General coefficient estimates for bi-univalent functions: a new approach

Abstract

We prove for univalent functions f(z) = z +∑∞k=nakzk; (n ≥ 2) in the unit disk U = {z : |z| < 1}) withf−1(w) = w +∑∞k=nbkwk; (|w| < r0(f), r0(f) ≥14) thatb2n−1 = na2n − a2n−1 and bk = −ak for (n ≤ k ≤ 2n − 2).As applications, we find estimates for |an| whenever f is bi-univalent, bi-close-to-convex, bi-starlike, bi-convex, or forbi-univalent functions having positive real part derivatives in U. Moreover, we estimate |na2n − a2n−1| whenever f isunivalent in U or belongs to certain subclasses of univalent functions. The estimation method can be applied for varioussubclasses of bi-univalent functions in U and it helps to improve well-known estimates and to generalize some knownresults as shown in the last section.