Generalized Differential Operator on Bistarlike and Biconvex Functions Associated By Quasi-Subordination

Abstract

Abstract. In this paper, the generalized differential operator is applied to derive some subclasses of function class  of bi-univalent functions defined in unit disk . We estimate the bounds of the coefficients 2 and 3 for all functions which belong to the derived subclasses of .

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

“…They have obtained non-sharp estimates on |d 2 | and |d 3 | for members of such families. For various subfamilies of the class , coefficient estimates and other properties of functions in these subfamilies, one can refer the works of [2], [5], [12], [17], and [22].…”

“…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].…”