THE COEFFICIENTS OF SCHLICHT AND ALLIED FUNCTIONS.
W.
K.
HAYMAN
Let 5 be the class of functions
w = f(z) = z + a2z2 + . . .
regular and schlicht (univalent) in | z | < 1, i.e. assuming distinct values w
for distinct values of z. The functions f(z) e S map | z | < 1 1 : 1 conformally
onto a simply connected domain D in the w plane. During the last international
Congress at Harvard Professors Schaeffer and Spencer [12] reported on the
progress, largely due to their own work, on the so-called coefficient problem
for S, i.e. the problem of characterising the region Vn in complex space of n — 1
dimensions occupied by the points (a2, a3, . . ., an) for f(z) e 5.
To-day I should like to discuss the behaviour of the an for large n and the
related problem of the behaviour of f(z) near | jar | ===== X - Many of the results
extend to more general classes of functions.
It was first shown by Koebe [9], that the distance d of w = 0 from the
complement of D is greater than an absolute constant, and that for fixed z
| f'(z) | and \ f(z) | are bounded below and above by positive constants for all
f(z) € S. The exact inequalities | a2 | ^ 2, d ^ J and the bounds for | f'(z) \,
| f(z) | and | f'(z)jf(z) | were obtained in (1916) by Bieberbach [2], Faber, Pick,
Gronwall and others. In each case the bounds are attained only for the functions
flz) =
n
J
__
— =
y nznel
(1 -- ze
zetdid)2f
(1
Z
71=1
-1)0
which we shall therefore call the greatest schlicht functions (g.s.f.). They map
| z | < 1 onto the w plane cut along a radial slit from — \e~lQ to oo. These results made plausible the famous Bieberbach conjecture, that \ an\ ^ n holds
for all positive integers n and f(z) e 5.
In (1923) Löwner [11] obtained a partial differential equation satisfied by a
class of functions dense in 5. By means of this he was able to show that
I #3 I = 3 and also to obtain the exact upper bounds for all the coefficients of
the inverse function z = f~1(w), again with equality only for the g.s.l.'s. His
method has also led to numerous other results, notably the exact bounds for
arg (f'(z)) and argj
»
\ . In these latter cases the extreme functions are not
102
the g.s.f.'s but map \z\ < 1 onto the plane cut along certain slits. Garabedian and Schiffer have just proved that |a 4 | ^ 4.
In (1925) Littlewood [10] proved that
1 C2n
I(r.t)=7r\
\f(rei6)\d6<-1
2n J
Hence taking r = 1 — 1/n,o
- I 2ni
— Jf
f(z)dz
zn+l
I(r, f) <
.0<r<l.
1— r
1
<
< en.
,,71-1(1 — r)
|s|-r
The g.s.f's. have I (r, /) ==
r
r
. The sharpest estimates to date are
\ — r2
r
\ — r2
h 55 and hence | an | < \en + 1.51,
due to Bazilevic [1].
00
Let us consider next the more general class of functions f(z) = 2 anz7l>
o
^-valent in | z | < 1, i.e. such that the equation f(z) = w has at most p roots in
| z | < 1. In (1935) Cartwright [5] showed that for such functions
M(r, f) = max | f(z) \ < A(p)jup(l - r)~2*>, 0 < r < 1,
(1)
\z\=r
where jup = max | an \ and A (p) is a constant depending on ^>. Hence, using a
generalisation of Littlewood's method, Biernacki [3] deduced that
I(r, f) < 4 ( 0 K ( 1 - ^ l 1 - ^ 0 < r < 1,
and so that
\an\<A(p)fz^-\
n^p.
(2)
(3)
The orders of magnitudes in these results are best possible, as is shown by the
p'th powers of the g.s.f s. The results were extended by Spencer [13] to mean
^-valent f(z), i.e. such that the equation f(z) = w has on the average at most p
roots, when w ranges over the disk | w \ < R, 0<R<co. Here p need no longer
be an integer. Then (1) remains true for p > 0, (2) for p > | and (3) for p > J.
It will be convenient to use a weaker notion of mean valency due to Biernacki [4].We shall say that a regular functions f(z) is mean ^-valent in a domain
A, if the average number, p(R) of roots of the equation f(z) = w, when w
ranges over the circumference | w \ = R always satisfies p(R) ^p.
Here
2nRp(R) is the total length of the arcs in the Riemann surface of w = f(z),
which lie over the circle | w \ = R. Alternatively
103
P(R)=^J7'n(Rei'P)d<p
0
where n(Rei(?) is the number of roots of the equation f(z) = Rei(? in A.
Further we shall consider functions f(z) analytically continuable with onevalued modulus throughout an annulus r0 < | z | < 1. If f2(z) is the branch
obtained from fx(z) by continuing once around the annulus, then | f2(z)lfi(z) I
= 1, and so by the maximum modulus theorem f2(z) = f1(z)eifX, where /u is a
real constant. Thus f(z)/z/J' remains one-valued in the annulus and possesses a
Laurent expansion
/ W = / ï V ,
—
r0< \z\<l.
(4)
GO
We define p(R) as above for a fixed branch of f(z) in the annulus cut along a
radius. Since replacing f(z) by a different branch gives rise to a rotation in the
w plane, p(R) is independent of the radial cut and the particular branch taken,
so that our definition of mean ^-valency is meaningful.
The following remarks may justify these somewhat artificial looking hypotheses. In the first instance if f(z0) = 0, then f(z) takes all sufficiently small
values w near z0. Hence if f(z) has q zeros in the annulus, then p(R) ^ q for
all small R and so q ^ p. Thus f(z) has only a finite number of zeros in the annulus r0 < | z | < 1, and so we can find a smaller annulus rx < | z | < 1, in
which f(z) is still mean ^-valent and f(z) ^ 0. Next, if v is any positive number,
the transformation W = wv maps arcs of angular length a on | w \ = R onto
arcs of angular length VOL on | w j = Rv. Thus if f(z) is mean ^-valent and not
zero in rx < | z | < 1, then [f(z)]v is mean y^-valent. For [f(z)]v can be analytically continued throughout rx < | z \ < 1, but need not be one-valued. Thus our
definitions enable us to discuss positive powers of f(z) together with f(z). This
is very useful in view of the following [1]
Regularity Theorem. Suppose f(z) is mean p-vaient in r0 < | z | < 1. Then
the limit
a = lim (1 - r)2pM(r, f)
(5)
r-»l
exists finitely. Further if p > \
lim (1 - r)2*-*I(r, f) =
r->i
^
„ ,
2r(iW(P)
6
and if p > J
Jim ULL =:
.
n»-i
r(2p)
Let us call a the limit constant of /(z).
n_+00
104
7)
One of the main advantages of this theorem lies in the close link between
the maximum modulus M(r, f), which is usually not too difficult to deal with
and the means I(r, f) and coefficients an, which are frequently less tractable.
Thus by our previous remark [f(z)]v is mean (^)-valent in the zerofree annulus
r
i < 1*1 < 1 °f f(z) a n d by (5) [f(z)Y clearly has limit constant av. Thus we
obtain at once information about the means and coefficients of [f(z)]v from our
theorem, by replacing a by av and p by pv.
Suppose for instance that f(z) eS, p = 1. Then we have
/'(«•«)
fire")
1+r
d
r(l — r)
dr
(1 — r)2
and it follows that except for the g.s.f's (1 — r)2r~xM(y, f) decreases strictly
with increasing r, 0 < r < 1. These results remain true if f(z) is merely mean
1-valent in | z | < 1 [6]. Hence if f±(z) = z + atf2 + • • • is mean 1-valent in
lim
n->ao
= a ^ 1
^
and a = 1 only for the g.s.f's. In particular the Bieberbach conjecture holds for
all sufficiently large n and any fixed f(z) e S.
More generally let
/,(*) = z*(l + b1z + b2z2 + . . . . )
be mean ^-valent in 0 < | z \ < 1, where p > I. Then fv(z) ^ 0 in 0 < | z | < 1
since the immediate neighbourhood of z = 0 contributes p to p(R) for small
positive R. Thus
fi{z) = [fv{z)?,v
=
z
{I +~~z
+ - • •)
P
is mean 1-valent in 0 < | z | < 1, and so in | z \ < 1, since fx(z) is evidently
singlevalued. Thus if oc is the limit constant of fx(z), the limit constant of fp(z)
is a? ^ 1, and we have
\K\
^
l
r
hm 2v-i = p(2p) -< r(2py
71—> 00 n
with equality only for the ^'th powers of the g.s.f's. An interesting special case
occurs, when p is a positive integer and fp (z) is ^-valent in | z \ < 1.
More generally suppose that p, k are positive integers, 1 ^k <4p, and
that
/,,*(*) = ^ ( ! + bi** + V 2 * + • • • )
105
is mean ^-valent in | z | < 1. The transformation zk -- Z maps the annulus
0 < | z | < 1 described once, onto 0 < | Z | < 1 described k times. Hence
fvAzm)
= z*lk(l + biz + b*Z2 + •••)
is mean (p/k)-valent in 0 < | Z | < 1 and so
\K\
J™ n2^-1
_
=
****
1
r(2p/k) ~r(2p/k)'
Equality holds in this case only if fv%lc(z) = zp(l — ^ V 0 ) - 2 ^ .
If p = 1 and fl3je(z) e S we may go further. In this case the function
/i(*) = / lt *(^ 1/fc ) = Z{1 + kbxZ +
...)
also belongs to S, and so \kb±\ < 2, except when fx(Z) = Z(l — Zß 10 ) -2 .
/ M (*) = *(1 — zkeie)~2lk. If IäOJI = c, where 0 ^ c < 2, it follows from a recent
result of Jenkins [8] that the limit constant a of fx(Z) satisfies
a ^ a(c) - 4[2 - (2 - c)*]- 2 exp {2 - 4/[2 - (2 - c)*]},
and that this inequality is sharp for every c, 0 ^ c < 2. Hence if & = 1, 2 or 3
and
fiAz)
= * + < w f c + 1 + ß2*+iz2&+1 + • •.
belongs to 5 and | ak+1 \ = cjk, then we have
,. I < W i I ^ [«(c)]1/ft
lim —nil ., <
,
_ * „ w2'4-1 -
r(2/k)
and this is sharp for every c, with 0 fg c fg 2.
Nevertheless if k — 2, the upper bound of | ä 5 | for varying odd f(z) e S
is ß~2/3-f \ = 1.01 . . . and | a n | > 1 is possible for every fixed odd n greater
than three and a corresponding function f(z).
I should like to conclude by indicating how one ma}' prove the Regularity
theorem [1]. We may suppose that f(z) is mean ^-valent and not zero in an
annulus 1 — 2ô < | z | < 1. One can then prove the basic inequality
4pô
fire")
t
1 - 2d < r < 1, | 0 | ^ n.
ie
f (re )
(1 - r)(r + 20 - 1
If we write f(reiQ) = Rea, this becomes for 6 fixed
d
4pô
d
(r + 2ô-l\
# -4. •loer
-log^
+ iX'(r\
a'w I <<: _(1 - r)(r
log(—Ïr——j.
w ^+ 2<5
ft> - _1)= 2 / > -^r
Hence I
\r
+
1 R decreases with increasing r and a fixed 0.
2d — lJ
106
,v
(8)
Suppose now that
a =~îïïn (1 — r)2pM(r, f) > 0
(9)
r->l
(If a = 0, our theorem is easily proved by classical methods.) We can then
find zn = rneidn, with rn -> 1, | 0n | ^ ut, such that
"-"^T^-ì'"•'"'"•
r + 20 - 1/
'
(2(5)^
v
n
It follows from the above monotonicity property that
t + M - l ) !/(«*•) I ^.. ! - « < ' < ' . ,
Hence if 0O is a limit point of the sequence 0n we deduce
/
1—r
\2P
V +
+ 22<50 -- 1 I/ / '
In view of (9) this implies
/ / ^ o ) I > _?L_
V
(2(5)^
1 - 20 < r < 1,
M(r, f) ~ | f(rei6°)\ ~
as r -> 1,
(10)
(1 — r) 2p
and this gives (5).
Next one can deduce from (8) that
~(r + 2d ~ l)2p~
(l-r)[X'{r)?£6p-log
L (1 - rf*R
dr
.
On taking 0 = 0O and integrating from 1 — ô to 1 we deduce
J
(1 - r) ß (r)]2dr ^ log j
î-ô
• [< oo
a
[
j
(1 — r)[X'(r)]2dr < s and rx < r2 < 1
in view of (10). If r± is so near 1 that
it follows that
I A(ra) - Afo) | g J
| A » ^ <£ j j
(1 - r) [A'(r)] 2 ^ j
— J
From this and (10) it follows that if rv r2 -> 1 so that (1 — ^i)/(l — ?2) re~
mains bounded, rx < r2 < 1, then
/(/2^0o)
~ / > .
•
107
-
Ö
By a "Normal Families" argument one can then deduce that
.„ / 1 - r \2*
as r -+ 1, uniformly while z remains in a region
A(r,e):e(l
— r) ^ 1 - ze-te» <
•
A
I
1
—
r
,
\
arg 1 - ze~lQA
/
.
f
l
\
I
<
n
s
for a fixed e, and that the area of the image by f(z) of the part of the annulus
1 — 2ô < | z | < 1 which lies outside A (r, e) is relatively small. In fact the
area over the w plane of the image of that part of A(r,s) by f(z), where
\f(z) | < i? = M(r, /), will already be nearly pnR2, the maximum permissible
amount for mean />-valent functions. From these facts (6) and (7) can be
deduced. In fact we obtain the asymptotic formula
/ r(i_j_L<><rL-*(«+A0<>
nan i—'
, as n -> + oo,
rpp)
and in view of (10), we have (7).
REFERENCES
[I]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[II]
[12]
[13]
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M. BiERNACKi, 'Sur les fonctions multivalentes d'ordre p ' , Comptes rendus 203
(1936), 499—451.
M. B I E R N A C K I , 'Sur les fonctions en moyenne multivalentes', Bull. Sei. Math. (2), 70
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M. L. CARTWRIGHT, 'Inequalities in the theory of functions', Math. Annalen, 111
(1935), 98—118.
W. K. H A Y M A N , 'Some applications of the transfinite diameter to the theory of
functions', J . d'Analyse Math. I (1951), 155—179.
W. K. H A Y M A N , 'The asymptotic behaviour of p-valent functions', Proc. Lond. Math.
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108