VOt.. 6 NO. 3
[19&3|
597
597-604
A NOTE ON SOLUTIONS OF
SCHIFFER’S DIFFERENTIAL EQUATION
RONALD J. LEACH
Department of Hathematics
Howard University
/ashington, D.C. 20059
(Received on February A, 1982 and in revised fora August 20, 1982)
ge consider Schlffers dlfferentlal equation for functions in the cZass of
ABSTRACT.
norsallzed unlvalent functions which axJaize the n---coeffclent.
By conslderln a
th
class of functiouals converging to the n---coefficient functional, we determine some
setrtes that extreaal functions possess.
AH PfiF.. Univalent function, Schtffer’s
addtttona/
L:Y
differential equation, varia-
tion, Harry relation.
1.
IN’L3I)DCTIOli
Let
$
the t dtc D.
S
,e
a2 z2 +...,
t
11- to be cact in the topology of unifo cver-
ve
lutt
S
ever
ts a conttnus fcttol on S.
By ct-
an t satisfy the differentl eqtt
(f)
p
q are ratil fcti d q(e
the fcti f
i6)
(1.1)
D
-q()
f()/ p(f(C))
pd
(f)
trl fction f for
lvt varti, Schiller [1, 2] shd tt
ere
sZytZc 8d univalent
cact subsets of D. erefore, variatiol probl of the fo
gce
CS
z +
denote the class of functions f(z)
0.
e
coefficients of p and q de-
therefore (.) is a fctiolifferential eqtion.
For les of the es of t vartiol td, see [-10].
foll the notation in Pr [8, p. 83-190].
soluti to the problm of zing Re
an;
p(f())
It is sho there that f is a
then
1
For consistency, we
n()
,
(.)
598
R.J. LEACH
n-i
q()-- (nI
n ()z)
Here
l)an +
J=IE
(n-j)
(Jaj
+
(1.3)
).
.aaj
i
is the familiar Faber polynomial of degree n for
Equation (i.i)
f()
with p and q defined by (1.2) and (1.3) respectively is the Schiffer differential
must satisfy.
equation which any function maximizing Re a
other solutions that are not extremal functions such as z(l-
(Unfortunately, there are
z2) -I
3.)
in the case n
In this note, we consider a class of functionals T r (f) that converge to (f)
Re a
We compute the Schiffer differential equation for each of these
as r + 0.
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.
THE MAIN THEOREM.
2.
We will need the following result whose proof is an immediate consequence of the
formula for the sum of a geometric progression.
Z
Let g(z)
Lemma.
n:0
THEOREM i.
(z
I)
ii)
2ij
n
r > 0.
(2iJn
n-i
Y. g\re
i=0
Then
n Y,
i--0
z
B2.3
2
j 0
as r
f(z)-B
j
1
2
bknrkn
r(f)
n-i
Re
nr
n
Z
J :0
f(zj)
where
Then
with the notation
i)
f (z
n
Let f be a function in S which maximizes T
re
where z
b z
n
B.
3
f(z.), f
must satisfy
3
O- z f’(zj) z’z.
Y.
j
j:0
i
7. z
j=O
]
]
j
f’(zj) l-’.z
3
n-i
Re E
J:0
f(zj)
(2.1)
0, the functions f (which may depend on r) approach a function in S
which maximizes Re a
PROOF
We follow the outline in Pommerenke [8, p. 183-190]
al of degree n and consequently
q()
f()/ P(f())
where
2
n-i
p(w)
r
J:O
and
w
B B.
]
T
r
is a linear function-
SOLUTIONS OF SCHIFFER’S DIFFERENTIAL EQUATION
+ zj
+
nl
i
q()
z f’
J
j=o
(zj
n-i
i
Z
z.
j=o
j
+
i
zjf’(zj)
n-I
zj
zj
i
Re Z
roves the first statement of the theorem if we replace
s
d statement, note that the lemma implies
i(o2ir
Z
f
n
nanr
T (f)
r
a
j=0
n
+
na2nr
(zj ).
f
j--o
Th
n-1
599
To prove the
by z.
2n
+...
and hence
n
+ 0(rn).
If g > 0 is given, we may choose r so that 0(r n) < g.
zing Re a
n
Then any function
fo maximi-
has
Re T
an +
< Re
t(f)
g
is the limit of functions maximizing Re T f as r+ 0
and hence
fo
REMARKS.
1.
It is well-known that a function that maximizes Re an actually has
2.
If f(z)
a
n
r
> 0.
+
z
functions
a2z2 +...+
2ij
n-i
n-i
Our technique of approx+/-ating Re a
of
n
+...
t\ef 1
_2kj
e
a z
n
n
z
z
e2ij
2
n-i
+
Tr(f)
by Re
maximizes Re a
a^z
Z
then so do the
n
+...+
n
a z
n
+
will yield only one of the rotations
f; the others can be obtained by considering replacing r by
e
r
2ij
n-i
r.
COROLLARY i.
This observation will explain some later results.
There is a function f e S which maximizes Re a for which
n
(n +
2a2an
l)an_ I
We fix r > 0 and expand
n-i
B
j
-
n-i
+ (-n- 2a 2 l B )z + 0(z 2)
j
j=o
inl
+
+
j=0
j =o
J
f’(zj)
f’(z
J
j=o
n-I
z.---z
Re Y.
j=0
I(i +
n I
i
l+z.z
J
z
+
zz/z.]
+
0 obtaining, since f(z)
both representations for F(z) about z
j--o
(n-
Let F(z) denote the expression appearing in (2.1).
PROOF.
E
l)an+I
2.z +
0(z
Bj
2)
i
zjf’ (zj
-i_
2
+
1-
nl[zjf’ (zj
j=0
Re l
j =o
f(zj)
2
a2z +...
)(I + 2__z +
z.
n-i
-iIm I
j=0
0(z2))
zjf’(z.)
3
]
R.J. LEACH
600
Re 7.
j--0
+
f(zj)
Equating coefficients, since
n-i
7.
f’(z
n-i
Y. f’
j=o
+
0(z2).
n-i
Re 7.
--o f’(zj)
(zj
z
J
we obtain
n-i
f
n-i
Y. f
j=o
2a 2
f(zj),
Bj
-Jim 7 z
--o (z
-n
+ z.
7. [-f’(z
j--0
j
+
(zj
f(z
(2.2)
--0
n-i
Y. --2
f’
(2.3)
(zj
z.3
j=o
Applying the argument of the lemma to (2.2) and (2.3), we obtain
-n
-nanrn + 0 (r2n)
-ilm n(n
+
l)an+ir
2a2nanrn + 0(r 2n)
-n- n(n
+
l)an+Ir
Upon dividing (2.5) by r
n
(n +
2a2an
REMARK.
and letting r
l)an+I
/
n
nanrn + 0 (r2n)
Re
n
+ n(n
l)an_ 1
r
n
+
(2.4)
0(r2n).
(2.5)
0, we obtain
(n
l)an_ I
The conclusion of the corollary is the well-known Marty relation.
It was
Hummel [6, p. 77] observed that the
originally derived by very elementary methods.
Marty relation can also be obtained by considering the Schiffer differential equation
for the functional Re a
3.
n
CONSEQUENCE OF THE MAIN THEOREM.
Suppose that a fixed function f maximizes Re
THEOREM 2.
r’s converging
to 0.
1
i
(z-; nn qbkn(’f’(z)) an
kn-i
E
n=l
kn(W)
PROOF.
is the
knt__h
(kn
(jajz
(kn-j
+
l)akn
jaj- z kn-3-’)
+
(3.1)
1,2,
k
Faber polynomial for W.
By Theorem i, f must satisfy the functlonal-differential equation
J_
f()7
f (z)
1
Bj
+
1
7.
Bj
f(zj)
fkre
n
;.
zjf’(zj)
n-I
l
j=O
where
for some sequence of
Then f satisfies the system of functional-dlfferentlal equations
’f(z)’2
where
Tr(f)
z
i
z
j
f’(zj)
-
zj
1
+
+
z
.z3
n-i
Re
7.
j=O
f(zj)
(3.2)
For fixed z, the expression F(z) defined by (3.2)
SOLUTIONS OF SCHIFFER’S DIFFERENTIAL EQUATION
is an analytic function of r for r in some small interval about 0.
in powers of r noting that the lemma insures that only powers of r
n
601
We expand (3.2)
kn
can appear.
We
since the computation for higher powers is
show the argument only for powers of r
similar.
(z)
(I +
j=0
L f(z) f(z) f(z)2
nl
+ 2zj +...)
zjf’(zj)(l
2
j=0
i
z
n-i
n-i
+
i j=07" --j f,(zj)(l + 2z +...)
fn
(zl)2 rn Ctl"
<zf’f (n)(z
C2(n)
1
(z) + f (z) 2 +" "+ f
i
2
n
=--nar + O(r
n
2
+ 0(r 2n) +
2n)
2
7.
j=0
7.
j
n-i
(jajz
j=l
.)
+
f’
(zj)
)]
0"r2n
z
+--3
zj
f’(z )z
n)a + n 7.
n
[z
n-i
+
j=0
[(n
(zin-I
Rej=0l f(zj)
f’(zj)
+ zj2
f (z
j) z
z
2
n-j
-(n-j) + j
jz
2
+.
n
)]r +
The coefficients C (n) are obtained in the following manner:
m
nrnc I (n)
n-i
l
j=O
2
Bj
n-i
7.
(f
j=o
(zj ))
n-i
7.
j=0
+
(zj
2
a2z +...)2
n-i
Z
J=0
nr
n
l
m=l m
7.
Z
a
a
l+m2---m ml m2
a
a
ml"2=n ml m2
+
0
(r2n).
+’"
"I
1
+-
Re na
n
O(r2n).
n
2-- n
ant
rn+ O(r2n)
R.J. LEACH
602
n-I
E B
nrnc 2(n)
j=0
n-i
3.
J
a2z +
(zj +,
y.
j=o
.)3
n-i
E
Ea
l
ml m2 m3
j=0 m--i
nr
lamlam2am3
+
The sum
m
I
+ m2
m
3
n.
n
Y.am
a
a
+
a a
I m2 m 3
0
is taken over all positive integers
m
2_ni"
/
\re
(r2n)
(3.4)
ml, m2,
m
3
with
This procedure yields in general
nrnc(n)
nr
n
Y.am
a
,
m+l
I
f-
i
i
n-i
We recognize
m +
n m > 0
+
C(n) as-n n f(z) an
i
I + 2
[8, p. 57]. This proves the result if k i. The other equations for k 2,... are
m
where m
Y’=l
obtained in a similar manner by equating coefficients of higher powers of r
REMARKS.
k(z)
i.
A result of Pfluger [7] shows that a Koebe function
ei0z)-2
z(l
2.
always satisfies (i.i).
The assumption that f is essentially the only extremal function for
the problem of maximizing Re a
is used quite strongly in this proof.
n
If there were
more than one function, the coefficients of the extremal function for T r would depend 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.
3.
i is of course the familiar Schiffer differential
The equation for k
equation for a function f maximizing Re a
suggests that, if f(z)
maximizes Re
a2n
Re
z
+ a2 z2 +
<_ i,
with
The nature of this family of equations
is a function which maximizes Re an, f also
If so, the Bieberbach conjecture would follow from a
a3n
result of Hayman [ii, p. 104].
lira
n
He showed that, if f e S,
I only if f(z)
z(l
eiz) -2"
k
THEOREM 3.
Suppose that f satisfies the hypothesis of Theorem 2.
the functional equation
Then f satisfies
SOLUTIONS OF SCHIFFER’S DIFFERENTIAL EQUATION
kn-i
1
1
,
nn *kn(f(z)
1
i
n nif(z)
(kn-
akn
an
(n
l)akn +
+
l)aEn
7. (ja.z -(kn-j)
3
+
n=l
n-1
603
jz kn-j
(3.5)
7. (ja.z
3
j=l
and hence f is algebraic.
PROOF.
Divide the kth equation in the system (3.1) by the th equation.
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
satisfies the
hypothesis
of Theorem 2.
Then there is a number
el@z) -2
@0
and that f
such that e
180
is
simultaneously a zero of
kn-I
qk(z)
l)ank +
(kn
Z
j=0
(jaj e-(kn-j)i$ + jje (kn-j)iS)
PROOF. Pfluger [7] has shown that if f
i
i
Re[ n((z))
k
1,2
is a function that maximizes Re a
< 0 unless f is a rotation of the Koebe function.
a
n
then
n
(He actually
proved this theorem for any linear functional and the rational function p related to
it by
It is well-known that I
(1.2).)
i
Cn(f(z)) an
(See [8, p. 194], [6, Theorem 13.6].)
function.
We consider equation (3.5) with
Re a
0 if and only if f is a Koebe
It is well-known that since f maximizes
i.
the function q defined by (i 3) must have at least one zero on z
n
the left-handed side of (3.5) is analytic by assumption, each zero e
THEOREM 5.
i)
ii)
is real k
2a2akn
(kn
qk(e i8),
k
1,2,
This completes the proof.
l)akn+l
(kn
(2.2) and (2.3) are valid for all r
rkn
in
Then
1,2,...
l)akn_l
PROOF. Since f is essentially the unique function maximizing Re
efficients of
of
+ jaje -(n-J)i8
(jaje-(n-j)i8
j=l
Suppose that f satisfies the hypothesis of Theorem 2.
akn
Since
n-I
(n- l)an + Z
ql(e i8)
must also be a zero of
i
zjl
an,
the equations
in some neighborhood of 0.
(2.2) yields, after an application of the lemma,
-nakn
in Im kn
akn-
or
-i Im
akn
-i Im k
nakn
n Re
akn
Equating co-
604
R.J. LEACH
which implies that
akn
is real.
Equating coefficients in (1.6) and applying the lemma, we obtain
-2a2n akn
-n (kn
+ i) akn+l + n (nk
i) ank_l
and the result follows after division by -n.
I would like to thank Professors J.A. Hummel and W.E. Kirwan for
ACKNOWLEDGEMENTS.
helpful conversations regarding the extremal functions for T (f).
r
to thank Professor
I would also like
A. Pfluger for pointing out an error in an earlier version of this
paper.
REFERENCES
i.
SCHIFFER, M.
"A Method of
Variation Within the Family of simple Functions",
Proc. London Math. Soc. 44 (1938), 432-449.
2.
SCHIFFER, M. "On the coefficients of Simple Functions", Proc. London Math. Soc.
44 (1938), 450-452.
3.
AHLFORS, L.
4.
BRICKMAN, L., and WILKEN, D. "Support Points of the set of Univalent Functions",
Proc. Amer. Math. Soc. 42 (1974), 523-528.
5.
GARABEDIAN, P. and SCHIFFER, M. "A Coefficient Inequality for Schllcht
Functions" Ann. of Math. 61 (1955), 116-136
6.
HUMMEL, J.A.
"Conformal Invariants", McGraw-Hill, New York, 1973.
"Lectures on Variational Methods
in the Theory of Univalent
Functions", Univ. of Maryland Lecture Notes, 1970.
"Lineare Extremal probleme bel schllcten Funktlonen", Ann. Acad.
Sci. Fenn. AI 489 (1971).
7.
PFLUGER, A.
8.
POMMERENKE, CHR.
"Univalent Functions", Vandenhoeck and Ruprecht, Gottingen,
1975.
9.
I0.
Coefficient Regions for schllcht Functions",
SCHAEFFER, A.C., and SPENCER, D.C.
Amer. Math. Soc. Colloq. Publ. vol. 35, New York, 1950.
SCHOBER, G.
"Univalent Functions
Selected Topics", Sprlnger-Verlag, Berlin,
1974.
II.
HAYMAN, W.K.
1958.
"Multlvalent Functions", Cambridge University Press, Cambridge,