Angular estimations of certain integral operators

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Internat. J. Math. & Math. Sci.

VOL. 21 NO. 2 (1998) 369-374 369

ANGULAR

ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS

NAKEUNCHO,INHWAKIMandJIA KIM

Department

of AppliedMathematics

PukyongNationalUniversity

Pusan608-737,KOREA

(ReceivedApril 1,1996andinrevisedformSeptember30,

1996)

ABSTRACT. The object of the presentpaperis to derivesome argumentpropertiesofcertainintegral operators. Ourresults contain some interesting corollaries as the special cases.

KEYWORDS AND PHRASES:

Argument,

integral operators,starlikefunctions,Bazilevi6 functions. 1991AMS SUBJECTCLASSHICATION CODES: 30C45.

1. INTRODUCTION

Let

A

denotetheclassof functions of theform

f(z)

z

+

oz

which areanalyticintheopenunit disk

U

{

z

Iz

<

1}.

If

f

and gareanalyticin

U,

wesay that

f

is subordinate tog,written

f

-

g, if thereexists aSchwarz function

w(z)

in

U

such that

f(z)

g(w(z)).

Afunction

f

e A

issaid tobeinthe class S*

[E, F]

if l+Ez

zf’(z)

-<

(z6U,-I<F<E<I)

f()

+

FZ

The class

S*[E,F]

wasstudied in[1,2]. Inparticular,

S’[1

2c,

1]

S’(c)(0 _<

c

< 1)

is the well known class ofstarlikefunctionsof ordera. We observe[2]that a function

f

is in

S"[E,F]

ifandonly if

f(z)

1-.F

<

1-.F

and

zf’(z)

}

1

E

Re

j(z)’

>

2

(z

6

U,F

1).

(1.3)

Afunction

f

e

Aissaidtobein theclassB(p,a,

)

if itsatisfies

Re{

Zf’(z)f

"-I

}

g,,()

>/(z

e

u)

for some

#(#

> 0), (0 _<

<

I)

and g

e S*(o).

Furthermore,wedenote

BI(/, a,/)

bythe subclass of

B(/,

a,

,O)

for g(z) =_z

e

S*

(a).

The classes

B(, a,/)

and

B

(/,

a,

)

are the subclasses of Bazilevi6 functionsinU

[3].

Wealsonotethat

B(I,

a,/)

C(a,/)

isanimportant subclassof close-to-convexfunctions[4].

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370 N.E.CliO,I. H KIM AND J. AKIM

Jcu(f)

c

+

#

tc-l

fu(t)dt

;(c

>

(14)

Kumarand Shukla

[5]

showed that the integral operator

Jc,u(f)

defined by

(1.4)

belongstothe class

S*[E,F]

forc

>_

u(g-:l)l_F,

whenever

f

S’[E,F].

The operator

Je.1,

whenc N

{1,2,3,

.},

was introduced by Bemardi [6]. Further, the operator

J.

was studied earlier by Libera [7] and Livingston[8].

In the present paper, we give some argument properties of the integral operator defined by

(1.4).

We also generalize the previous results ofLibera [7], Owa and Srivastava

[9]

and Owa and Obradovi6 10].

2. MAIN RESULTS

Inprovingour mainresults,weshall needthefollowing lemmas.

LEMMA

1([11]). Let

M(z)

and

N(z)

beregularin

U

with

M(0)

N(0)

0, and

let/

bereal. If

N(z)

maps Uontoa (possiblymany-sheeted)regionwhich is starlike with respect tothe origin, then

Re

N’(z)

>

(z

e U)

=

Re

M(z)

and

Re

N.t{z)

<

,O(z

U)

=

Re

N(’z)

<

(z

U).

LEMMA

2

([12]).

Let

p(z)

beanalytic in

U, p(0)

1,p(z) :/:.0 in

U

and supposethat there exists apoint

zo

U

such that

Iv()l

<

fo

I1

<

Iol

and

where/

>

0. Thenwehave

o’(o)

ik,

where

whvn argp(zo)

r

2

and

1(1)

k<-

a+-a

when argp(zo) 2

where

p(o)

+ m( > o).

Withthe help of Lemma andLemma2,wenowderive

THEOREM 1. Letcand _# be real numberswithc

>

0,

>

0 and 1

< F

<

E <

1 andlet

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ANGULAR ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS 371

-

<

T(0_<

<1,0<_<1)

for someg

S"

[E, F],

then

where

arc,.

isthe imegraloperator definedby(1.4)and

r/(O <

r/_<

1)

is the solution of theequation

{2(rsin(1-t(E,F)))

6

7+-Tan

-1 _for

F

:/: 1

7r c

+

_I+F_+rlcos

’(1

tc(E,F))

"

r] for

F=

-1,

when

tc(E,F)

28in_

(

E- F

7r

c(1-F

2)+I-EF

(2.1)

(2.2)

PROOF. Letus put

where

and

M(z)

p(z)-N(z)’

1

zf,

t-If"(t)dt

t1

tC-tg(t)dt

M(z)

_

(z)

c

N(z)

#

tc-

gu(t)dt.

Thenp(z)isanalyticin Uwithp(0) 1.

By

asimple calculation,wehave

M’(z)

N’(z)

N(z)

zp’(z))

p(z) 1

+

zN’(z)

p(z)

1(zf’(z)f"-(z) )

Sinceg

e

S*[E,F],

J.(g) e

S*[E,F]

[5]

andhence

N(z)

is(possibly many-sheeted)starlike function with respecttothe origin. Therefore, fromour assumptionandLemma1,p(z) 0 inU.

Ifthere exists apoint

zo

6Usuch that

[argv(z)[<

--

for

Izl

<

Izol

and

then,fromLemma2,wehave

0p’

(o)

where

when argp(zo

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-372 N.E. CHO, I. H.KIMAND J.AK/M

and

when argp(zo) 2 where

p(zo)

i.(.

>

0).

Since

Jc.z(g)

E

S’[E,F],

from(1.2)and(1.3),wehave

g’()

(&,(g))’

N()

J,.()

/c

pe’T,

where

1-E

I+E

c

+

I .F

<

P

<

C

+

--tc(E,F)<<t(E,F)

forF#

-I,

when

t,(E, F)

isgiven by(2.2),and

1-E

c+

2 <p< oo,

--1<<1

for F= -1.

At first, supposethatp(zo)

ia(a

> 0).

Forthe case

F

:

1, we obtain

z0f’(z0)f"-I

(z0)

_/)

(1

)M’(zo)

arg

N’(zo)

(

1

zo/g(z,o))

argp(zo)

+

arg 1

+

_{z(y.,.(g))’}

p(zo)

&(g)

+

c

rr7-

+

arg

(

)1

+

(pe’

],,

-’

irlk

-

+

Tan-1 g

+o

g(-

)

_>

r?-"

/Tan_

(

sin (1- tc(E,F))

c/

+EI+F

+cos

’(1.

t(E,F))

_r_,

2

where

t(E,F)

and6aregiven by(2.2)and(2.1), respectively. Similarly, forthecaseF 1, we have

(f()F-()

)

Theseare acontradictiontothe assumption ofourtheorem.

Next,

supposethat

p(zo)

ia(a

>

0).

Forthe case

F

4=

1,applyingthe samemethodasthe

above,wehave

arg(zf’(z)fz-(z)

<

"Tr

Tan_

(

nsin(1-t(E,F))

-[-

I+EI+F

/7]C08

"(1

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ANGULAR ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS 373

/’()/"-

()

)

’

arg

g"(zo)

B

<

2

whichare contradictionstothe assumption. Thereforewecompletetheproofofourtheorem. Taking

E

1

2a(0 <

a

< 1)

and

F

1inTheorem 1,wehave

COROLLARY1. Letc

_>

0,#

>

0 and

f

EA. If

z’f’(z)f"-(z)

<

(0 <

<

1, 0

<

< 1)

arg

g(z)

forsome g ES*

(a),

then

arg

where

dc,u

isthe imegral operatordefinedby

(1.4).

REMARK1. For6 1,Corollary isthe resultobtainedbyOwaand Obradovi6

[10].

Setting

E

1,

F

1, 1, 6 1 andg(z) z inTheoreml,wehave

COROLLARY2. Letc

_>

0and

f

A. If

.Re

ft(z) >/(0

<_/5

<

I),

then

e

(s,.

(I))’

>/, where

Jc,1

isthe imegral operatordefinedby(1.4).

Letting# 1 inTheorem 1,wehave

COROLLARY3. Letc

_>

0and 1

_<

F

<

E

<

1and let

f

A. If

)1

arg

g(z)

/

<-(0_</<1,0<6<1)

forsomeg6S"

[E,

F],

then

where

Jc,1

isthe integral operatordefinedby(1.4)and

r/(0

<

r/<

1)

isthesolutionof the equation

(2.1).

Taking

E

1

2a(0

_<

a

<

1)

and

F

1inCorollary3,wehave

COROLLARY4. Letc

>_

0and

f

A. If

then

arg

Jc,

l(f)

cz

<

-,

where

Jc,1

is theintegral operatordefinedby

(1.4).

PuttingE 1

2a(0 _<

a

<

1), F

1 and//= 1 inCorollary3and Corollary 4,weobtainthe following result ofOwaandSrivastava[9].

COROLLARY

5. If the function

f

definedby

(1.1)

isin the class

C(c,/),

then the integral operator

Jc,1 (f)(c

>

0)

definedby

(1.4)

isalsointhe class

c(a,/).

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374 N.E.CliO,I. H.KIMAND J.AKIM

By

using thesametechnique asinproving Theorem 1,wehave

THEOREM2. Let cand #berealnumberswith c

>

0,/

>

0 and 1

< F

<

E <

1 and let

fA.

If

,qV(z)

<

--

(/

>

1, 0

<

6

<_ 1)

forsome g ES*

[E, F],

then

where

Jc,,

isthe integral operatordefinedby

(1.4)

and

r/(0

<

r/<

1)

isthesolutionof the equation(2.1)

Putting g 1

2a(0 <

a

<

1), F

1,# 1and 6 1inTheorem 2,wehavethefollowing resultby OwaandSrivastava

[9].

COROLLARY6. Letc

>

0and

.f

EA. If

()

<(>1)

forsome g S*

(c),

then

z(Jc,1

(f))’

}

where

,Jc,1

istheintegraloperator definedby

(1.4).

ACKNOWLEDGEMENT. The authors wouldlike tothankProfessor M.Nunokawaforhisthought

encouragement and much valuable advice in the preparation ofthispaper. This work was partially supported by NonDirected ResearchFund, Korea Research Foundation, 1996and theBasic Science Research

Program,

Ministry of Education, ProjectNo.BSRI-96-1440.

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H. and

SILVIA,

E.M,

Subclasses of starlike functions subordinate to convex functions,Can.d. Math. 37(1985),48-61

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R.,

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Close-to-convexSchlichtfunctions, Michigan Math. 1(1952),169-185.

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S.L.,

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755-758

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H.M.,

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