Coefficient Bounds for Certain Subclasses ofm-Fold Symmetric Biunivalent Functions

Abstract

We consider two new subclasses Σ ( , ) and Σ ( , ) of Σ consisting of analytic and -fold symmetric biunivalent functions in the open unit disk . Furthermore, we establish bounds for the coefficients for these subclasses and several related classes are also considered and connections to earlier known results are made.

Coefficient bounds for certain subclasses of $m$-fold symmetric bi-univalent functions

“…In fact, the aforecited work of Srivastava et al [17] essentially revived the investigation of various subclasses of bi-univalent function class ∑ in recent years. Recently, many authors investigated bounds for various subclasses of bi-univalent functions (see, [1], [2], [7], [8], [13], [17], [19]). Not much is known about the bounds on general coefficient |a n | for n ≥ 4.…”

Initial coefficient estimates for a new subclasses of analytic and $m$-fold symmetric bi-univalent functions

“…Let S be the subclass of A consisting of the form (1.1) which are also univalent in U. The Koebe one-quarter theorem (see [4]) states that the image of U under every function f ∈ S contains a disk of radius 1 4 . Therefore, every function f ∈ S has an inverse f −1 which satisfies f −1 ( f (z)) = z, (z ∈ U) and f ( f −1 (w)) = w, (|w| < r 0 ( f ), r 0 ( f ) ≥ 1 4 ), where g(w) = f −1 (w) = w − a 2 w 2 + 2a 2 2 − a 3 w 3 − 5a 3 2 − 5a 2 a 3 + a 4 w…”

“…respectively. Recently, many authors investigated bounds for various subclasses of m-fold bi-univalent functions (see [1,2,5,13,[15][16][17]).…”

“…It should be remarked that the classes E Σ m (δ , γ, λ ; α) and E Σ (δ , γ, λ ; α) are a generalization of wellknown classes consider earlier. These classes are:(1) For δ = λ = 0 and γ = 1, the class E Σ m (δ , γ, λ ; α) reduce to the class S α Σ m which was considered by Altınkaya and Yalçın[1].…”

On certain classes of bi-univalent functions related to m-fold symmetry

“…(iv) For λ = 0, and µ = 0, we obtain class which consists m-fold symmetric bi-univalent function [2].…”