Certain Subclasses of Bistarlike and Biconvex Functions Based on Quasi-Subordination

Abstract

We introduce the unified biunivalent function class M , , ( , ) defined based on quasi-subordination and obtained the coefficient estimates for Taylor-Maclaurin coefficients | 2 | and | 3 |. Several related classes of functions are also considered and connections to earlier known and new results are established.

Special Subfamilies of Holomorphic and Bi-univalent Functions related to Quasi-subordination

“…Robertson [15] innovated a concept called quasi-subordination, which generalizes both the concepts of majorization and subordination. For holomorphic functions s(z) and τ (z), s(z) is quasi-subordinate to τ (z), indicated as s(z) ≺ q τ (z), z ∈ D, if there exists two holomorphic functions ς and ψ with |ς(z)| ≤ 1, ψ(0) = 0 and |ψ(z)| < 1 such that s(z) = ς(z)τ (ψ(z)), z ∈ D. Observe that if ς(z) = 1, then s(z) = τ (ψ(z)), z ∈ D, so that s(z) ≺ τ (z) in D. Also note that if ψ(z) = z, then s(z) = ς(z)τ (z), z ∈ D and hence s(z) ≺≺ τ (z) in D. There are more studies related to quasi-subordination such as [1], [7], [8], [11], [14], [16], [19] and [21].…”

Coefficients assessment for certain subclasses of bi-univalent functions related with quasi-subordination

“…and it is said that f (z) is majorized by h(z) and written as f (z) ≪ h(z) in D. Hence it is perceptible that the quasi-subordination is a popularization of the usual subordination as well as majorization. The labor on quasi-subordination is very extensive and that includes some recent investigations [2,4,9,10,12,14,19,20].…”

A Generalized Subclass of Alpha Convex Bi-Univalent Functions of Complex Order

“…It is obvious to see that the quasi-subordination is a generalization of the usual subordination. The work on quasisubordination is quite extensive which finds interesting dimensions in some recent investigations [1,5,7,12].…”