Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator

Ibtisam Aldawish,B. A. Frasin,Tariq Al-Hawary

doi:10.3390/math8050783

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.

Highlights

(λ, β, s, t) of the function class Ω, that generalize the previous defined classes. This subclass is ζ defined with the aid of the new differential operator Dm,λ of analytic functions involving binomial series in the open unit disk U
Two new subclasses of bi-univalent functions related to a new differential operator of analytic functions involving binomial series in the open unit disk U were introduced and investigated
The novelty of our paper consists of the fact that the operator used by defining the new subclasses of Ω is a very general operator that generalizes two important differential operators, ζ

Summary

Introduction

Let A be the class of all analytic functions f in the open unit disk U= {z : |z| < 1}, normalized by the conditions f (0) = 0 and f 0 (0) = 1 of the form The classes S ∗ (α) and K(α) of starlike and convex functions of order α(0 ≤ α < 1), are respectively characterized by In [7], Frasin defined the subclass S(α, s, t) of analytic functions f satisfying the following condition

Results

Conclusion