奈良教育大学学術リポジトリNEAR
On the Coefficients of Certain Schlicht Functions.
著者 SAKAGUCHI Koichi
journal or
publication title
奈良学芸大学紀要
volume 4
number 2
page range 23‑28
year 1954‑12‑25
URL http://hdl.handle.net/10105/5038
(23)
On the Coefficients of Certain Schlicht Functions.
Koichi RAKAGUCHI
Department of Mathematics, Nara Gakngei University (Received September 1, 1954)
In this paper two kinds of schlieht functions will bo discussed-
Let us denote by F(z) a function regular in the unit circle and with the
expansion
F(z) =z+ ^anz*
about the origin. "="
§1. It is well known that if F(z) is sehlieht and if cither it is starlike with respect to the origin or its coefficients are all real, then
(1-1) |«n|^», n^2.
This theorem has been extended by S. Ozaki (l~) us follows:
THEOREM A. Let F(z) be regular for \zl<L1 and starlike in the direction
of a diametral line, then F(z) satisfies (1. 1).
(In this theorem F(z) is not necessarily schlicht.)
On the other hand it has been shown by M. R. Robertson £_2~} that if F(z) is
schliclit and starlike with respect to a point other than the origin, then
1 J-f/7\2|
* I V"!
,F(a)
a . ( F(a)\ 1
1"~3+3 I rta/V F(a) a ) GF{a) a \^,
whore F(d) is tho star-centre point of the image domain.
But lie has not touched there the coefficients an «aoh that n^>4.
In this section I shall show by extending Theorem A that the schlicht function starlike with respect, to a point other than the origin satisfies (1. 1) also for W^>4.
DEFINITION 1. Let /(z) be regular for lzj<U and let $ be an arbitrary point such that |#|-1, then tlie straiglit line /(f), f(-f) i* ('ailed a semi-diametral lino of /(Z).
REMAKK. When a semi-diametral line passes through the origin, it becomes a
diametral line, whose idea was introduced by S. Ozaki [1J and N. G. DeBrniju £4~].
DEFINITION 2. Let /(2) be regular forJ2i<;i and let the image carve of"|z|=l by /(z) be cut by a semi-diametral line at exactly two points, then /(z) is said to be starlike with respect, to the semi-diametral line.
Theorem 1. Let F(z~) be starlike with respect to a semi-diametral line and let d denote the distance from the origin to the semi-diametral line, then
(1.2) |fls,+1 |^2»+1, n^J,
(1.3) l«s.. L^2ii+( Iasj-2), n^l,
(1. 4) |fl. |^2(l +rf).
PROOF. Let F(l), ~F(-£), |ll=lj be the semi-diametral line and let «=argf.
Further let & denote the angle formed by the line and the positive imaginary axis, and let Wo be the foot of the perpendicular from the origin to the semi-diametral
Journal of Nara Gakugei University, Vol, 4 No. 2, Dec. 20, 1954
( 24 ) Ivdichi SAKAGTJCHI line. Then for a sign <? (+or-) suitably chosen
(...) *-(*..)-..) sg z :f:f,i:+2,
On the other hand, putting
g(z)=if($-z) (-G-z)/z,
wehave
' *K J V ; (^0 for a+7t<fi<ia+2n,
which combined with (1. 5) gives
MeeiP (F(z)-wo)gizfeO, \z\ =l.
Here
"e& (F(z)-u>o) g (z)
=«i--aT' -5a-(wo+«aoa)«+ S («n-?2«,,+3) zn+1]
'å Z n=l J'
where s = aiei(@ - a).
Consequently by Robertson's lemma (3~) we have
( 1- 7) i Ifhvo-*(«,., + £%3) i^29l( -«£*),
From (1. 7)
which gives (1. 4), since
I* £|=1, \wo\=d, 9l(-««a)^l.
Next from (1.8) we have
(1.9)
Putting «=1, 2, in (1. 9), we obtain successively I«« !:£KI+2^3, |a+|^|aa|+2,
Thus we get (1.2), (1.3.).
Coeollary 1. (Theorem A). Let, in Theorem 1, the semi-diametral line
pass through the origin, then F(z) satisfies (1. 1).
PEOOF. Putting rf=O in Theorem 1, we get this at once.
Corollary 2. Let F(z~) be schlicht in Theorem 1, then F(z) satisfies (1. 1).
PROOF. Berawso in this ease | tf2 |<I2.
Corollary 3.-Let F(z) be schlicht and let D denote the image domain of
the unit circle by F(z). If either (1) D is starlike with respect to a point which is not necessarily the origin or (2) D is convex in one direction, then F(z) satisfies (1. 1).
PROOF. We may assume without loss of generality F(z) to be regular for |2 |<11.
The case (1). Let c be the centre of starlikeness of D, and put f(z)=F(z)-C.
Then y(z) has at least one diametral line, which cuts the image curve of \z\=l by f(z) at exactly two points. This shows that F(z) has such a semi-diametral
On the Coefficients of Certain Schlicht Functions. ( 25 ) lino passing through the point c as cuts the boundary of D at exactly two points.
Accordingly by Corollary 2 F(z) satisfies (1. 1).
The case (2). Let / denote a straight line which shows the direction of convexity.
Then we can easily verify that there exists such a semi-diametral line /' as is parallel to /. And /' cuts the boundary of D exactly two points. Thus this function also satisfies (1. 1).
§2- Recently A. Dvoretzky (_5~) has established the following theorem :
Let F(z) be schlicht and let A(R) denote the radius of the largest circle whose centre lies on \w\=R and whose interior lies in the image of |z|<^1 under W-F(z)-
If A(R)=O(R*), then
(2. 1) \an \=O(tUK*-V) for 0<^l,
( 2.2) \an
(2.3) (2. 4)
=O(logw) for ^=0,
=0(1) for -2^<0,
=O(»-l/2+2/Ca-A)) for /i<-2.
These estimates show a gradual passage from \anl=O(n) for the general schlieht
function to |an\=o(n-lW) for the bounded one.
This theorem has been improved by W. K. Hayaman ifi~) for the two oases ^=1,
0 as follows:
fl) When/t=l,
(2. 5) |«B |=O(»-i/a+A*c)
on the condition that A(R)<jR (c<2/3) for R>R0, where K is an absolute constant, and
(fl) when A=0,
(2. 6) |aB|=O(n-Va logw).
(2.5) shows that \an\ can be estimated by the same order as (2.4) under a
weaker hypothesis on A(R).
In this section I shall improve the result of Dvoretzky for each value of X and
at the same time show that (2.5) can be made more precise and further the
restriction c<^2/3 is unnecessary.
Lemma. Let F(z) be schlicht and let A(R) denote the same quantity as in
the above theorem, then
(2-7) \F'(z)\<g4A(R)/(-i-\z^), R=\P(z)\.
PROOF. The function
<=g(z^£F(f±-Z'-)-F(zo)1/F'(zo) (1-|zop)=z+ , |zo|<l, is regular
-I~rZoZ
and schlicht in |2;<1. Moreover g(z) omits somo values on every circle |C|=P>
A(Ro')/\F'(zo)\ (l-\Zo\2), Ro=\ F(Zo)\, from tho definition of A(R). we liave therefore by Koebe's 1/4-theorem
A(Ro)/\F'(zo) \ (l -\ zo \*)2>l/4, which gives (2. 7).
Theorem 2. Let F(z) be schlicht and let A (R)=O(RX). Then, (I ) in the
case X=l, on the condition that A(R)<^cR for R^>Ro(where c, Ro are constnts),
( 26 ) Kfiichi SAKAGTJOHI
(2. 8) |flB|=O(ii-i+aC) for~<c<l,
(2. 9) |an|=O(l»g«) /orc=|,
(2. 10) |«n |=O(l) /br^c<~
( 2.ll) l«,,|=O(»-i/a+ac) /or c<i
4 (JI) m £/te cosg ^<1,
(2. 12) |fln|=O(«-V2 OgraJi/a-,0).
REMARK.Wo should remark in the case ^=1 that for the general schlioht function F(z), Kr^tA(R)/R^<^.
K->oo
PEOOF. Since A(R)=O(R*), there are two constants c, /?o such that
(1) A(R}<cRl for i?>7?Oj
where c is not necessarily no larger than 1.
From the nature of our problem we may assume that F(z~) is not bounded, and
so there is a constant ro ("O) such that max 'F(z)\=Ro-
Now lot P ("\1) be a number larger than ro, then we can find a real a, depending on p, such that |F(p0«) |-max|F(z) \^>Ro•E
Putting \ F(teia) =R(j), we have
(2) ^^p- <\ F'(t&«) \.
Take Po slioh that |F(poe'a) \=R0 besides |F{teia)\^>RO for all t in Po<^t<P, then by (2.7), (1), and (2) we find
yivrr\ a At i?r+\\ a~t>(+\*
~dt ^ 1-W^T^Pa
Henoe
I. When /l=:l, from (3)
R(p)<Ro f(l+p)/(1-p)>, ro<p<l.
Henw
(4) max|F(z)l=^(P)<^o(l^)° forevery pin0^»<l.
\z\=p V1 ^-y
Consequently by the area-principle we Imve
S » !«» l^2^(max | F(2) |)2<i?o(i±£).
Hence |fln |<i?ow-l/2 p- n T^-)2' »^2, 0<><l.
On the Coefficients of Certain Schlicht Functions. (27)
Put here p=»/(»+l), then
(5) | on|<ieo«-l/2 (] + -)n (2«+l)2<^oW-1/2(2«+l)2r, n^2,
which proves that (2. ll) holds for an arbitrary non-negative real number C- Next we shall show that (2. ll) can be made more precise for l/4<e<l.
Set G(2) = 1/F(z2) =21/p(z2)722=2 + y
then G(z) is regular and sehlioht for |z|<^l, and further we have
27T 2K 2K
(6) (IF(rei<>) [dd=\ \F{re^)|dO=\ \ G(/-V2 eie) *dO.
o o o
Let J(r) denote the area of the image of \z\<r (<O) by G(z~), then from (4) wo have
J(r)^n max\ G(z) V=n max|F(z*) IO#o(^W*(y^^)•E
On the other hand by the distortion-theorem eoneerning schlicht functions we have J{r)<jz max !F(22) l<7r^/(1-^2)2.
Oonsequently for l/4^p<^l,
271271 å P 1 /4: fi
£j,C(w.,*.=1JZCrl*<j^^w^
o o o 1/4
1 \2cdr
r) r
(7) iMih)
o -l+2c
for O>~, for c=±,
Ao log-:
for c<A,
where A-,, An are constants depending only upon Ro and c, and A2 is a constant
depending only upon Ro-
"We have now by Cauehy's theorem
271
M zF(z)F(z)n+l dz -
2npn F(pe>») [ d9,
u ni-j 2ni ) z"+1
and it follows from (K) and (7) that for W^>2, T./4<:p<^l,
1 / 1 \-l+2c
{Ai Pn ^i_/y;
\ an ^
2n
27T
iSr ^r5lG(pi/a ^)Pd»<«
Iog T-7T
u 3 nn
for e.>-sr, for c=-s-,
for ,<f
Putting here p=(l--)2, we obtain for n>2,
( 28 ) K6ichi SAKAGUCHI
l'4 e 'iA l n ‑ ^ + 2e f o r c > ‑ｫ,‑,
4 e '2A 2 ¥o g n f o r c = Y , 4 e 2 ^4 Q f o r c < 4 i I «»J<J
from which (2.8), (2.9), and (2.10) follow.
I. When ;<1, from (3)
2e(p)<{#ow+2c log14/9-}1/CW\ ro<P<l.
Hence max |F(z) \=R(p)<A,(log--1-)!/(W) for evely p ini-<p<l,
\z\=p i-P å *
where Ai is a constant depending only upon Ro and c- Therefore by the area-principle we have
Honoo
|2 'in
n =i
i/a-;).
,-va^-n,
«B|<J44,,-'/V"(log5^-)Va-J), w^2, y^P<l.
Putting here p=l («^i), we obtain (2.12).
We thus complete the proof of the theoerra.
References-
en S. Ozaki, On the coeffifients-prohlem in the theory of scHicht func'ions, J. Olsuka Math. Soc., 6 (1937), pp. 23-29.
[2J M. S. Kobemwn, Star antre points of multvalenl functions, Duke Maih. J., 12 (1945), pp.669-684.
C3J , The rariaiotl of the sign V for an analytic funaiou U+iV, Duke Math. J., 5
(1939), pp. 512-519.
C4] N. G. DeBruiJN, Kin Sate iiber schlkht Funktionm, Neder. Akad. Wetensch., 44 (1941), pp 47-49.
15J A. Dvobetzky, Bouvfh for tie coeffi ien s </ wival nt functions, Proc. Amer. Math. Soc., 1 (19G0), pp. 629-635.
[6J W. K. Haiman, Functions with ralues in a given domain, Proc. Amer, Math. Soc\, 3 (1952), pp.
428-432.
