into annihilators of the ideals of banach algebras
- Characterization of nonlinear mixed skew lie and jordan n-Type derivation on â-Algebras
- Linear and uniformly continuous surjections between Cp-spaces over metrizable spaces
- A goodness-of-fit test for testing exponentiality based on normalized dynamic survival extropy
- Monotonicity results of ratio between two normalized remainders of Maclaurin series expansion for square of tangent function