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 g}^are^analyticⁱⁿ

^U,

^we^{say 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}

^is^{said to}^beⁱⁿthe class S*

[E, F]

^if

zf’(z)

l+Ez

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

f() + FZ

The class

S*[E,F]

^was^{studied 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]

^if^andonly 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

^A^is^said^to^be^{in the}^classB(p,a,

)

^{if it}^satisfies

Re{ ^Zf’(z)f

^g,,()^"-I

} ^>/(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].

Forapositiverealnumber_#

>

0andafunction

f ^A,

^wedefine the integral operator

Jc.,

^by

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

Jcu(f)

^c

+

#

tc-l fu(t)dt ;(c >

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Kumarand Shukla

[5]

showed that the integral operator

Jc,u(f)

defined by

(1.4)

belongstothe class

S*[E,F]

^for^c

>_ u(g-:l)l_F,

^whenever

_f S’[E,F].

The operator

Je.1,

^when^c 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/

^be^real.

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

.fA

^If

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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)

²^7r^8in_

( c(1-F

^{E- F}

2)+I-EF

(2.1)

(2.2) PROOF. Letus put

where

and

p(z)-

M(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

⁶^U^{such 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),^we^have 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

c+

1-E

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)

_/)

^arg

⁽¹ N’(zo) ^)M’(zo)

(

¹

^zo/g(z,o))

argp(zo) +

^arg ¹

+

z(y.,.(g))’

p(zo)

&(g)

+

^c

rr7- ⁺

^arg

( )1 ⁺ ^(pe’ ^],, ^-’ ^irlk

- ⁺

^Tan^-1

^+o

^g

^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,^applying^{the same}^method^as^the

above,wehave

arg(zf’(z)fz-(z) ^< "Tr

^Tan_

( nsin(1-t(E,F))

-[-

I+EI+F

^/7]C08

^"(1

where

t(E,F)

and 6aregiven by

(2.2)

^and

(2.1),

respectively and for thecase

F

1, wehave

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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

ÊÂ. Îf

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 the}integral 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

^defined^by

^(1.1)

^isin the class

C(c,/),

then the integral operator

Jc,1 (f)(c > 0)

definedby

(1.4)

^is^alsoⁱⁿ^{the class}

c(a,/).

REMARK

2. Takingc=/ 0 and c 1 in Corollary 5, weobtain the result given earlier by Libera[7]

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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)}

for_{some g E}S*

[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

ÊÂ. Îf

() <(>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.

REFERENCES

[l]

JANOWSKI, W.,

Some extremal problems for certain families of analytic functions, Bull. de L’Acad Pol. des

ScL

21(1973), 17-25.

[2]

SILVERMAN,

H. and

SILVIA, E.M,

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

[3]

SINGH,

R.,

OnBazilevifunctions,Proc. Amer.Math.Soc.l(1952),

169,-815.

[4]

KAPLAN, W.,

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

[5]

KUMAR,

V. and SHUKLA,

S.L.,

Onp-valent starlike functions withreferencetotheBemardi integraloperator,Bull. Austral.Math.Soc.30(1984),^37-43.

[6]

BERNARDI,

S.D., Convexand starlike univalentfunctions,Trans. Amer.Math. Soc. 135(1969), 429-446.

[7] LIBERA, R.J.,

Some classes ofregularunivalent functions,Proc. Amer. Math. Soc. 16

(1965),

755-758

[8]

LIVINGSTIN, A.E.,

Ontheradiusofunivalenceofcertainanalytic functions,Proc. Amer.Math.

Soc.17(1966),352-357.

[9] OWA, S.andSRIVASTAVA,

H.M.,

Someapplications of the generalizedLiberaimegraloperator, Proc.

Japan

Acad 62,Ser. A (1986), 125-128.

[10] OWA,

S. and

OBRADOVI(, M.,

Certainsubclasses ofBazilevi/:functions of typec,Internat. d.

Math. Math.

ScL

9

(1986),

^347-359.

11] MILLER,

S.S.and

MOCANU, P.T.,

Second order differemial inequalitiesinthecomplex plane, o Math.Anal.

AppL

⁶⁵(1978),289-305.

12] NUNOKAWA, M., Onthe order ofstronglystarlikenessofstronglyconvexfunctions,Proc.

Japan

Acad69,Set.A (1993),^234-237.

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Mathematical Problems in Engineering

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

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Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009

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ANGULAR ESTIMATIONS OF CERTAIN INTEGRAL OPERATORS