Univalent functions which map onto regions of given transfinite diameter

By a variational method, the sharp upper bound is obtained for the second coefficients of normalized univalent functions which map the unit disk onto regions of prescribed transfinite diameter, or logarithmic capacity. 0. INTRODUCTION Let S be the usual class of functions f(z) = z + a2z2 + analytic and univalent in the unit disk 1D . Many years ago, Pick [7] used Bieberbach's inequality Ia2I 1. The extremal functions map the unit disk onto the disk Iw I < M minus a radial slit of suitable length. Our purpose is to solve a related problem: to find the maximum of Ja2j among all functions f E S whose range has a given transfinite diameter R, 1 < R < 00. This problem is considerably harder than Pick's, but we shall see that for "large" R the solution is somewhat similar. The sharp inequality is 1a2l<2-e 2/4R and the extremal function maps the disk onto a certain Jordan region minus a radial spire. For small R the situation is rather different. The sharp bound is then determined implicitly through elliptic integrals, and the extremal function maps the disk onto a certain Jordan region with analytic boundary. The division occurs at the transfinite diameter R = e27r2/64 = 1.139.... This unexpected phenomenon, the dual nature of the solution, gives the problem a particular interest. A more detailed summary of our results appears in the theorem stated at the end of the paper. The proof uses a special variational method devised to Received by the editors October 18, 1988 and, in revised form, January 20, 1989. Presented to the American Mathematical Society at Louisville, Kentucky, on January 18, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 30C70, 30C50, 30C85, 33A25. KeY words and phrases. Univalent functions, second coefficient, extremal problems, variational methods, quadratic differentials, transfinite diameter, logarithmic capacity, elliptic integrals. The research of the first author was supported in part by the National Science Foundation under Grant DMS-870175 1. (D 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page