New Subclasses of Bi-Univalent Functions Associated with Exponential Functions and Fibonacci Numbers

Authors

DOI:

https://doi.org/10.21123/bsj.2024.9954

Keywords:

Bi-univalent functions, Coefficient bounds, Exponential function, Fibonacci numbers, Komatu integral operator, Subordination

Abstract

Lewin discussed the class  of bi-univalent functions and obtained the bound for the second coefficient, Sakar and Wanas defined two new subclasses of bi-univalent functions and obtained upper bounds for the elementary coefficients |a2| and |a3| for functions in these subclasses, Dziok et al. introduced the class  of -convex shell-like functions, and they indicated a useful connection between the function   and Fibonacci numbers. Recently, many bi-univalent function classes, based on well-known operators like Sãlãgean operator, Tremblay operator, Komatu integral operator, Convolution operator, Al-Oboudi Differential operator and other, have been defined. The aims of this paper is to introduce  two new subclasses of bi-univalent functions using the subordination and the Komatu integral operator which are involved the exponential functions and shell-like curves with Fibonacci numbers, also find an estimate of the initial coefficients for these subclasses. The first subclass was defined using the subordination of the shell-like curve functions related to Fibonacci numbers and the second subclass was defined using the subordination of the exponential function. The Komatu integral operator was used in each of these subclasses. Limits were obtained for the elementary coefficients, specifically the second and third coefficients for these subclasses.

References

Duren PL. Univalent Functions. Grundlehren der Mathematischen Wissenschaften (GL, volume 259). New York: Springer; 1983. 384 p.

Lewin M. On a Coefficient Problem for Bi-Univalent Functions. Proc Am Math Soc. 1967; 18(1): 63-68.

Sakar FM, Wanas AK. Upper Bounds for Initial Taylor-Maclaurin Coefficients of New Families of Bi-Univalent Functions. IJOPCA. 2023; 15(1): 1-9.

Wanas AK, Páll-Szabó ÁO, Coefficients Bounds for New Subclasses of Analytic and m-Fold Symmetric Bi-Univalent Functions. Stud Univ Babes-Bolyai Math. 2021; 66(4): 659-666. http://dx.doi.org/10.24193/subbmath.2021.4.05

Juma AS, Al-Fayadh A, Shehab NH. Estimates of Coefficient for Certain Subclasses of k–Fold Symmetric Bi-Univalent Functions. Iraqi J Sci. 2022; 63(5): 2155-2163. . https://doi.org/10.24996/ijs.2022.63.5.29

Mahmoud MS, Juma AS, Al-Saphory RA. On Bi-Univalent Functions Involving Srivastava-Attiya Operator. Ital J Pure Appl Math. 2023; (49): 104–112.

Shi L, Srivastava HM, Arif M, Hussain S, Khan H. An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function. Symmetry. 2019: 11(5): 1-14. . https://doi.org/10.3390/sym11050598

Zaprawa P. Hankel Determinants for Univalent Functions Related to the Exponential Function. Symmetry. 2019: 10(11): 1-10. . https://doi.org/10.3390/sym11101211

Dziok J, Raina RK, Sokol J. On α-Convex Functions Related to a Shell-Like Curve Connected with Fibonacci Numbers. Appl Math Comput. 2011; 218(3): 996-1002. https://doi.org/10.1016/j.amc.2011.01.059

Altinkaya Ş, Yalçin S, Çakmak S. A Subclass of Bi-Univalent Functions Based on the Faber Polynomial Expansions and the Fibonacci Numbers. Mathematics. 2019; 7(2): 1-9. https://doi.org/10.3390/math7020160

Singh G, Gurcharanjit S, Gagandeep S. A Subclass of Bi-Univalent Functions Defined by Generalized Sălăgean Operator Related to Shell-Like Curves Connected with Fibonacci Numbers. Int J Math Math Sci. 2019; 2019: 1-7. https://doi.org/10.1155/2019/7628083

Altinkaya Ş, Yalçin S. On the Some Subclasses of Bi-Univalent Functions Related to the Faber Polynomial Expansions and the Fibonacci Numbers. Rend Mat Appl. 2020; 41(7): 105-116.

Rashid AM, Juma AS. Some Subclasses of Univalent and Bi-Univalent Functions Related to K-Fibonacci Numbers and Modified Sigmoid Function. Baghdad Sci J. 2023; 20(3): 843-852.: http://dx.doi.org/10.21123/bsj.2022.6888

Shehab NH, Juma AR. Third Order Differential Subordination for Analytic Functions Involving Convolution Operator. Baghdad Sci J. 2022; 19(3): 0581-0581. http://dx.doi.org/10.21123/bsj.2022.19.3.0581

Challab KA. Study of Second Hankel Determinant for Certain Subclasses of Functions Defined by Al-Oboudi Differential Operator. Baghdad Sci J. 2020; 17(1(Suppl.)): 0353-0353. https://doi.org/10.21123/bsj.2020.17.1(Suppl.).0353

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New Subclasses of Bi-Univalent Functions Associated with Exponential Functions and Fibonacci Numbers. Baghdad Sci.J [Internet]. [cited 2024 Jul. 22];22(1). Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/9954