Re !편2... ) > a

KyungαlOk Math. J Võlume 26. Number 2 Ilecernber, )986

A NOTE ON FUNCTIONS SATISFYING Re(f(z)/z}> a

By Shigeyoshi Owa

1. Introduction

Let S(a) denote the class functions of the form

∞

(1.1) f(%) =z+ C a^“z

”

n=2

which are analytic in the unit disk U= (z : 1%1 <1) and satisfy the following condition

(1.2)

Re !편2... ) > a

^(zEU)

for some a(O듣a<l)

This class S(a) was studied by GoeJ [5] and Chen [3]. In particuJar, The class S(O) was studied by GoeJ [6] and Yamaguchi [15]

\Ve introduce the class SG(B) (a) of funtions f(z) ES(a) such that G (z) ES(ß) •

.l~.l ", ' . ... ,)

where C(z)=r (2- ).)z^D;f(z) and D;f(z) means the fractionaJ derivative of

。rder ). of f(z). The object of the present paper is to show some distortion theorems for f(z) beJonging the class SG(B) (a)

Now

,

there 8re many deIinitions of the fractional calculus

,

that is

,

the fractional integrals and the fractìonal derivatives. We fjnd it convenient to reslrict ourseJves to the following definitions of the fractionaJ caJcuJus used recentJy by Owa [9].

DEFINTlON [. The fractionaJ integraJ of order ). is defined by (1.3)

qinz)=돼>1.‘꿨쁨T '

where ),> 0, f(z) is an analytic function in a simpJy connected region of the

'- 1

z-pJane containing the origin and the muJtipJicity of (z- Ç)"-' is removed by requiring Jog (z-Ç) to be reaJ when (%-Ç)> O.

DEFINITlON 2. The fractionaJ derivative of order ). is defin어 by AMS(MOS) subject c1assifications (1 98이 26A24 , 30C45.

168 Sh;g~yoshì Owa

1 d

r ‘ Æ

ldC

(1.4)

D;f(.) =꺼덕「 깅끽 강펴y-.

where 0<2<1. f(z) is an analytic function in a simply ccnnected iegion of the z-plane containing the origin and the multiplicity of (z-C) -1 is removed by requiring log(z-C) to be real when (z- C)> O

DEFINITION 3. Under the hYFotheses of Definition 2. the fractional derivative of order (n+시 is defined by

.n

(1.5)

대+1

f(z) =

주r

D:nz),

η*

'where 0<.<1 and nε NU(이.

REMARK. For other definitions of the fractional calculus. see Agarwal [1]. Al-Salam [2]. Erdélyi. Magnus. Oberhettinger and Tricomi [4]. Nishimoto

[7]. Osler [8]. Ross [!l]. Saigo [12]. Sneddon [13] and Srivastava and _Buschman [14].

2. The .13SB 8_G(p) (a)

For the function f(z) of the form (1. 1). with a simple computation. we get (2.1)

G(.)

=

F(2

-.)삼여μ)

∞ F(n+l)F(2-.)

=z+등，

F(시

+

1

-.)

. a꺼

for 0<.<1

THEOREM 1. Let 11늘 2. (n-l)/n르 a<I.0흐ß<1 alld n (1-a)득1-β Theη there exists the function of S(a) de，까ned by

(2.2) f(z) =z+anz"

such that G (z) εS( (J)

PROOF. Let the function f(z) defin어 by (2.2) belong to the class S(a).

Then we can see that (2.3)

and (2.4)

Re(an:

”

^{l)>a l}

r

(n+l) F(2- .) G(:)=z+ F(a+l- A)

%Z

Consequently. we obtain

(2.5) Re [ G;z) _V~~J

) _}= = 1+ _{1 +}

^L(n+1 r~~í \Gi)~) Re(a상 -1) ^{1)F(2- .l)}

A Nou on Functions Satis!y;ng Rt ν(%)/%J>a

r

(n+ 1) r(2-À)

>1+ • \;'::_~;~~l\ _r(n+l-À)

'"

^(a-l)

르a

for (n-I)/n등a<1 and n(l-a)듣I-ß. This shows that G(.)ES여)

169

1n view of Theorem 1, let Sc얘/a) denote class of functions f(z) ε S(a) of the form ^(l. 1) such that G (z) ES(ß)

We need the following lemma by Chen [3J.

LEMMA 1. L.t th. function f(%) of th. 끼orm (1. 1) belong to the class S(a).

Then

w.

have

1+(4a-2) Izl +(2a-l) 1%1²

(2.6) Re {f' (z)} 르 ^?

(1+Izl)‘ for 0듣 Izl<I/2 a찌

a- 2al. 1²+(2a-1) 1.1

‘

(2. 7) Re {f' (z)} 르 ”

(1- 1. 1‘)‘

for 1/2츠 Izl <1. These results aπ sharp

Further we need the following lemma by Owa [10].

LEMMA 2. Let the jiωlction f(z) of the fr01ll (1. 1) belong 10 the class S(a) Then tνe have

(2.8) and

(2.9)

|파| 드 받띤희

• 1= 1-1.1

e[

^f~)

^J;;;; _‘르

^1-(냥a)

^{1 .1}

• J= 1+lzl for .EU. Th. results are sharp

THEOREM 2. Let the function f(') of the form (1. 1) belong to the class SC(ß) (a). Th.n

w.

have

n.1 _{rJ_\} 1_1-11_ 1+(1-2β) 터

(2.10) ID;f(.)/.'-ì득，...수+탄T뚱i:늑、

ond

(2.11) Re

{여 f(Z)/ZI-치르 th!}:앤)”lz!”

Jor 0<^,1<1 and zEU. The results are sharp.

PRCOF. Since G (.) defined by (2. 1) is in the class S얘)， in virtue of (2.8),

170 Shigeyoshi Owa we can show that

(2. 12) ^{Ir(2- ,1)z^-'D}

;J

^{(z) 1르~슨구투피 1+ (1- 2ß) Izl}

from which (2.10) follows at once. Further the second estimate (2.11) of the theorem follows from (2.9) ^‘

Finally, (2. 13)

the results of the theorem are sharp for the function J(z) defined by

d _-;I

_(z)

-

^{{1+ (1-2ß)z)z}

,

^-1

(1- z)r(2- ,1)

COROLLARY 1. Let the Junction f(z) of the for11l (1. 1) belong to the class SC(o/a). Then we have

(2. 14)

1 껴ηz) /z'→ 1 르

^/’

1 션괴L l、

and

, ... 1 1" \ I 1-1, __ 1-Izl (2.15) ReID;f(z)_{ι zJ} '-f/"" /z'-') j

능

= (1+ IzJ)r(2-시

Jor O<l<1 aηdzε U. The results are sharp.

THEOREM 3. Let the functioη f(z) of the frorm (1. 1) belong to the class

SC(이 (a). Then we have

1 ... 1+1 N ,, ___ (l - l)+2(2ß-ß,1- 1) Izl

+

(1-,1) (1- 23) I%J'

(2.16) Re{z^D:T^f(z)) 르

’

‘

(1+lzl)"r(2- l)

for 0<^,1 <1 and 0 등 1 z 1 <1/2, and (2.

1끼 Re 낭여

+1 f(z))

늘

(ß-,1) +U(I-ß) Izl-2ß(1- ,1) Izl'- U (1-ß) IzJ3+ (2,8•1)(1 - ,1) Izl'

" ,

(1-lzIT r(2- À) for 0<,1 <1 and 1/2등 Izl<1

PROOF. Since G(z) defined by (2.1) is in the class S(ß)

,

Lemma 1 implies that

(2.18) _{Re{G'(z)) 늘} ^{1+ (4ß-2)} Izl + (2ß- l) Izl'

”

(1

+ 1 % 1 )

^‘ for 0흐 Izl <1/2 and

(2.19) Re{G'

(z)) 르 a-쟁 I z l'+ (앤: l) l z | 4

(1

-1 % 1

‘)‘

for 1/2흐 Izl <l. Hence

,

by using of (2.11) and (2.18)

,

we get

(2. 때)

^Re

(z' _D~

+1 f(z)

J 늘

^{1+(4a-2)lzlf(23-1)lzj}2 -ARe{DK(z)/:1

•

l

‘ (1+1%1)ιr(2-^,1) ‘

A Note On Funct

‘

^{on' Sati}

,

^{!ying Re{f(z) /z} >a 171}

능

^{(1- 1)+2(2ß-ß1- 1) Izl + (1- 1) (1- 2}/l) Izl2 (1+ Izl)2

r

^{(2- 1)}

for 0<1<1 and 0르 1=1 <1/2. Further

,

in virtue of (2.11) and (2.19)

,

(2.21) Re{z'D!+'f(z)}

능 a- 2{l

^lzI

2셋 2ß- l)

^Izl4 -ARe{Dfnz)/zl-A}

(1-lzl‘)‘ F(2- 1)

능

(ß-1) +21(1-ß) Izl-2{l(1-1) IzI2

- 21(I- {l) Izl'+ (2{l- I) (1- 1) Izl

‘

(1-lzI2

)2F(2-1) for 0<1<1. 1/2:::;1%1 <1. Thus we have the theorem.

COROLLARY 2. Let the function f(z) of the form (1. 1) belong to the class SG(O) (a). Then we haνe

(2. 22)

2 -

z -

4 4

1i -。ι

치 셰 + ’끼

r그

에

n

끼

l n

늘

-

야

(

j fJ

나

Z

i D 터 z R

for 0<1< 1 and 0등 1=1 <1/2

,

aπd

(2.23) l ... l+l r I _ \ ' __ - 1+211%1-211=1'- (1- 1) Izl' Re{z"D:" " f(%)}르 ”

‘ (1-1%1‘)‘F(2-1) for 0<1<1 and 1/2르 1=1<1.

REFERENCES

’[lJ R. P. Agarwal. Certain fractional q-integrals and q-derivatives, Proc. Cambridge Philos.

Soc .• 66(1969). 365-370.

ι[2J W. A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc. Edinburgh Math Soc .• 15(1966). 135-140

[3J Ming-Po Chen, On the regular functions satisfying Re{f(z)fzJ>a, Bult. [nst. Math Acad. Sinica, 3(1975). 65-70.

•[4J A. Erdélyi, W. Magnus. F. Oberhettinger and F.G. Tricmoi, Tables of Integral Transforms, Vol. D. McGraw-Hill B∞k Co., New York, Toronto and London, 1954.

;[5J R. M. Goel, On functions sat객fying Relf(z)/z}>a, Publ. Math" 18(1971). 111- 117.

1[6J R. M. Goel, The radius of convexity and starlikeness for certain classes of analytic functions with fixed second coefj

‘

^cients, Ann. Univ. Mariae Curie-Skl여owska， 25(l971}, 33- 39

tC7J K. Nishimoto, Fractional derivative and integral. Part 1, J. Coll. Engng. Nihon Univ., B-17(1976), 11-19

172 Shigeyoshi Owa

[8J T.J. Osler. Ltihniz rule for fractional der;vative gentraliud and an application /0.

infinite series, SIAM J. Appl. Math., 16(1970), 658-674

[9) S. Owa, On the distortion th

,

^{ortms. 1,} Kyungpook Math. J., 18(1978), 53- 59 (10) S. Ow" On jllnction

,

satisjying Re!J(z)jz}>a, (preprint)

[11J B. Ross, A brief histor

,

and exposition 01 tht fundamtnlal theory 01 fractional calculus, in Fractional Calculus and lts Applications (B. Ross, ed.), Springer-Verlag, New York.

HeideJberg and BerJin, 1975, 1-36.

[l2J M. Saigo, A remark on integral operators involving 1M Gauss hyþergtOmetric functions, Math. Rep. Kyushu Univ., 11 (1978), 135-143

[I3J l.N. Sneddon, The use in Mathematical Phys;cs 01 Erdélyj.Kober 0，μralors and 01

some 01 IM;r generalizations, in Fractional Calculus and Jts Applications (B. Ross, ed.).

Springer-Verlag, New York, Heidelberg and Berlin, 1975. 37-79.

.

[14) H. M. Srivastava and R^‘G. Buschman, Convolution integral equations with special function kernels. John Wi1ey and Sons, New York, London, Sydney and Toronto, 1977 [J5) K. Yamaguchi, On junction

,

satisjying Re{j(z)jz}>O, Proc. Amer. Math. Soc., H

(1966), 588-591.

Kinki University Osaka, Japan