A NOTE ON SOLUTIONS OF SCHIFFER ’ S DIFFERENTIAL EQUATION

ge consider Schlffers dlfferentlal equation for functions in the cZass of norsallzed unlvalent functions which axJaize the n---coeffclent. By conslderln a th class of functiouals converging to the n---coefficient functional, we determine some addtttona/ setrtes that extreaal functions possess. L:Y AH PfiF.. Univalent function, Schtffer's differential equation, varia-


I i
Here n ()z) is the familiar Faber polynomial of degree n for f() with p and q defined by (1.2) and (1.3) respectively is the Schiffer differential equation which any function maximizing Re a must satisfy.(Unfortunately, there are other solutions that are not extremal functions such as z(l-z2) -I in the case n 3.) In this note, we consider a class of functionals T (f) that converge to (f) r Re a as r + 0. We compute the Schiffer differential equation for each of these n functionals and obtain new conditions that the extremal functions must satisfy.In certain cases, we show that the extremal function must satisfy an infinite system of differential equations.The equations in this system are of the form (i.i) and have the unknown coefficients of the extremal function appear in the equation.
We will need the following result whose proof is an immediate consequence of the formula for the sum of a geometric progression.
n-i (2iJn f(z.), f must satisfy ii) as r 0, the functions f (which may depend on r) approach a function in S n-i Re E f(zj) (2.1) J:0 which maximizes Re a

PROOF
We follow the outline in Pommerenke [8, p. [183][184][185][186][187][188][189][190] T is a linear function- r al of degree n and consequently f()/ P(f()) q()   If g > 0 is given, we may choose r so that 0(rn) < g.Then any function fo maximi- zing Re a has n Re T t(f) < Re an + g fo f as r+ 0 and hence is the limit of functions maximizing Re T r REMARKS. 1.It is well-known that a function that maximizes Re a actually has n a > 0. n z n

If f(z)
z + a2z2 +...+ a +... maximizes Re a then so do the Our technique of approx+/-ating Re a by Re Tr(f) will yield only one of the rotations n of f; the others can be obtained by considering replacing r by 2ij n-i r e r.This observation will explain some later results.
Upon dividing (2.5) by r n and letting r / 0, we obtain where Bj f(zj) fkre n ;.For fixed z, the expression F(z) defined by (3.2) is an analytic function of r for r in some small interval about 0. We expand (3.2) kn in powers of r noting that the lemma insures that only powers of r can appear.We show the argument only for powers of r n since the computation for higher powers is similar. (z f'(z )z +--3  +, a2z + .)3j=o (zj The sum lamlam2am3 is taken over all positive integers ml, m2, m 3 with m I + m 2 + m 3 n.This procedure yields in general . This proves the result if k i.The other equations for k 2,... are obtained in a similar manner by equating coefficients of higher powers of r REMARKS.i.A result of Pfluger [7] shows that a Koebe function k(z) z(l ei0z)-2 always satisfies (i.i).

2.
The assumption that f is essentially the only extremal function for the problem of maximizing Re a is used quite strongly in this proof.If there were n more than one function, the coefficients of the extremal function for T would de- r pend upon r, making the functional-differential equations even more complicated.It seems reasonable to suppose that there is essentially one extremal function (apart from rotations) for each n but we are unable to prove this.The following result is of interest only if the Bieberbach.conjecture is false.
THEOREM 4. Suppose there is a function f not of the form z(l el@z) -2 and that f 180 satisfies the hypothesis of Theorem 2. Then there is a number @0 such that e is simultaneously a zero of qk(z) (kn l)ank + kn-I e-(kn-j)i$ Z (jaj j=0 + jje (kn-j)iS) k 1,2 PROOF.Pfluger [7] has shown that if f is a function that maximizes Re a then n i i Re[ n((z)) a n < 0 unless f is a rotation of the Koebe function. (He actually proved this theorem for any linear functional and the rational function p related to it by (1.2).)It is well-known that I i Cn(f(z)) an 0 if and only if f is a Koebe function.(See [8, p. 194], [6, Theorem 13.6].) We consider equation (3.5) with i.It is well-known that since f maximizes Re a the function q defined by (i 3) must have at least one zero on z i Since n the left-handed side of (3.5) is analytic by assumption, each zero e of n-I -(n-J)i8 ql(e i8) (n-l)a + Z (jaje-(n-j)i8 + jaje n j=l must also be a zero of qk(e i8), k 1,2, This completes the proof.
ii) 2a2akn (kn l)akn+l (kn l)akn_l PROOF.Since f is essentially the unique function maximizing Re an, the equations (2.2) and (2.3) are valid for all r zjl in some neighborhood of 0. Equating co- efficients of rkn in (2.2) yields, after an application of the lemma, or -nakn in Im kn akn-n Re akn -i Im akn -i Im k nakn Let f be a function in S which maximizes T r(f) Re n Z f(zj) where nr the notation B.
+ l)an+ I (n l)an_ I REMARK.The conclusion of the corollary is the well-known Marty relation.It was originally derived by very elementary methods.Hummel [6, p. 77] observed that the Marty relation can also be obtained by considering the Schiffer differential equation for the functional Re a n 3. CONSEQUENCE OF THE MAIN THEOREM.THEOREM 2. Suppose that a fixed function f maximizes Re Tr(f) for some sequence of r's converging to 0. Then f satisfies the system of functional-dlfferentlal equations 'f(z)W) is the knt__h Faber polynomial for W. PROOF.By Theorem i, f must satisfy the functlonal- -j) + j jz )]r n + O(r2n).

3 .
The equation for k i is of course the familiar Schiffer differential equation for a function f maximizing Re aThe nature of this family of equations n suggests that, if f(z) z + a2 z2 + is a function which maximizes Re f a n, also maximizes Re a2n Re a3nIf so, the Bieberbach conjecture would follow from a result of Hayman [ii, p. 104].He showed that, if f e S, lira <_ i, with I only if f(z) z(l eiz)-2" k THEOREM 3. Suppose that f satisfies the hypothesis of Theorem 2. Then f satisfies the functional equation kn-i (kn-l)akn + 7. (ja.z -(kn-j) the kth equation in the system (3.1) by the th equation.