Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator

Abstract

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z < 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

Certain classes of bi-univalent functions associated with the Horadam polynomials

“…In 2010, Srivastava et al [28] revived the study of bi-univalent functions by their pioneering work on the study of coefficient problems. Various subclasses of the bi-univalent function class Σ were introduced and non-sharp estimates on the first two coefficients |a 2 | and |a 3 | in the Taylor-Maclaurin series expansion (1) were found in the very recent investigations (see, for example, [1,2,3,4,5,6,7,8,9,10,12,13,16,17,18,19,20,21,22,23,24,25,26,27,29,30]) and including the references therein. The afore-cited all these papers on the subject were actually motivated by the work of Srivastava et al [28].…”

Fekete-Szegö Inequality for Analytic and Biunivalent Functions Subordinate to Gegenbauer Polynomials

“…Thus, clearly, the image of the domain does not contain the unit disk U. For a brief history and some intriguing examples of functions and characterization of the class Σ, see [7][8][9][10][11][12][13][14][15].…”

“…where the function f −1 ðwÞ is defined by ( 7) and H α is the generating function of the Gegenbauer polynomial given by (10).…”

A Comprehensive Family of Biunivalent Functions Defined byk-Fibonacci Numbers

“…The recent research trends are the outcomes of the study of functions in Σ based on any one of the above-mentioned polynomials, which can be seen in the recent papers [21][22][23][24][25][26][27][28]. Generally, interest was shown to estimate the first two coefficient bounds and the functional of Fekete-Szegö for some subfamilies of Σ.…”