Early Coefficients of the Inverse of a Regular Convex Function

UBing known properties of functions of positive real part it is shown that the first seven coefficients of the inverse of a regular univalent function mapping the disk onto a convex region are bounded by 1. 1. Introduction.If f(z) is in the class 5, then (1.1) f(z) = z + a2z2 + a3z3 + ... in the open unit disk A and f(z) is one-to-one on A. The inverse of f(z) has a Maclaurin expansion in a disk of radius at least |, say (1.2) f(w) = w + l2w2 + i3w3 + It has been conjectured [1] that the Koebe function k(z) = z + J2k'=2 hzk provides sharp bounds for all coefficients of f(z), (1.1) and it has been shown that its inverse does provide the best bound for all \, k = 2,3,..., over all members of S, [8].Recently, Smale [9] made use of these ideas in developing a method for finding a zero of a complex polynomial.Let K be the subclass of S containing all functions f(z) for which /[A] is a convex region.It is known [6] that for f(z) in K, \ < 1 for all k and that (except for a rotation of A) the function T(z) = z/(\ -z) = J2kL=i z>c renders all these bounds sharp.Since T(z) appears to have the same relation to K as k(z) does to 5 it is reasonable to expect T(z) to provide bounds for the inverses of functions in K.However, CO converges in A, whereas there are functions f(z) in K for which (1.2) converges only in disks of radius $ + e, e > 0; and for these, (1.2) cannot have bounded coefficients.These observations have been discussed elsewhere [3 and 4].In an explicit demonstration Kirwan and Schober [3] show that for some f(z) in K, ∈\ > 1 for n > 10.It is our purpose to examine bounds for \^n\ when n is small.THEOREM.For all f(z) in K, |-y"| < 1, n = 2,3,4,5,6,7, and these bounds are sharp.Our method of proof is classical and highly computational; in 2 we outline our methods and reproduce earlier and known results as lemmas and in 3 we outline the computations themselves.