奈良教育大学学術リポジトリNEAR
A note on close‑to‑convex functions (2)
著者 OGAWA Shotaro
journal or
publication title
奈良学芸大学紀要
volume 8
number 2
page range 11‑17
year 1959‑02‑15
URL http://hdl.handle.net/10105/4837
(11>
A note on close-to-convex functions ( 2 )
By Sh6tar6 OGAWA
(Received October 25, 1958)
I. Introduction.
In this paper we shall show other proofs of theorems concerned in the partial sums of schlicht functions and also that of close-to-convex functions.
A. Kobori QJ and Y. Miki C23 proved the following theorems.
THEOREM 1. (A.Kobori) Let the power series
*>(z)=z+S 6vzv
v=2
be analytic and schlicht in the circle |zj<l and starlike with respect to>
the origin- Then any one of the partial sums
an(z')=z+ 2j 6v2v, (»=2,3, )
V=2
is schlicht and convex in the circle jzj< - o . The constant - cannot be 8 replaced by any greater one.
THEOREM 2- (A. Kobori) Let the power series
<p(2)=Z+ S ftvZv
V-2
be analytic and schlicht convex in the unit circle |z!<O- Then any one:
of the partial sums
<r,,Cz)=z+S6vZv, n O=2,3, )
å v-V
is schlicht and starlike with respect to the origin in the circle |zi<- - The constant - cannot be replaced by any greater one.
THEOREM 3. (Y.Miki) Let the function
/(Z)=Z+S «vZv
be close-to-convex* for |z|<l with respect to the function
<^(z)=z+S 6vZv.
Then the n-th partial sum
Sn(z):=Z+'S «vZV (>=2,O, )
* we call an analytic function /(z) to be close-to-convex for |z|< R with respect to <P(z), if there
exists a function?>(z), convex and schlicht for [z|< R, such that /'(z)/ip' (z) has positive real part
for(z|</?.
< 12 ) Sh6tar6 OGAWA
offQz) is also close-to-convex for |z|<- with respect to the n-th partial
4
sum
n
<rn (X)=2+S JvZv
V-2
of <p(z)- The constant - cannot be replaced by any greater one.
The principle of the proofs of these theorems is as follows. For example in the theorem o next inequality holds, that is,
j R^) > 9{/'(,) Jp!HIS +\r'n(2)\ Ift*)=5B (2T)+rn(z) \
V (.*)=>On (Zi+PnieV ,
( /(-
W
<
rV(z) ' <P 'Gz) \ <P '(z)H(>'» CzX
where 31 "p^>0. /(z)«z+2z3+3z3+ , <pCz)«z+z^+z^+
So we can estimate each term of the right side of the inequality. Thus we know when n^4 the right side is positive for Izl<-. Therefore we have to verify concretely for the case n=2,3å
Kobori's theorem is proved by the similar way, and for this theorem it is needed to show concretely that the statement is true for the case n=2, and for the theorem 2,n=2,
3,4.
Though the proofs of these theorems for concrete cases are very skillfull, it seems to us that according to these methods there are many difficulties to extend these theorems to the general problem.
So we shall show here other proofs of concrete cases i.e. n=4,3 in the theorem 2 and n=3 in the theorem 3- (at these theorems the proof for the case n=2 is trivial.)
One of the authors already announced at the general meeting of the Mathematical Society of Japan (October 25, 1958) that we can extend Kobori's theorems 1 and 2 to the close- to-convex functions by the same method we shall show here, and the detailed contents of the announcement would be published on another occasion.
jf. Lemmas.
Following lemmas are needed in the proof.
1'. Let the function ^(z)=2+S b-, 2V be schlicht and convex for |z[<l, then the coefficients satisfy the following equalities with suitable ei (JeibsLl) and e2 (leafeil ),
(1) b-2=b (|6S<1),
262+Si
(2) bs=
(3) b±=
3 2ft3+3hei+ ®2
6
PROOF. From the convexity of <p(z) we have
* Ci+«f3g)>0.
A note on close-to-convex functions (2) ( 13 ) Hence, by Carath6odory-Toeplitz's theorem, we can put
)_ 2b2z+6ba z2+ 12b4,z s + ) ~ l +2b2z+3baz2+Abtz3+
= l -\-diZ+d2z2+dsz&-\- , \dv\^2- 2 (p' Qz}~ l+2b2z+3baz2+Ab4,z3+-
On the other hand,
2b2z+6&s22 + 12b4.zs + -
1 + 1 +2b2z+ 3bsz2 + 4b4zså å
=1 + (2b2z+6b3z*+126*23+ )
x {l-(2b2z+3bsz2+4b4,z*+ )+(2b2z+3bsz*+ >- }
=1 + 2b2z+(Qbs-4Z>f>3+(12*4-1863&3+8*J>3 +
So, replacing d\-2b, d2-2s\, ds=2s2, and reducing bs tob, b± tobs, wehave finally (1),(2) and (3).
2'. Let the function f(z)=z+ Savzvbe close-to-convex for |z|<l w«i& respect to the function ^(2)=z+S 6vzv , then with suitable <?i (l^il^l) and d2 (l^al^l)
*?e^if equalities hold, v-2
(4) a2=b2+3i, (5) 3«s=3&a+4&2<5i+2£2.
PROOF. From the definition we have
Hence, by the same reason in 1', we can put
1^^*^=1+2^+2,,,.+ , (,,^i).
Sowehave
l+2aaz+3aaz2+ =(l+2622+36sz2+ ) (l+2512+252za+: ).
From this we have (4) and Co)-
1i. Proof of the theorem 2 for n = 4.
Weshall prove 3t?^^>0 for \z\<\.
Now,
^ ztf'tO) _ ml +2?>2Z+3l>3Z2+4b4.z3
-26*z3
=2-5R
>2-
l+b2z+b3z2+biz3 l -bsz2-2b4,zs
l +b2z+bsz2+bizs
That the denominator never vanishes is easy to verify. So by the maximum principle we may prove with \z\ =-. Furthermore, considering e/(ez) instead of f(z) with suitable s (|e|=l), the proof is reduced to the case with z=-. Thus it is sufficient to prove
4-&S-&4
1 >> 8+Ab2+2b8+b±
( 14 ) Sh6tar6 OGAWA By CD.C2) and (3)
)_2&2+ei_2ft8+36si+e3
A_». t..
*±-i/g -U4=
8+4&2+26g+64
0+40+ ~ r ~
o b
24-4b2-2*s-(2+36)ei-e2 48+24*+8*a+2/>s+(4+3*>i+s2
Again by the maximum principle it is sufficient to prove with \b\=l, |ei[=l, and |ea|=l- So finally we have the sufficient condition as follows.
TC6)=2 |24+126+462+^|-4 \Q-b2\-\4+3b\-\2+3b\-4>0, (|*|=1).
Or putting 916=x, we have
C6) T(x^2(521 +53Qx +432x^ +192x^2 _ 4C49-24^2)V3 -(25+24^)^
- C13+12x)V2-4>o, (-l£*£i).
From (6) we have
("} T C -)= 8C67+108X+72JC3) 96« 12 6
and
f S") T"r A=§^iZg9+9378^+4824JC2-t2592^s+964x4)
1 ^ U; (521+536*+432»3+192jc3)?2
4704 , 1 44 , 36
"faa_oa-i-2^%"f"foc_i_o^vMSi"I"/å (49-24jc2)% (25+24x)8/2~ra3+12x)% '
Now for 0 2S*2S1, considering the maximum or minimum of each term in this interval, we have
Tir^ 8x67*l 12 6
o
and for -1^x^-- 4
^C'X^*'-7fc-^-f=^-><»-
Consequently, we have the minimum value of T(x) for -l^^^l in the interval
Moreover, from (8) we have for -r-^^^0
r ;/f^>_64x3Jl:2^+4Z04+i44 , 36
1 W-> (,/281)S +?8+53+G-13)3 1U'84 U"
*i At the first term (for convenience let us put this term a'(x)=/>(x)/^(x)) on the right side of (8), pix) and q(x~) are both monotonic increasing and />(0)>0,<7 (-1)>0. Therefore o'(x)>0 for xS0: Thus the first term (let us put this terma (x) ") on the right side of (7) is monotonic increasing for xgO.
3 3
*2 P( )<0. Then j>Cx)<0 for -l^Lx SJ- -. This implies that a'(x)<Q and a(a) is monotonic
4 4
decreasing in this interval.
3
*8 In this interval p(a;) is monotonic increasing and p( )> 0,/>(0) >0, and?(x) is positive and 4
3 3 3
monotonic increasing. Thus the Min (/>(-<0/o(*) ( S^xSjO) is equal to if--)/g( )å
4 4 4
A note on close-to-convex functions (2) CIS) Nowby (7)
T ( lv_8x31_ 48 12_ _6_ -_
1K 1}~ ^337 v/43 >/l3 ^/y-"-09'-
So that we know that the MinT"(A-) takes place for --^x^--.
By(6)
T(- y) =2i/33T-4v/43-/i3-/ 7^-4=0- 236-
For --^x^-- , noticing T"(jc)>0, we have
TW>T(-|)-C-i-^)T'(-|).
Therefore considering the case x=-~7at above inequality, we have
Min T(x^>TC-h-~T(-h =0-236-~l>0.
-l^ar^l 2 4 2 4
Thus the proof is completed.
I2. Proofofthe theorem 2 for n = 3.
tt _i__ii mstfs'(z)^ ^ r._ 1.1 ^ 1
nere we snail prove Jt f~^-yj ior |»|»
We have
m
zO'sjz) 08 O)"
<78<»
' l+bzz+bgz2
=2-3*
^2-
l+b2z+bsz2 l-ftsz2 l+bes+bsz2
As the case n=4, we may prove reducing this case to that with z=-, \b\=\ and ]ei|= 1- So it is sufficient to prove that
2> 4-*s Or from (l),(2)
2>
4+2*2+63
2(6-&2)-si 2(6+3&+&2)+ei or
T(/>)^4 \e,+3b+l>2l-2 |6-£2|-3>0 (1*1=1).
Putting ^Rb=x, we have the following sufficient condition that
( 9 ) T(*)=4-\/34+42a-+24x2-2v/49-24xS!-3>0 (~!^*=l)- From (9) we have
19f7 4-S^^ 48x
( 10) T(x)=
fOA_LAOvA-
OA^2\ */£ +?
and
By (10)
(34+42c+24a:2)/^ (49-24*2)V2
n -n T"r i- ^5 2352
UU W (34+42*+24*2;%+(49-24*3)%
( 16 ) Sh6tar6 OGAWA
T'C-|) =7fe-7j3 =4.59>0.
r'C-f^T-^= -3-M<0-
Therefore considering (ll) we know that the Min T(^) (-l^^^l) takes place at the
O 1
