Univalent functions with large late coefficients

Abstract

A major problem in geometric function theory is discussed in this paper. The problem is to describe the solutions to the extremal problems [f(N)(O)/f'(O)l = maximum among one-to-one analytic functions f(z) on the unit disc. For large N, the problem is solved approximately; but the description is not sufficiently sharp to improve the known estimates of the quantities to be maximized. The same methods are used to discuss other extremal problems. These problems involve derivatives of the univalent functions near the boundary of the unit disc.