奈良教育大学学術リポジトリNEAR
A note on close‑to‑convex functions (3)
著者 OGAWA Shotaro
journal or
publication title
奈良学芸大学紀要
volume 9
number 2
page range 7‑23
year 1960‑02‑15
URL http://hdl.handle.net/10105/4815
Jour. Nara Gakugei Univ., Vol. 9, No. 2, 1960
A note on close-to-convex functions (3)
By Shotaro OGAWA
(Department of Mathematics , Nara Gakugei University) (Received November 25, 1959)
1. Introduction
Recently Y.Miki [ID proved the following theorem : THEOREM 1. (Y.Miki) Let the function
f(z)=z+ S «v2v
be close-to-convex* for |z|<l with respect to the function
CO
(p(z)=Z+ S *vZ".
Then the n-th partial sum
V=2
off(z) is also close-to-convex for |zj<- with respect to the partial sum
(Tn(z)=Z+ XI £V 2V
of <pCz). The constant - can not be replaced by any greater one.
4
This theorem corresponds to the following theorem proved by G.Szego. £3) THEOREM 2 (G.Szego) Let the function
f(z)=Z+ S «v2V
V=2
be analytic and 'schlicht' in (z[<l. Then any one of the partial sums
n
z+ S «v2v (m=2,3, )
v-2
is also 'schlicht' in \z\< - , awrf /fee constant 4- caw «o^ 6e replaced by any greater
4 4
owe.
In this theorem the word 'schlicht' may be replaced by 'star-shaped with respect to the origin' or by 'convex', both in the hypothesis and in the conclusion in corresponding manner.
*Following W. Kaplan (2] we call an analytic function /(z) to be close-to-convex for \z\ <CR with respect to <pCz~), if there exists a functin ^>(z), convex and schlicht for |z]<J?, such that/' CO./- i)?' (z) has positive real part for \z\<_R.
8 Sh6tar6 OGAWA
Here we recall following theorems proved by A.Kobori £4} related to the above theorem.
THEOREM 3. (A.Kobori) Let the power series
<p(z)=Z+ S #v2v
be analytic and schlicht in the circle \z\<\ and starlike with respect to the origin.
Then any one of the partial sums
<rn(z)=z+ 2 Z>vzv (n=2,3, )
v=2
is schlicht and convex in the circle \z\< ~ . The constant - can not be replaced by
8 o
any greater one,
THEOREM 4. (A.Kobori) Let the power series cp(z)=Z+ S 6v2v
V=2
be analytic and schlicht convex in the unit circle |2|<1. Then any one of the partial sums
<rn(z)=z+ S S»2V (n=2,3, )
V-2
is schlicht and starlike with respect to the origin in the circle \z\<-. The constant å ycannot be replaced by any greater one.
Kobori's theorems may be modified by the same way as Miki modified Szego's theorem.
The author C^D showed another proof of Theorem 1,4 and the principle of this proof may be used also in the generalization of Theorem 3,4 in this paper. Here we aim to prove following theorems.
2. Theorem Theorem I. Let the function
f(z)=Z+ S fl»2v
be analytic and satisfy next inequality 9tifS>0 ()z|<1) for some suitable regular function
V(Z)=Z+ S 6vZv,
v=2
schlicht and starlike with respect to the origin in the unit circle \z\<l. Then any one of the partial sums
f n(z)=Z+ J] « v2V O=2,3, )
satisfies next inequality
^X) <W<i>
A note on close-to-convex functions (3) 9 for the -partial sums
<pn(z)=Z+ ^ *vZv (M=2,3, )
v=2
and the constant 4- can not be replaced by any greater one.
THEOREM n. Let the function f(z)=Z+ S ft2"
be analytic and satisfy next inequality
®Cjwlr >0 (l2|<1) for some suitable regular function
000=2+ S 6VZV,
schlicht and convex in the unit circle |z'<L. Then any one of the partial sums
fn(z)=Z+ S «vZv (W=2,3, )
satisfies next inequality
/or i/ze partial sums
?n(z)=Z+ithZ" («=2,3, )
awrf tte constant ~-can not be replaced by any greater one.
At Theorem I , let the function /(s) be starlike especially, then the associating function
<p(z) can be replaced by /(z) itself. So the conclusion means that
*^' =^+ $g2 ) >0forN<A-,
namely the partial sumfnCz) is convex in the circle \z\<^ -. So we have Theorem 3 as
o
a corollary of Theorem I. Similarly we have Theorem 4 as a corollary of Theorem II.
Remark 1. Hereafter in this paper the notation cp(z) is used as a function starlike with respect to the origin, and the notation fi(z) is used as a function convex.
Remark 2. Let the class of functins which satisfy
m~{W>° (|2(<1)
be denoted by iT(kappa) and the class of functions which satisfy
namely the class of close-to-convex functions, dy 2. Then corresponding to the relation appointed by Alexander C63
f(z>® å £. sFW&t, we have the relation
Kz)sK -^ zf'iz-)sS.
10 Sh6tar6 OGAWA
Since there holds J?Z)<3«, sowe have Kzi$t (of course 2^>K). As for the class K few properties are known. For example, for fQz^eS we have
f(z) < z+2zz+3zs+ +nzn+•E•E•E•E•E•EiT}, so from the above relation we have immediately
/(Z) < 2+22+28+ +Zn+
for f(z)eK. The geometrical character is remained unknown.
3. Lemmas
Following lemmas are needed in the proof.
1. Let/(z)= S 0vZvbe regular and have positive real part for |z|<l. If «o be real, then v=o
"we have
(|a|=r<l)
l~r -rnw ^m-rr~\^' l+»*
. j\vj isvv4;st^z JyvJ>
1+r' l-r'
and
i-;~/co) ^i/cz)i< ±±£ /co), (Polya-Szego C83) aH\ <2o«. (C.Caratheodory C9D)
2. Let <p(z)~z-)r 2j ^vZv be regular and starlike with respect to the origin for |2|<1, then
V-2 V iz-)
< r 1+r
=\-r(|zi=r<l).
<e'U)
PROOF. From the hypothesis there holds 3tz ^^2 >Q. So from Lemma 1 we have 1-r < gy'O)<P(z) I .
\+r This implies Lemma 2.
3. Let f(z) and cpQz) be both regular and satisfy
then and
(*-0)
-
I >0 for |z[<l,
1+r£s Ji ,^^
^(2) - i-r
<(1-r)2 (|*|=r<l).
PROOF. The first evaluation is evident from lemma 1.
As for the latter, from the hypothesis, adopting Lemma 1,
and
F(z)=,=*/'O)0>O)< l+2z+222+ +2z"+-
F 'Cz)" (SS"})' <2(1+22+3z3+423+- 0
= (l-a)a
4. Let /Cz) and <?>CZ) be regular and ^(z) be starlike with respect to the origin and satisfy SftZfCz} >Q for |Z)<1) then we haye
A note on close-to-convex functions (3) l -6r -t-2r2-6r3+r4
ll
reCzf(Z)y and
Proof.
Th erefo re
</ >'o ) = ( l ‑ r ) 3 C i + r )
( * / ' (* )) ' < r ( l + r ) ( l + r a ) 0 >'O ) ( 1 ‑ r ) 3
0>'(z) (p(z) i/)(z>'(«)
=<y(z). (z///(z)^/(z))<P(z)-g/'(z>/z) = ff(g) \?f^2\'
<P(.z)» <p'(z) «p U)J.
</>'U )
Sft (z/'(s))' 5ft 5£« ig. }(p1.. ^^z)(zj\zj\å ,/ as ^\'
ifJ KZJ \y^zj
(z/'CsoY
#>U)
Furthermore we have
re(g/O))' > tog/(») _
P'(«)
yWi.
\-rl+r'
*>U)
l+r
1-r Q-r)
\Viz))
*>'O)
_l-6r+2r3-6r3+r4
2
Similarly
<j5TW
<f>Lz)+
Cl-r)3 Cl+r)
(W'ooy
\ 9>(z)
(Lemma 2,3)
1-r^ 1-r (1-r)3
_(1+r) (1+r2)
(Lemma 2,3) (1-r)3
5 Let /(2)=2+ S «v2Vbe regular and schlicht for |z|<l (so of course including the case
v=2
starlike), then as is well known
1-r ^ |/'(z) l for |z|=r<l. (R. NevanlinnaClO]) (J+r)3
6 Letf(z}=Z-\- S 2vavbe close-to-convex and <» C2}=a+ Siv 2V be starlike with
V-2 V-2
respect to the origin, then
«v| ^v (v=2,3, ) (ReadeC73)
\by\ ^v Cv=2,3, ) (R. NevanlinnaCll^), and so for
and v=»+l
there holds
(1) | Qzrn'(z)y |<Sr« ^(l-r)«+3n'Cl-gl^O-~^+i+#+*
(2)
PROOF, (the latter half) From the first half of this lemma, we have r,,O)= S «vzv (i.e. /(«)=/»(«)+/•E»(2))
V-«+l
P»(2)= S^vZv (i.e. «s(2)=?>»(2)+p»C2))
Qzrn'(2))' ^ 21 Qv+iy«v+i2v| g S(v+l)!>rv =(the right hand side of (1)),
and
Pn '(Z)
^s O + l)6v+lZv
^s O+l)2rv = (the right hand side of (2)).
12 Sh6tar6 OGAWA
7. Let the function ^~^ be regular and ^~l >0 for \z\<R, then
9?^~Q >0 for |z| <i?, where $?(z)=z+ S byZvis starlike with respect to the origin (and
<P\.Z) V = 2
so of course including the case convex).
PROOF. For arbitrary r (</?), "^p^Q is regular and bounded Qz\ts=r). So for sufficiently- large t, we may assume that if^> h'es m the circle whose center is t and radius is p (p<t) for \z\<r.
Putting |gl =^+^2), (naturally ^(z)|<p , (|Z|^r)) we have /'(z)=^ ^'(2)+fi(2VC2)
and so f f'(z)dz=t [' <p'{z)dz + f ^(z>'Cz)rfz,
Jo Jo Jo
/(2° = t+ Jo
pO)
Integrating along the mapping curve L on the Z-plane of the linear segment 0$?(z) (from the hypothesis the segment OtpQz) lies in the mapping domain of p(z)). we have
fJo gQz Wtz^dz
Kz) < Jo
(0 \<p'(z)\\dz\
i
p\ \d(p(_z}|
eOO
d<P<Lz) I
= P-
P(z) ^ p <t,
So that we have and this implies
L et the function /(z) satisfy 9fc^^-' >0 and f^W^z)/«=o = 1for m ff-l > 0 for arbitrary r<R.
then CD (2) (3)
3t 000 >o
> 1-r
1+y zf<Lz)\
U) < 1+r\-r
**>I ^ IT? (K. L6wnerC12;i) (where ^(2) is a convex function (Remark 1))
PROOE. (the first three evaluations) From Lemma7 we have immediately (1), and from (1) and Lemma 1 we have (2) and (3).
9. Let /(2)=2+ S «vz" and $Qz')=z+ S b-,zvbe regular and satisfy
31izKzjf >0 for|Z|<1, then we have
A note on close-to-convex functions (3) 13
and so for
we have
[«v|<l, (v=2,3, )
r»O0= S «vzv (i.e./(z)=/M(2)+rre(z))
V=M+1
^(2)= S Jv2v (i.e.cS(z)=!inCz)+(0M(2))
for |z|=/*<l respectively.
PROOF. From Remark 2, we have |av|^l. As is well known |6v|^l (Remark 1).
So we have
Z Tn'CzM VI va^z'^S ^J vr"=rn+i
V=W+1
n(l-r)+l Q-Oa , and
Pn(.z) S 2, |6vzv| S 2_, rv =~
v=»+l 1-r .
zfCz) i
for]2|<l, then we have following evaluations with suitable Cv , e, and ex i.e.
10. Let/(z)=2+ S«V2V and <pCz)=z+ S bvzvbe regular and satisfy KSA5J >o
v=2 V=2 VK.Z)
(2a2 =b2-t-ci f Z>3=2e W^i
CD
Z>3=2e ,|^2 (2) \2bs^2b2s+2Sl
=4e2+2ei (3as=bs+b2Ci+c2
PROOF. From the hypothesis and Lemma 1, we can put
, [Cv|^2.
<1
z(l+2a2z+3a-iz2+•E•E•E)_
--i-Ti'l*Ti'2* i "
z+b^z2+bsz3-\-
Multiplying both sides 2 + baZ2 + bsZs + and comparing each coefficient of zv , we have(1).
From Remark 1 there holds 9f -y^y >0. So putting /(z)=^(z) namely putting av =bv in (1) we have (2), where Ci,c3 is replaced by 2e,2ei respectively.
ll. Let f(_Z~)=Z+ S «vZv and «i(3)=2+ S 6V2V (notice Remark 2) be regular and satisfy !R å *^" >0 for [«|<1, then we have following evaluations with suitable cv ,Sv and
e. i.e.
(1)
(2)
4^2 =2&a+Ci 9as =3ba+2b2Ci+c2 16a4=4b±+3b3cx+262C2+cs
25«5 =5*5 +42^! +3*31?2 +2<5>2^3 +^
36«8=6^6+5&5C! +4&4C2+363C3+262C4+c5 263 =2e
6&3 =462s+2ei
1264 = 653s+463si +2e2 20b5=8bis+6bss1+Ab2s2+2ss
30be=10boS +8bis1+6bss2+4b.2ss+2s4.
(3)
14 ShStard OGAWA
bi=-~e!i+±ss1+±e2
o Z b
&5=A£,+|s3si+^E3+i_Es+]L£l2 be=^+^e1+^£2+5ee8+5eet^+~e1s2+2ei.
PROOF. From the hypothesis and Lemma 1 we can put
ifgrlife =1+^+^.+ , ^2.
From this we have (1) just as Lemma 10. From Remark 2 there holds 3?^^~.>0.
So we have (2) putting <2V =6V , Ci=2e, cv =2sv-i in (1). Substituting each equation of (2) into the following one successively we have (3).
Remark 3. Futhermore we can express «v with s,ev and cv substituting (3) into (1), but the detail is omitted here.
4. Proof of Theorem I
1°. It is easy to verify that-j- can not be replaced by any greater constant. In fact, the function .F(z)= y\ \a" = S vZvis regular and starlike with respect to the origin, namely satisfies 9t?|~5i ^>o. Hence FQz~) satisfies the hypothesis of Theorem 1.
For the partial sum F3(z)=Z+223 , 1+2 Fj"^zl=^^ and this value vanishes for 2=--5-
r2U) 1+4^ o.
2°. For/Cz)=/»Cz)+rM(2) and v>(z)=<pn(z)+,oB(2), we have IQs/W I
SR Czfn '(z))' ^ sft (zf(z)Y [^/(z)|-[ y'(z) 14
|P'(«)I~|P» 'C p'(z) +Kzr,/y\
'O)|
For |z|=r<l, substituting each evaluation of Lemma 4,5,6 into the right hand side of above inequality, we can verify that this value exceeds
,-, >. l -6r+2ri -6>-3 +r4
k J Cl-r)3 Cl+r)
n2tl-ry+2n(l-r^+l->~r (1+Q (1+r2) . n «3Cl-r)3+3«2(l-r)2+3«Cl-y)+l+Ar+r*
' (l-r;a ' (l-r)» + ' (l-ry-
1-r _ n2<H-r)2+2nil-r)+l+r ~\
Q +r)s (l-r)»
which is not less than 0.272 for n=4 and \z\ = r =~. Moreover the fact that (1) takes
o
a positive value for n>5 is almost evident. Thus we have
(2) ^^g)'>0forW =j, ^4.
A note on close-to-convex functions (3) 15 It is easy to verify that ^,,'(2)^0 for |z|^4" and ri>2 from the fact that
o
for \z\ < i-and «;g2.(£)(>) means the denominator of (1), i.e. the formula in the bracket
o
C Dof(l).)
Hence by the maximum principle for harmonic functions, the inequality (2) holds also in
So we have only to prove for the case w=2,3 concretely.
3°. (the case n=2) Since the function
is harmonic for jzj^- , we have only to prove
8
By cnonsidering s/Csz) in place of/(z) with a suitable e(|e|=l), the proof of (1) is re- duced to that of (1) with z=-, i.e.
o
st j-~ >o.
Now from Lemma 10 we have
1 + ^r 2(22-6J ci
SR u*4 = 1+3t =i+ m
1 + 4+&2 4+62
1- 4+62A+b2 -^-4ry=0.
4°. (the case 72=3) In this case we have to prove
<KT+262Z+36S^>U tOr |2|< T.
By the similar reason in 3°, we have only to prove
1 9
SR 1 f >0.
1+T*2+646s
From Lemma 10, the left hand side yields
1 1 9 3
1 t. , 3
1 + ~6a+ ^6s 64+1662+3*3
=1 +* igfi^^rir cw^i,i«ii^i,|c1|^2,[Csj^2).
Although ei and cvare dependent on each other, but assuming that s,si and cvare in- dependent of each other and so applying the maximum principle for harmonic functions,
16 Shdtar6 OGAWA we are able to prove this case putting |e[=|ei|=l, (ci|=|c2|=2.
Hence we have next sufficient condition
1> (16+6e)ci4 3c2 (|e|=|ei(=l, Ic\i=\c2\=2),
I 32+16s+3s2|-2|8+3e| -| >0 ( |e[=l), or putting <3ls=x
(1) T(x}= V 1097+1120*+384x2- 2-/ 73+48/- - >0 (-I^at^I).
We can prove (1) as follows.
560+384*
T(x) =
V 1097+1120a;+384*2 V 73+48*
T,,M= 107648 +- 1152 >o.
(V1097 + 1 120a: + 384x2)3 (-\/73 +48*)3
So 3»= T(jt) has a graph which is convex downwards. On the other hand,
T(-l)=19-10- J- =4.5,
T'(-l)= ^-f =-0.337.
So cosidering the tangent drawn at the point (-1,T(-1)) on the graph of y=T(x), we have
Min T(a0>T(-l)+T'(-l)x2=4.5-0.674>0.
Thus the proof of Theorem I is completed.
5. Proof of Theorem II
(The main point of the proof is similar to that of Theorem I. So in places we show only the brief sketch of the proof.)
1°. For the first we verify that the constant ---can not be replaced by any greater one.
In fact the function 7-^- = S zv satisfies the hypothesis of Theorem H. For the partial
1 Z v=l
sum fzCz)=Z+z2, we have
z/g'Cz) _ 142z
/aO) 1+z
and this vanishes for z= -
2°. Now we consider the following inequality:
zfn 'Qz) z/OO /0» (z)
tfO»\ ,
ZVn'iz)
^ yj + |*ww f/t»=/. oo+>-» oo
co ' ~> va
O)=jS» O0+P» (z)
y.i.k^j rvj i tk^j ] - i fn\*-j
A note on close-to-convex functions (3) 17 We can evaluate each term of the right hand side by Lemma 8,9. Thus we have
mzfn'(z) > \-r_ _ r"(l+r)Ml-r)+2+r) ,, _r^-n
^i(z) - 1+r (l-r){l-r-r"Cl+r)} ^-r^-U
The right hand side takes a positive value for n^7 and ir=\ å So we have to prove for each case n=2,3,4,5,6 concretely.
3°. (the case n=2) For this case we have to prove
*f3?=*nS>° for^<1•E
With the same reason in 3° of 4, the proof is reduced to the following, i.e.
3? 1+as
1 +-rf ^0.
From Lemma ll, we have
SR l+a2 1 +
b2
= i+ m 2a>-b22+bt ^ >1-4+2&3Cl ^>1 _21 -04-2 "U- 4°. (the case n=3) For this case
tozfs'jz) _ m l+2a2Z+3a-iz2 _ -, , sy? (2gg-62>+(3as-&3>2
So it is sufficient to prove
i+62i;+6sz
1^
-V(2a3-&2)+ ~ (3a3-68)
1+1 **+ }*>
or applying Lemma ll ,
i:> C3 +_262)ci + C2_
I2+662+2622
By the maximum principle this sufficient condition yields (N^l, |cvl^2).
yields CN=D,
T(x)=V34+42x+24x* - -/13+12.V - -y>0 Q-l^x^l).
)ve this inequality as follows.
7*00 =
6+3e+s2 |-! 3+2s |-4>0 or putting 3fs=x, we have finally
In this case we can prove this inequality as follows.
21+24*
-/34+42X+24*3 375
Vl3+12x
T"0O=(>/34+42x+24xs)3 +----(/l3+12x)336 >o
T'C-y)=-0.203, T'(0)=1.937, T(- -|-)=0.213.
So considering the tangent drawn at the point (å --, T(--)) on the graph y=TQx), ehave
Min 0X^)=Mm T(x^>TC -i)+-l^(-i-)=0.213-0.102>0 .
l I Z 2
18 Sh6tarfi OGAWA
5°. (the case «=4) For this caae
jv. zU'jz) _m l+2a2Z+3atszz+Aa4Zs _ -å , m (2g2-62>+(3a8-fes)z2+C4a4-64)z3
H-62Z+Z>32:2 +£*£3
So we have following sufficient conditions putting Z= \- and applying Lemma ll (1),
1 >
1 1 2 1 1 3 1
1 +Y&2+~&3+Y&4
or applying Lemma ll (2),(3)
|48+24e+8e2+2e3+(4+3e>i+£3 [
> lC12+8e+3e3+|-eOci+C4+3e)c2+-|c8| , |s|=|ev|=l, [cv|=2, or
48+24e+8ea+2e8-K4+3e>i+e2 l~l 24+16e+6e2+3ei |-J8+6sr)-3>0.
Now there holds
24+16e+6e3+3ei = {48+24e+8e*+2e'>+(4+3e>i} (y+y^O
hence we have
-c|+-i->'+ci-T--H>1 '
{|48+24e+8s2+2£s |-]4-|-3e]> •E{!- |\+~z\}
3 6
as a sufficient condition.
Putting yte=x, this yields
",-U» -|8+6s!-4>0 01=1),
T(#) ={2v/ 521 + 536#+432#3+192ff8-V25+24#} {l-iv/37+12ff}
-^v/101+20*-^a/709-396:j;+144x2_2a/25+24#-4>0b lz (- l^x^ 1).
We may prove T(x)>0 by a elementary method as follows.
We notice that each function of ^4(x)=2 ^521+536^+432^2+192x3-V25+24#,
-B(#) = --v/lOi+20^, and D(#)= 2-/25+24i is monotonic increasing with # and each
6
of P(aO=l--^-v/37+12^ and C(ar)=^- -/709-396«+144ar2 is monotonic decreasing w ith So we can estimate the lower bound of T(x) at each partial interval as follows.
For instance
TXx-)> ^C-l)P(-|)- JBC-|)-CC-l)-JDC-|)-4
=16.202-13.421 >0.
Similarly we can justify T^x~)^>0 at the intervals
A note on close-to-convex functions (3) 19
These proofs are elemental, so the details are omitted here and we only note some values of T^x) for reference, i.e.
_h-,
2T1)=10.339, T(0)=2.151, TIC-±-)=1.843, T(-^)=1.898, T(-l)=6.832.
6°. (the case w=5) For this case
mz/s'O)_m 1+2a2z+3asz2+4cnz3+5g^z4
= 1. to (2g2-62>+(3g3-63>2+(4a4-64)z3+(5a5-6s>4
"*"""å l + baz + fcs^ +ft-^ +fis^4
By the same reason as the foregoing cases, we have following sufficient conditions, i.e.
putting 2=- and applying Lemma ll ,
1>
1 +y62+|68+|64+^6.
240+120s+40£2+10£s+2£4+(20+15s+6e2>i+(5+43>3 +-!(Sl2+£8)
{60+40e+15e3 +4e8+(^+6e)ei15 +2e2}ci+(20+15e+6e2+3ei)c2+(±£+ 6e)c8 +3C4|
C |e|=l, |ev|=l, |cv|=2).
Consequently we have the following sufficient condition, i.e.
| 240+120e+40e2+10e3+(20+15e+6s2>i+(5+4e>2
120 +80e+30e3+8e8+(15+12e>i+4es|-|40+30e+12ea+6ei | -i 15+12e |-ll>0.
Now we notice next relations, i.e.
120+80e+30e2+8s«+(15+12e>i+4s3
={240+120£+40£2+10e3+(20+15£+6e2)Sl+(5+4e)£:,} (1+^s)
40+30e+12£3+6ei
={240+120£+40e2 +10e3 +(20+15s+6e2>i +(5+4e>2} C-l+J-e)
6 24
Hence we have the following sufficient condition,
{j240+120e+40e2+10£s|_[20+15£+6£2|-|5+4e|} {1-| i+^e[-] -^+^e |}
>o
-Ij+f* *+?-?•E•EHf-!f.~Hi i-f-^- l
T -T-!•E''- -|5+4«|{3+l|+i«|}-ll || >0.
20 Sh6tard OGAWA
(the term -f~T e~¥ £2-7£3 is modified for the convenience of calculation.)
o o o 4
We denote the left hand side of this inequality with T(x) as a function of x=^Rs. The fact that TQx~)>Q for -l:S#<:i can be shown by the same way as the preceeding case.
For instance, at the interval O^xg-
4 ,
Kx)>A(0)P(±)-B (i-)-C (^|)-JDC0)-JB(|)-FC0)-GC|)-ll.792
=58.648-57.761>0,
where A(x)=\ 240+120e+40e2+10es |- -|5+4e|
PW=^^-i-It+s-i >bc*)=i|+|.i.
G(*)=|5+4«|{3+||+i«| }.
At other intervals -1<>x<--,--<X< , , -<X<1, wecan verify TO)>0
4 4 2 4
as above. The details are omitted. For reference we note here some values of TOO, i-e-
TC1)=21.572, T(i-)=9.763, T(0)=9.433, T(-j)=11.508, T(-l)=36.558.
7°. (the case »=6) For this case
nj zfe'jz) _gx 1+2a2Z+3asz2+4QAZ3+5ar,z±+6a6Z5
= 1 j. SB- (2a3-62>+(3g8-63>2+(4aA-64.>3+(5«5-6:-,)z;t+(6a<i-6n)z5
A I J.__2li_A._3_l_A._4 i_"A.._5
l +b2z+bsz2+bi:Zll+br,z'l+bt;za
So it is sufficient to prove 1>
Y (202-62)+~ C3as-bs)+~ Ua4-bi)+j^ (5a5-65)+^(6ae-6e) 1 +^-b2+~bs-i- ^b4+^br>+^,bl16"
32
Or applying Lemma 11C1).
>
l+i-68+4-6s+-s-6*+^6+i b
16 32
t6»ci +~ca+~bsc1+±b,c2+±cs+±bici+^bsci+±bscs+^C4
12 gCi 16 32
4Cl
40 80
+ W*«Cl+^3+^«+^*»^+W192""""- ' 48" 64 2C<L C*
or applying Lemma ll(2),
720+360s+120e2 +30e3 +6s4+e5 +(60+45s+18£2 +5es>>1 +(15+12e+5£3>3+(-§-+jO es+(f-+f£>i3+f-sis2+fe4
> {180+120e+45ea+ 12es +^+C~+lSs+f£^e1+C6+5s')s1+fes+fs1^} d
+ {60+45£+18e3+5e3+(9+15£)£l+A£3}C3+{^+i8s+i|S3+^£l} c8
.15. 15
+ (9+^S>4+^ Cb
A note on close-to-convex functions (3) 21
Hence as a sufficient condition we have
720+360e+120s2+30£3+6e*+(60+45e+18s2+5e3>1+(15+12s+5e3>2
+ Cf+^«>8 +(f+f«>X
360+240e+90e2+24e3+5£*+(45+36£+15e2>i+(12+10e>2+ ^es+-si2
4 4
120+9(te+36e2 +10s3 -«18+15e>i +5e2 45+36e+15es +^ei 18+15e| -i4>o.
Now denoting
^(£)=720+360s+"-+(60+45£+-"+5e3>i+(15+"-+5£2>2+C-|-+^s>s+(-^+i5e)e13
2 4 2 4
15. 15
B(s)=360 +240s+ +^£s+j£i2
C(s)=120+90£+ +5e2
JD(e)=45+36e+ +yei-
we notice next relations, i.e.
C=^xC|+iT£)+Cl-Ae3-|£3>3+(8+5£_^£2_^£3_Aeit>1
24 12 24
+ C-I-2-4*' -^3>-c|+^xU^>8- 4+^xl+^>24 >,- 24 ISll
_i(15+12.+5..>,- i c |+^>.-i(|.+^>1..
So we have as a sufficient condition,
{I^h^H^i-^J} {i -) i+i-sj-|X+J-£|_l}1
2 24 16J
_| ! +i£+j£2 | _ ,15+lZ£+|£3_4£3-A£,Hl+^-|£-A£3
3-f£-jes Hl-^-> H8+5£-f£3-^-J^
-ii-¥*-le3-24T+T e| f+£-
1 2^2 8 8 4 16 8 16
-i | 15+12e+5e2 ) _i_ JJL+I5 e ;-|18+15e|-12-^
Id o *i 4 2
å 0.
Hence we have finally,
Sh6tar6 OGAWA
(1) UA^-lA.HAsl-^A^} {|-\\+±s\-l|+^l >
-Il_!*»|-|8+5«-5c3 |-| j-fe-fe3 ' 2 2 15.
Representing the left hand side with T(x) as a function of x=9is, we can show IX^)>0 for -l<Lx<ll by the similar method as preceedings.
For instance, at the interval -<at<1,
{| ^1-1^,1-1^,1-21^41}{|-I|+is |-||+^£ I} =A0(x) •EP(x)>A0 (|) •EP(l)
=131.536 ,
where A0(x) denotes \A±\-[A2\-\AB\-2\A±\ and PQx) denotes j|- 2+12 'e+24 as the function of x, while the sum of other terms in (1) can not exceed 131.231.
By the same way we can justify X(#)>0 f°r -1^X<1. Merely for -<x^-j, a some detailed examination is needed.
In this case A0(x)P(x)>A0(±)P (-|-)=134.179, while the sum (=S(») of the maxi- mumvalues of each other term is equal to 134.511. So we remark the term
3 I 6+5sl {^+j-+S7i £}=£>O). and | 8+5s-5e3 | =R(x). (?(*) is monotonic
32 6 Z4
increasing and RQx^) is monotonic decreasing in this interval. Since
<?(-§0=33.614, Q(-|)=34.938,i?(i)=13, i?(-|)=12.258, hence for \<X^,^- S(x) can not exceed
134.511-34.938+33.614-133. 187, and furthermore we have
T^x) > ^o(J-) - P (4)-133.187>^o C|)P(i)-133.187=134. 511-133. 187>0.
5 a o 4 / 4
For ^<^<~ , S(x) can not exceed 134.511-13+12.258=133.769.
2 o
Thus we can verify T^x)>0 for this case. For reference we note some values of 2X#):
1(1)=39.180, :ZTC-|)=27.179, T(|-)=25.139, T(^-)=27.970, T(-l)=104.111.
Thus the proof of Theorem n is completed.
I wish to express my hearty gratitude to Mr. Shigeyoshi Watanabe for his assistance to these computations.
A note on close-to-convex functions (3) 2!
References.
(1) Y.Miki: A note on close-to-convex functions. Jour. Japan. Math. Soc.8 (1956), pp. 256
-268.
( 2) W.Kaplan: Close-to-convex schlicht functions. Michigan Math. J. 1 (1952), pp.169-185.
( 3 ) G.Szego: Zur Theorie der schlichten Abbildungen. Math. Ann.100(1928), pp.188-211.
( 4 ) A.Kobori: Zwei Satze fiber die Abschnitte der schlichten Potenzreihen. Mem.Coll.Sci Kyoto Imp. Univ. A,17(1934),pp.171-186.
( 5) S.Ogawa:A note on close-to-convex functions (2). Jour. Nara Gakugei Univ.8(1959),pp.
ll-17.
(6) A.W. Alexander: Fuctions which map theinterior of unit circle upon simple regions. Ann.
Math.17(1915),pp.12-22.
( 7 ) M.O.Reade: The coefficients of close-to-convex functions. Duke Math.J. 23 (1956),pp-459
-462.
( 8 ) G.P61ya-G.Szego: Aufgaben und Lehrsatze aus der Analysis.I.(1925), p. 140.
( 9 ) C.Caratheodory: Uber den Variabilitatsbereich der Koeffizienten von Potenzreihen welche gegebene Werte nicht annimmt. Math. Ann. 64(1907),pp.95-115-
(10) R.Nevanlinna: Ober der schlichten Abbildungen des Einheitskreises. Oversikt av Finska Vetenskaps-Soc. Forh. (A)62(1919-1920), pp.1-14.
(ll) R.Nevanlinna: tJber die konforme Abbildung von Sterngebieten. Oversikt av Finska Veten- skaps-Soc. Fr6h. (A)63(1920-1),pp. 1-21.
(12) K.Lowner-' Untersuchungen uber die Verzerrung bei konformen Abbildungen des Einheits- kreises [z| <1, die druch Funktionen mit nicht verschwinder Abbildung geliefert werden, Leipziger Bericht,69(1917),pp.89-106.
