ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS

Abstract

This chapter discusses some classes of bi-univalent functions. It presents several classes of functions f(z) = z + ∑anzn that are analytic and univalent in the unit disc U = {z : | z | < l}. The class of all such functions is denoted by S. The σ denotes the class of all functions of the form f(z) = z + ∑anzn that are analytic and bi-univalent in the unit disc, that is, f ∈ S and f−1 has a univalent analytic continuation to {| w | < l}. The chapter also introduces the following classes: (1) the class S*σ [α] of strongly bi-starlike functions of order α, 0 < α ≤ 1; (2) the class S*σ(β) of bi-starlike functions of order β, 0 ≤ β < 1; and (3) the class Cσ(β) of bi-convex functions of order β, 0 ≤ β < 1.