On concave univalent functions

Conformal maps of the unit disk onto convex domains are a classical topic. Recently Avkhadiev and Wirths discovered that conformal maps onto concave domains (the complements of convex closed sets) have some novel properties, namely that there are non-trivial lower estimates, a rare thing for univalent functions (Avkhadiev, F.G. and Wirths, K.-J., 2002, Convex holes produce lower bounds for coefficients, Complex Variables, Theory and Application, 47, 556–563). We prove an inequality involving two variables and introduce a lower order for concave functions. §Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.