New Subclasses of Bi-Univalent Functions Associated with Exponential Functions and Fibonacci Numbers (Baghdad Science Journal, 2024)
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.