On certain classes of p valent analytic functions

(1)

ON

CERTAIN

CLASSES OF p-VALENT ANALYTIC FUNCTIONS

NAK EUNCHO

Department

of Applied Mathematics

National FisheriesUniversity ofPusan

Pusan

608-737

KOREA

(Received

November5,

1991)

ABSTRACT.

The objective of the present paper is to introduce a certain general class

P(p,a,3)(pEN {1,2,3,...},0_<a<p and 3 > 0) of p-valent analytic functions in the openunitdisk V

andweprove that if

I

eP(p,a,3)then

$v,c(I),

definedby

c+P/:

tc-’f(t)dt

(cN),

Jv,

c(.f)

--e

belongs to P(p,a,/). We also investigate inclusion properties of the class P(p,a,3). Furthermore,

weexaminesomeproperties foraclass

Tv(a,3

ofanalyticfunctions withnegative coefficients.

KEY WORDS AND

PHRASES. p-valent analytic function, Hadamard product,integral operator, multipliertransformation, p-valentlyconvexof order6.

1991AMS

SUBJECT

CLASSIFICATION CODE.

Primary30C45.

1.

INTRODUCTION.

LetA_vdenotethe class of functions of the form

f(z)

zP+

an+

pzn+p(pN (1,2,3 })

(1.1)

n=l

which are analytic in the unit disk V {z: Iz[ < 1}.

We

also denote by S

v the subclass of Av

consisting offunctions whicharep-valentin V.

A

function

f

e

Av

is said to be in the class P(p,a) (0<a_<p) if and only if it satisfies the inequality

R4}

>a (O<_a_<p,z

.

U).

(1.2)

The classes P(1,0) and P(p,0) wereinvestigated by

MacGregor

[7]

and

Umezawa

[11],

respectively.

In fact,

theclassP(p,a) isasubclass of the class

Sv

[11].

Let

I

andgbeinthe class

Av,

withl(z) givenby

(1.1),

andg(z)definedby

z"+p.

(1.3)

()

z"+

.+

n=l

(2)

n--]

For a function

y

A_v given by

(1.1),

Reddy and Padmanabhan

[10]

defined the integral

operator

Jt,.c

(P,* N)by

/

pj

tc_y(t)d

Jp,

c(f)

"-oo

// _"+t,.

(1.5)

=zt,+

E

_c+p+n_an+t,

n=l

The operator

J,c

wasintroduced byBernardi

[2].

In

particular, the operator

J,

were studiedby Libera

[5]

and Livingston

[6].

Clearly,

(1.5)

yields

fG..

Ap

=

Sp,

.

At,

(1.6)

Thus, by applyingtheoperator

Jt,,,:

successively,wecanobtain

Jp,

c(J’,(f))

Sv,(f)

[

_l(z) (n_{(n- O).}N),

(1.7)

Wenowrecall thefollowingdefinition ofamultiplier transformation

(or

fractionalintegraland fractional

derivative).

DEFINITION

([3]).

Letthe function

z"+t,

(1.8)

()

.+

n----O

be analyticin Uandlet Abeareal number. Then themultipliertransformation

I’

isdefinedby

l’Xb(z)=

E

(n+p+l)-’Xcn+t,

zn+t, _(zq.U).

(1.9)

Thefunction

lXtk

is clearly analyticin U. It mayberegarded as afractional integral

(for

,

>

0)

or

fractional derivative

(for

a

<

0)

of

.

Furthermore,in termsof theGammafunction, wehave

IX4,(z)

r-

f

o(tO)x-

4,(zt)dt

(,x>0).

(1.10)

DEFINITION

2. The fractional derivative

DXb

of order

,

>0, for ananalytic function given by

(1.8),

isdefinedby

z"+p (,>O, zeU).

(1.11)

O()=-(z)

(-+p+l)

,:.+p

n-’0

Makinguseof Definition 2, we nowintroduce aninteresting generalization of the class of functionsin

At,

which satisfy the inequality

(1.2).

DEFINITION

3.

A

function)’e

At,

is said tobeinthe classP(p,a,3) ifandonlyif (p+

)-a

oa

I

eP(t,,,) (0<_a<

(3)

n

I+I

a.+n

(1.12)

which shows that (p+1)-a

Day

6_:A_nif_feA_v.

In

particular, the class _P(1,a,3) wasintroduced by

Kim,

Lee,

and Srivastava

[4].

2.

SOME INCLUSION

PROPERTIES.

In

ourpresent investigation of thegenera]classP{p,a,3) (0_<a<p,3_>0),weneed thefollowing

lemma.

LEMMA

2.1([1]).

Let

M(z) and N(z) be analyticinU, N(z) mapUontoamanysheetedstarlike region of order7(0_<7<p)and

Thenwehave

M(0)- N(0)=O, p,

R,f

M(z)’

.kN(z)]>

(o_<<,_>1).

By

using

Lemma

2.1,we canprove

THEOREM

2.1. Letthe function f(z)be inthe class P(p,a,B). Then

Jt,,c(f)

definedby

(1.5)

is

alsointhe class P(10,a,8).

PROOF.

A

simplecalculationshows that

dDa(Jn,

Zp- zC+pJo

(2.1)

where the operators

’/n,

(c N) andD (,>0)aredefinedby

(1.5)

and

(1.11),

respectively.

In

view

of

(2.1),

weget

M(z)= c+p

[**,{Dai(t)}dtandN(z)=zp+

(p+

1)aJo

(2.2)

sothat

fM’(z))

{(p+

1)

ad--Daf(z)

(2.3)

Since, by hypothesis,

I

-

P(p,a,B),the secondmemberof

(2.3)

isgreaterthana,andhence

fM’(z)]

Re

N--j

>a (o<a<p).

(2.4)

Thus,by

Lemma

2.1,wehave

(vtz’J

fM(z)){(P+l)-b-’Da(Jpc(f)))

z p-whichcompletestheproofof Theorem 2.1.

Let f A_nbegiven by

(1.1).

Suppose

also that

(4)

(c

+

p)...(c,.

+

p)

z"+

z

+

(cj N(j 1,2 ,),,. N).

,

(c

+ +

,)...(c,.

+

,)0-+

(2.6)

Then, byTheorem 2.1,wehave

COROLLARY

2.1. Let the function f(z) be in the class p(p,a,). Then the function Fm(.f)

definedby

(2.6)

isalsointhe classp(p,a,).

The next inclusion property of the class p(p,a,), contained in Theorem 2.2 below, would

involvetheoperator

J,(

>0)definedby

Jp,l(f)----

(1

+ p)lAf(z)

(A>O,f

Ap).

(2.7)

For m N,wehave

Clearly,wehave

Jpm,

l(f (1

+

p)mlmf(z)

(m-

1)!’

(log{)

lf(t)dt.

(2.8)

f

At,=J,(f)

Ap

(>0).

(2.9)

(> 0)

THEOREM 2.2. Let thefunction f(z)be inthe class P(p,a,). Then the function

Jp,

definedby

(2.7)

isalsointhe classP(p,a,).

PROOF. Makinguseof

(1.9)

and

(1.11),

thedefinition

(2.7)

yields

(p+

1)-D(jXp,

_l(f)

jAp,((p

+

1)-

Df)

Therefore,setting

(B _>0,A>0,!

A)

(2.10)

wemust show that

A

g(z) (p+

1)-Df

and G(z)

S,(g),

(2.11)

fG’(z)l

RV_,]

> (0< <)

(2.t2)

wheneverf P(p,a,).

Fromthe integral representationin

(1.10),

weobtain

sothat

(P+ 1) f*-G’(z)

r(z)

]o(iT

-t’(zt)dt

(A > 0),

(2.13)

Sincef P(p,a,/),wehave

(A >0),

(2.14)

’(t)

(5)

andhence

(2.14)

yields

R

e/z-’T-I

fG’(z)/

(p+l) It

f

F(A)

J

(ld)-

ltpdt (0<_ <p,A> 0),

(2.16)

whichcompletestheproofof Theorem 2.2.

COROLLARY

2.2. If0_<a<pand0_< <7,thenP(p,a,7)cP(p,,).

PROOF.

SettingA

--

>0inTheorem2.2, weobserve that

.f P(p,a,

7)=J,

a(l)

P(p,a,7)

:(p

+

1)-’rD’r(J,,’

(f))

P(p,o) ,(p+

1)-Df

*!

P(p,,,),

andtheproofofCorollary2.2iscompleted.

A

by

h(z)=

z’+n=

(n+

p+

ln+"

Then,interms of the Hadarnardproductdefinedby

(1.4),

wehave

(zU).

(h.f)O)

+

l{f(O

+

(2.18)

(2.19)

which,whencomparedwith

(1.11)

withm 1,yields (h,.f)(z)

p---i-D

I.

Wenowneed thefollowinglemma for another inclusionpropertyof the classP(p,a,/).

LEMMA

2.2([8]).

Let(u,v) beacomplexvalued function such that

(2.o)

/p:D--,C, DCCxC(Cisthe complezplane),

and letu _ut

+

iu2, v

v

+

ivy.

Suppose

that the function /(u, v) satisfies

(i)

/(u,v)iscontinuous inD,

(ii)

(1,0)DandRe{(1,0)}>0,

+

]

d*(,,,)}

<

o.

(iii)

for all (its,

v)

D such that

v

<

---,

Let

p(z)

+

pz

+

paz

+

be analyticinthe unit diskVsuchthat (p(z),zp’(z)) Dfor allz V. If n,{(p(,),,p’(,))}>0

thenRe{p(z)}> 0(z V).

THEOREM

2.3. If0_< <pand _>0,then

P(p,,

+

1)cP(p,,)

PROOF. Let

thefunction

(z V),

=2,,0,+

)+

,

20,+)+

I

(2.21)

First,weshall show that

(6)

Jf’(z)/_,

(+1)+

(O_<a<p,z{5U),

(2.23)

whenever

fF’(z)l

(0_<a_<p,z{5U).

(2.24)

By

the differentiation of F(z), weobtain

F’(z)

p--{2f’(z)+

zff’(z)}.

(2.25)

We

define the functionp(z)by

/’(z)

p-

3’

+

(1 3,)p(z)

(2.26)

with 3’

2,,(+1}+r2(+i)+

(0<_3’<1). Then p(z)

l+plz+pz+

is Mytic in U.

By

using

(2.25)

d

(2.26),

weobtMn

F’(z)

{(

+

P)(7

+

(1 7)p(z))

+

p(1-7)zp’(z)}.

(2.27)

Hence,

in viewof

ne]

> (0 < ), wehave

Re{(v(z),zp’(z))} >O,

(2.28)

where 6(u,v)isdefinedby

qb(u,v)

-i-{(p

4-p)(’r4-(1-3’)u)4- p(1-3’)v}-a

withu u

+

iu andv

v

+

_ivy. Thenwe seethat

(i)

(u,v)is continuous inD=CxC,

(ii)

(1,0){5Dand Re{C(1,0)} -a>0,

(iii)

for

all(iu,h){5

Dsuch that v < -0

+-22)

Re{b(iu2,vl)}

p__((p2

+

p)3’4-p(1-

3’)Vl}

_<

_L_+

l{(p2

4-p)3’_

p(1-3’)(12

4-

u:2)}-,a

_<0

(2.29)

,(r+ 1)+t,

Consequently,(u,v)satisfiestheconditions inLemma2.2. Therefore,wehave for3’ _r(+ii+.

RJl’(z)

2a(p+I)+p

(2.30)

-t-

>3’

2(V+ )+

Next,

inviewof

(2.20)

and abovearguments,wehave

.f{5P(p, ct,/+ l):(p

+

I)-B-

1DB+

if

{5P(p,a) =h {(p+ I)-

tn/}

{5P(p,a) =(p+1)-

BDf

{5P(p,l.t)

f{5P(p,,), whichevidentlyprovesTheorem2.3.

2a(p+l)+

2--(-

)

/

(7)

REMARK.

Since

o

< c*<p, wehave and henceP(p,g,)c

3.

THE CONVERSE

PROBLEM.

Let

T;,

denote theclassoffunctionsof the form

f(z)-’z n=l

c*(p

+

)+

2(p+

1)4-(p6N=(I,2,3

},an+p>0

whichareanalyticin Uand let

T;,(c*,)=

In

thissection, weinvestigate the converse problemofintegrals defined by

(1.5)

for the class

LEMMA

3.1. Let

!

T;,.

Then

I

e

T;,(c*,)

if andonlyif

E

o (n+p

<n+p+l)

1

p+l

a.+o

<p-c*.

(3.1)

n=l

PROOF. Suppose

that

n+p+

(n+p a <

n=l p+l

+;,-It

is sufficient toshowthat the values forIv+

1)-(Oal)’

_zp- lie inacirclecentered at pwhose radiusis p c*. Indeed,wehave

-p=_ (.+p)

,n+p+

n p+

an+;

zn

Conversely,assumethat

(

)

<-

E

(n+p) n+p+l

n= p+

a.+,l

zl

(

,=I p+l

a.+;,

<p-c*.

(3.2)

> a(o_<c*<p),

(3.3)

whichisequivalent to

R (,+p) n+p+l z" <p-c*.

(3.4)

I,n p+

Choose values ofzontherealaxis sothat

’

Xn+p+I)

1 n=l

(n+p

p+l

a.+;,z"

isreal. Letting z-lalongthe real axis,weobtain

oo

,In+p+

<p-a.

E

("+P

p+l

a.+,_

n=l

(8)

THEOREM

3.1 Let

F.Tp(a,)and

y(z)

[-c]

[zCF(z)

(ceN).

belongstothe class

Tp(,)

(0_< <p)for [z[ <r,where

Then the function l(z)

(3.5)

The resultis sharp.

PROOF. LetF(z)=z

’-

=a.+

z

n+.

Thenitfollows from

(1.5)

that I()

LF-6J

[F()]

=z_

(++)

+

n p+c n+

(3.6)

Toprovetheresult, itsuffices toshow that

(p

+

1)

(Of)

zp-

(3.7)

for

zl

_<r. Now

(p

+

’(Df)

Zp-1 --P (n+p p+l p+c n+p

n=l

(

_< (n+/

n++

.++

n=l p+l p/c

,+lzl".

Thuswehave

+

1)

(Df)

zp_

-P <

p-(n+p)(n+p+ly

n=l p+l p+c n+o

But Lemma

3.1 confirms that

_

(n+p) n+p+

n= p+

a.+

;,_

Therefore

(3.10)

will be satisfied if

n

+ p’

(n

+

p

+

c)

[n

+

p’

p--_ )

p/c z <_jy=-sj

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

for eachn N,orif

I=I_< _.+p+=

(3.13)

The required result followsnowfrom

(3.13).

Sharpnessfollows ifwetake

F(z)=zP- _nTp+l

(3.14)

for eachn$N.

THEOREM3.2. LetF

.

T(a,B)

and f(z)

=L+eJ

z-]

[zF(z)] (e

_

N). Then thefunction f(z)

(9)

izl

<r,=inf[.

p(p-6)

(n+p+c)(n+p+l)/3]

n> (n+ p+)(p-a) p+c p+

(3.15)

Theresultis sharp.

PROOF. Toprovethetheorem,it is sufficient to show that

14-if(z)

)

-19 <-p-

(3.16)

for

zl

_< r’.

In

viewof

(3.6),

wehave

Thus

_yOn=in(n+p)(n+p+c)

n+pzn+p-I

(p-E=,(n+p)

("+P+C)a,,+c

_.+

z")z-’

5- ln(n

+

P)

+

_n+ Z

<

(3.17)

-E=(+/

+

,1.

<p-

(3.18)

oon=

In(n+

P)(n+l+c)

ap+c

n+p]zl

<p-&

(3.19)

or

(n+p)(n+

p+6)(n+p+e)a

n- P(P-6) p4-e

,+lzl"

<1.

(3.20)

But fromLemma3.1,weobtain

(n+p(n+p+l)/

<1

(3.21)

n=

\p-a/ p+

an+-Hencel(z)is p-valentlyconvexof order (0 < <p) if

’"+P)(n+P+6>(n+P+C)

[n+p’(n+P+’)

1,

p(p 6 p 4-

Izl"_<U=-a]

p4-)

(3.22)

or

P(P-"

(p+c

II__< _(n+p+)(p-,) _n+p+e _p+

(3.23)

for eachn N. Thiscompletestheproofof the theorem. The resultis sharpfor thefunctiongiven

by

(3.14).

REFERENCES

1.

AOUF, M.K.,

OBRADOVIC,

M.

&

OWA,

S.,

On certain class of p-valent functions with

referencetotheBernardiintegraloperator, Math. Japonica 35

(1990),

839-848.

2.

BERNARD, S.D.,

Convex and starlike univalent functions, Trans.

Amer..Math.

Soc. 135

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3.

FLETT,

T.M.,

The dual of an inequality of Hardy and Littlewood and some related

inequalities,J.Math. Anal. Appl.38

(1972),

746-765.

4.

KIM,

Y.C., LEE, S.H.,

&

SRIVASTAVA, H.M.,

Someproperties ofconvolution operatorsin

the classP,,(#),submitted.

5.

LIBERA, R.J.,

Someclassesofregularunivalentfunctions,Proc. Amer. Math. Soc. 16

(1965),

755-758.

LIVINGSTON, A.E.,

On the radius ofunivalenceofcertain analytic functions, Proc. Amer.

Math. Soc. 17

(1966),

352-357.

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Functions whose derivatives has a positive real part,

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Math.

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104

(1962),

532-537.

8.

MILLER,

S.S.,

Differentialinequalities and Caratheodory functions, Bull. Amer. Math. Soc.

81

(1975),

79-81.

9.

OWA,

S.

&

NUNOKAWA, M.,

Properties ofcertainanalytic functions, Math. Japonica 33

(1988),

577-582.

10. 11.

REDDY, G.L.

& PADMANABHAN,

K.S.,

On analytic functions with reference to the

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Soc.

25

(1982),

387-396.

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T.,

Multivalently close-to-convex functions,

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