The Coefficients of Schlicht and Allied Functions

regular and schlicht (univalent) in | z | < 1, i.e. assuming distinct values w for distinct values of z. The functions f(z) e S map | z | < 1 1 : 1 conformally onto a simply connected domain D in the w plane. During the last international Congress at Harvard Professors Schaeffer and Spencer [12] reported on the progress, largely due to their own work, on the so-called coefficient problem for S, i.e. the problem of characterising the region Vn in complex space of n — 1 dimensions occupied by the points (a2, a3, . . ., an) for f(z) e 5. To-day I should like to discuss the behaviour of the an for large n and the related problem of the behaviour of f(z) near | jar | === X Many of the results extend to more general classes of functions. It was first shown by Koebe [9], that the distance d of w = 0 from the complement of D is greater than an absolute constant, and that for fixed z | f'(z) | and \ f(z) | are bounded below and above by positive constants for all f(z) € S. The exact inequalities | a2 | ^ 2, d ^ J and the bounds for | f'(z) \, | f(z) | and | f'(z)jf(z) | were obtained in (1916) by Bieberbach [2], Faber, Pick, Gronwall and others. In each case the bounds are attained only for the functions