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

We prove for univalent functions f(z) = z + Sigma(infinity)(k=n) a(k)z(k); (n >= 2) in the unit disk U = {z : vertical bar z vertical bar < 1}) with f(-1)(w) = w + Sigma(infinity)(k=n) b(k)w(k); (vertical bar w vertical bar < r(0)(f), r(0)(f) >= 1/4) that b(2n-1) = na(n)(2) - a(2n-1) and b(k) = -a(k) for (n <= k <= 2n -2). As applications, we find estimates for vertical bar a(n)vertical bar whenever f is bi-univalent, bi-close-to-convex, bi-starlike, bi-convex, or for bi-univalent functions having positive real part derivatives in U. Moreover, we estimate vertical bar na(n)(2) - a(2n-1)vertical bar whenever f is univalent in U or belongs to certain subclasses of univalent functions. The estimation method can be applied for various subclasses of bi-univalent functions in U and it helps to improve well-known estimates and to generalize some known results as shown in the last section.