Sasaki, Yasuharu

University of Fukui Repository

On some Family of Multivalent Functions. 
On some Family of Multivalent Functions .. 
Yasuharu SASAKI 
§ 1. Family Ep of regular functions. 
Let the function 
ao (1.1) F(z) = 1,' anzn• (ap == 1) 
R=P 
be regular in the unit-circle. where p is any positive integer. (1) 
We denote by Sp or Kil respectively the family of functions Fez) by which the 
unit-circle is mapped into a star-like region with respect to the. origin or a Convex 
region, then the following theorems are well known. C2J 
Theorem 1. 
The necessary and sUfficient condltion that FC z) should belong 
to the f amz"ly Sp is 
R C z F' (~)J > 0 ( I z I < 1). F (z) 
Theorem 2. 
The necessary and sufficlent condition that F(z) should belong 
to the /amlly Kp is 
l+R [zF,,(z)J"l - p-1 R [z~'(z)J >0 
F' (z) p F (z) (I z) < 1). 
Now we denote by Ep the family of functions F(z) having the following two properties. 
1 Q The mapped region of I z I < I by F(z) is p-valent. 
2° At any point of the mapped curve of I z I = r. where 
r is any positive number less than 1. by F(z). the 
curvature is positive and finite determinate. 
And we call F(z) a quasi-convex function. We can understand. by this definition. tha,t 
a quasi-convex function F(z) is convex on Riemann surface of F(z). Then we have 
Theorem 3. 
Proof: If 
then 
and we have 
The necessary and sufflct"ent condition that F(z) should belong 
to the family Ep t·s 
l+R (z Ff!(z)] >0 (I' 1) F'(z) z i < . 
l+R Cz F"(z)] == R [z (zf~.(z)'j"l >0 
, F' (z) zF' (z) • 
eF' (:?J = P =\= o. zP 1 z=o 
Thefore F'z~~ does not vanJsh in I z I < 1 and F'(z)=\=o in 0 < I z I < 1. (3) 
Now we denote by p the 'curva ture at any point on the mapped curve C of I z I = r 
(0 <r <1). then 
- 1 [F,,(z)J 
p - 1 zF' (z) 1- R l+zF'Cz) >0. 
which is the property 2°. 
As F' (z)=\=o on I z 1= r, C is a regular curve and the angle from the real-axis to the 
tangent line at any pnint on C is given by argizF'(z). ~o that we have. as z describes 
on I z I:::: r, J dargizF' (z) = J dargZP + J dargF~~zJ. = J dargZP = 2prr, 
F'(z) 
where -----zr-t does not vanish. as cited above. 
Therefore the mapped Curve C is closed and p-valent, and. being r arbitrary. the mapped 
region of I z I < 1 is p-valent which proves the property 1°. 
Conversely. if we have properties 1°, 2°, then the curvature 
- 1 r F"(z)J 
p - 1 zF'(z) I R 1,1+z F ,(z) 
at any point z on I z I = r (0 <r <1) is positive, and, being r arbitrary, we have 
(I z I < 1). 
Thus our theorem is completly proved. 
§ 2. Relations among Sp, Kp , and Ep , 
If Fez) belongs to anyone of Sp, Kp and Ep. then Fez) is p-valent in ! z I < 1, and 
consequently F(z) does not vanish in 0 < I z I < 1. Hence there exists the regular function 
h(z) such that 
h(~) = z P /F(z) F(z) = Chez)]p, heo) = 0, h'(o) = 1 (I z I < 1 ), 
'V zP , 
and we have, for h(z), the relations 
R [z F' (Z)J' = pR [z ~' (z)J F (z) h (z) 
[ FII(Z)] _ l" h"(z)] .--- h' (z), (2.1) 1 + R z F'(z) - l+R z h'(z) + (p-1) R l z h (z).) 
!,1 + R [z ~~~:~J - P;l R [Z~' ~:~J = l+R [ Z ~~~:~J . 
We get immediately, from these relatins, the following 
Theorem 4. 
Suppose that F(z) ESp, then h(z) E S1 and suppose that 
F(z) E Kp. then h(z) E Kt. 
If we assume that F(z) E Kp and therefore h(z)eK1, then 
R l ...... z h' (z)J >l 
h (z) 2 
On some Family of Multivalent Functions. 
by the theorem due to M. Strohacker (4J. and. from (2,1), 
C 
F"(z)'l 1 
l+R· zF'(z)J >2 (p-l) ~o, 
This proves the following 
Theorem 5. 
R r z F' (z )J >-E 
, F (z) 2 . 
Suppose that F(z) E K p, then F(z) E Ep and F(z) ESp. 
3 
Next we assume that E(z) E Ep and denote by tp the angle from the real·axis to the 
tangent line at F(z), where Z = re i9, on the mapped curve of I z I = r, then 
dtp _ d ., _ r F'll z)l 
dO - dO ar.glzF (z) - l+R ,z F' (z)"" > o. 
So that the tangent line rotates such that tp increases as Z describes in positive sense 
on I z 1= r. And. as the curvature p is positive, the radius vector F(z) rotates in positive 
sense. Hence we have 
R ( z F'(zlJ 
l. F (z) :0 arg F(z» o. 
which reduces to the following theorem. 
Theorem 6. 
Suppose that F(z) E Ep, then F(z) ESp. 
We can represent the theorem 5 and 6 symbolically by 
and, especially for the case p= 1, 
by (2.0. 
Lastly we have the theorem as follows, from the relation 
R C z (zF'(~J = l+R lr z FIf(z)J 
zF' (z) . F' (z) 
Theorem 7. 
if F(z) E Epo then ! zF' (z) E Sp and 1/ F(z) ESp, then 
p JF~) dz E Ep. 
§ 3. The circle of quasi-convexity for tha functions E Sp. 
Suppose that F(z) E SI" then 
R [ z F'(~J >0 F (z) , 
and 
[ .F'(z)] z--· =p F (z) co . 
Therefore, 
(Iz[~r), 
ahd we have, by the well known theorem of G. Julia, 
4 
! Z ~f~CZ) _ Z F' (z) + 1 i L 2 _, z_l_ R [z fffiJ F' (z) F (z) i = 1- \'z!Z I z ~r ~:~ I 
Combining these inequalities, we get 
1 + R [Z F"(z)] :::::,. prZ-2(p+l)r+p 
F'(z) = 1 - rZ C)zILr), 
and hence, for Izl< all ~ ~ (p+l-V2p+1). 
r F"(z)J 
1 + R l z F' (z) > o. 
and that, for the function 
we have 
1 + R· [z F"(z)J' = 0 F'"(z) 
Thus we conclude the following 
Theore.m 8. 
1 --Suppose that Fez) ESp, then F(z) E Ep for \ z 1 < ap = p(P+l-J'2P+l). 
And ap is the greatest number for quasi-convexity. 
§ 4.' Distortion theorem and Coefficient problem for Ep• 
For the function Fez) E Ep , we have, by the theorem 7, 
~ z F' (z) E Sp , 
and consequently, 
I zip I) t I I Z \ll (I +1 z 1)21' ~ P z F, (z) J L (1-1 z I )ZP • 
From tMs relation, we can prove the following theorem by the analogous method for 
the case p =:; 1. 
Theorem 9. 
Let Fez) = l' a"z71 (ap=l) E Ep, then 
n=p 
And the equality sz"gns are true lor 
J z Zp-l F(z) = P 0 el-zYP dZ E Ep. 
We have already known, for Fez) ESp, that 
On some Family of Multivalent Functions. 
. zP 
and the equalzty holds for F(z) = (l-z)2I)' (5J 
We can represent this theorem, by Majorant symbol. as follows. 
Zll 
F(z) -< -(-,;-:l-_-Z----=oy-ll. 
Now, if F(z) E EQ, thenl ZF' (z) E S" and therefore p . 
1 Z" P z F'(z) -< (l-Z)2P • 
From this relation, we have 
j z Zp-l F(z) -< p 0 (l-z) p dz 
i, e, 
F( ) e- JZ [ p-l + ).~ 2p(2p+l)-··(2p+ k - 1) z 'Z." P, 0 Z K=l k ! 
Consequently the following result is obtained~ 
Theorem 10. 
Let F(z) = .1 a"z" (ap= 1) E Ep, then 
u=p 
Z p+K-l J dz 
I a I L 2P(2P+1)···(2p+k-1) (k=1,2, ••. ) 
p+k = k! (p+k) , 
and equaUty holds true for the function 
J% ZP-l F( z) = p 0 ( 1-Z )2P dZ. 
§ 5. Some lemmas. 
5 
In this section, some inequa~ities, all of which hold for I z I L r, will be given for the 
preparation of next section. 
Let F(z) be bounded (I F(z)! < M), then we have 
Lemma 1. 
mrp1- Mr L.I Fez) I L MrJll+ Mr M-r = = M+r' 
Now putting cp(z) = Fi~), then i cp(z) 1 ~M, tf(o) = 1 and therefore 
I,· £()'(z) I,' ~ M2 - I so(z) I 2 
,.. M(l-r2 ) • 
i,e, 
Combining (5,1) with lemma 1, we have 
Lemma 2. 
I F' (z) I ::::::,. Mr p-I p(r). 
, = (M-rY~' 
where per) = pM - {(p+I)'M2 + (p-l)}r + pMr2. 
and 
Lemma 3. 
M2 r 2p 
I J(z) I ~ (M--r)2 (1- 2Mr+r2 ), 
where J(z) = zF' (z) - (p-1) F(z). 
From Lemma 1, 2. (5,1), we get 
Lemma 4. 
,where J.!(z) = M2 l' 2(~_=1) I F(z) i 2. 
and 
Lemma 5. 
Next we have, applying (5,1) and lemma 1 to 
I J(z) [ _ I F(z) I [F' (z) I -=M---=-r-P~-l-"-+ I-F(z)! - MiP-=i-+liCz)T I Z FCz) - (p+1) 1 , 
Lemma 6. 
__ I J.(z)J __ ::::::,. r(1-2Mr+r2) 
Mr p - l + I Fez) I == (1-1'2) (M-:"'r) . 
LasfIy lemma 2 and 4 conclude 
Lemma 7 . 
. The equality sign in lemma 1 """ 7 holds at z=r for the function 
F(z) = MzI) 1-Mz . 
M-z 
§ 6. The circle of quasi - convexity for the bounded functions. 
Let Fez) be bounded (I F(z) I <M), then the auther has proved [6) that the circle of 
convexity of F(z) for KI =: EI is given by : z! < PI. whre PI is the positive root less than 
I of the equation 
M - (4M2 - 1) Z + 3 MZ2 - Z 3 = o. 
Now we shall generalise this theorem for the functions of Frl • 
If we define the function ¢(z) regular in I z I < 1 by 
zp-1F( w) - wP-1F(z) s 
¢(z) = M2 - -' W = = .. + z. (z I < 1) Mil Z!l-lWp - 1 _ F(z)F(w) 1 - z S I , 
On some Family of Multivalent Functions. 7 
then I ¢(s) I <M, ¢(O) =0 and, by the theorem of bounded functions, 
I ,f.,' (0) 112 1 M- 'f' M ~ 2 11/'(0)1. 
Thie inequality is reduced to 
( (p- I)M21 z 1
2(1'-1l_ zF(z)F' (z) A (z) J v(z) 
+ 2R ).I(z) • zF' (z) - 2M(1- rll)ll i z 11'-2 J F' (z) I • 
where F' (z)=\::o and ).(z)=zF' (z) -(p-l)F(z), ).I(z)=M2 1 z 12Ip -': 1 ) -I F(z) \ 2. 
The fourth term of the right-hand side in (6.1) is reduced to 
2 R r ( -1- F(z)l.(z) ) _ ).(z) 'I 
" p ).I(z) zF' (z)J , 
which is not less than 
2 (p-l)R [z~~(~)J - 21 F(~~z~(z) I ! z~\ZZz) I , 
and 
F(z) ).(z) 
(p-l) zF' (z) = 1 - zF' (z)" 
Therefore we have 
(6.2) R [ z ~:2~SJ ~ p -2 + Ci~~ ~ll:2 +p J R Cz}>~~)J +21 z~\t{z) [ _~ir:St~i F(z) I 
).I(z) 
- 2 M(I- 'z :2)2: z 11'-2,F'(z)[. 
ApplyIngl emma 1 ...... 7 in § 5, the inequality 
(6.3) 1 + R ( Z F'I (z) J :::::". ---=-:;--K~(,,=-r-==-)--,-l F' (z) p(r)(M - r) 
holdsfor : z' L r, where 
(6.4) K(r) =M2p2_ M {(p+ 1)2M2+2p2 -2p-l}r + {(2p2+2p-l)M2+ (p - I)2}r2 - Mp2r3~ 
The equality sign in (6.3) holds at z =r for the function 
F(z)=MzP I-Mz. 
M-z 
As it is easiy proved that the equation K(r)=O has only one positive root (Jp [ess than 
I and K(r) and Per) and positive for o~r <PP' we have the result as follows. 
Theorem II. 
Let F(z) = l' a"z" (ap=l) qe regular and bounpep ([ F(z)! < M) 
n-p 
in 1 z: < 1, then F(z) is .quasi-convex in i z! < Pp, where pp is a 
posit~ve rooot less than 1 of the equation K(r) =0. And pp is 
tbe possible number for quasi - convexity. 
8 
If we assume that p is any real mumber, then we can prove 
dpp :> 0 
dp , 
so that pp increases with p and hence 
PI < pz < ...... <PP < P.+I < ...... . 
Consequently we have the following corollary of theorem 11. 
Corollary. 
Let F(z) be any function in theorem 11, then F(z) is quasi ~ convex 
'in : z i < PI for all posiUve integer P. where PI is a posiUve 'root less 
than 1 of the equatz'on 
M - (4M 2 - l)Z + 3MZ 2 - Z 3 = 0 . 
In conclusion. 1 express my heartly thanks to professor Akira Kobori of Kyoto University 
for his kind advices. 
Faculty of Technology, Fukui University. 
References. 
(1) In this note, the function F(z) is always of the form (1. 1) . 
(2J A. Kobori, Sur les Fonctons Multivalentes, Proc. phys- Math. Soc. Japan. Vol. 
23. No.6. (1941) . 
(3) A. Kobori, Uber die notwendige undo hinreichende Bedingung dafiir. dass eine 
Potenzreihe den Kreis bereich auf den sohlichten sternigen bzw, Konvexen 
Bereich abbildet, Mem. ColI. Sci. Kyoto Imp. Unv. (A) 15 (1932) . 
(4) M. Stroh~cker, Beit~ge zur Theorie der schlichten Funktionen. Math. Zeitschr. 
37 (1933). 
(5) A. Kobori, loc.cit. (2J 
(6J Y. Sasaki, Theorems on the Convexity of Bounded Functions. Proc. Jap. Acad. 
Vol. 27 (1951). 


On some Family of Multivalent Functions .

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