An inequality for certain schlicht functions

Abstract

Let S denote the classical family of schlicht functions f on the unit disk E which have the Taylor expansion f(z) =Z+ j'=2 Ajzi. Recently Ozawa [5] used the Grunsky inequalities and an inequality of Jenkins to show that if A2 >?0 then RA6 ?6 with equality occurring only for the Koebe slit function Z/(1 -Z)2. In this note we shall show that if Aa is real for j?p then (RA_ <n for n_ 2p+1 with equality occurring only for one of the Koebe slit functions Z/(1 + z) 2. This will be established by using a continuity argument to deduce from Jenkins' General Coefficient Theorem [i] that the extremal functions for this coefficient problem have real coefficients. Let Sp = {fES: Aj is real for j_p}, S -= {fES: A, is real for every j}. Set Vp,n= {(RA1, , 6A n): fCSp} Let Hn, (e= +1) denote the metric space of symmetric pairs (Q, g) defined as follows. First Q ==P(w)dw2 is a quadratic differential on the Riemann sphere R of the canonical form