ON o:-CONVEX FUNCTIONS OF ORDER

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 4 (1997) 769-772

769

ON o:-CONVEX FUNCTIONS OF ORDER

SEIICHIFUKUI

Department

of Mathematics Facultyof Education Wakayama University, JAPAN

(ReceivedNovember 1,

1995)

ABSTRACT. In

1969Mocanu introducedand studiedanewclassof analytic functions

conmsung

of a-convexfunctions

Many

mathematicianshave studied and shown thepropertiesof this class Nowwe will define newclasseslikethat

Mocanu

classandtheninvestigateandgivesomeresults The class ofo- convexfunctionsoforder partiallyincludes

Mocanu’s

class

KEY WORDS AND

PHRASES: a-convexfunction, starlike function, convex function 1991AMSSUBJECT

CLASSIFICATION

CODES: Primary30C45

1.

INTRODUCTION

Let

U

be the unit disc in thecomplex plane,

U

zE

C’ Izl ^{< 1}}

^{and let}

^A

^{be a}class of analytic functions,

f(z)

^z

+

a2z

+

ⁱⁿ

^U

^Wedenote three subclasses of

A,

as

S, S"

and

K,

thesetsof univalentfunctions, starlike functionsand convexfunctions, respectively Also we denoteby

S" (o)

^and

K(a)

theclassesofstarlike functionsof orderaand convex functions oforderowhichareanalytically expressedasfollows.

S’(a)= f(z) EA’Re

f(z) >’ zEU

(1

1)

K(a) I(z)

E

A Re

1+

f,(z ] >

a, zE

(12)

whereoisarealnumber and 1

>

^a

>

0

We

canput

S’(0)

,.q" and

K(0) ^K

Itiswell-known that

K

C

S"

C

S

C

A, S’(al)

^C

S’(a2),

and

K(al)

C

K(a2)

for0

<

_a2

_<

a,

<

^1.

Thefollowing resultwasshown by

Marx [2]

independently of Strohhacker

[3]

THEOREM

A.

It

holds that

K

C

S*().

In

1969

Mocanu

defined a newsubclass of

A

consisting ofa-convex functions asfollows DEFINITION 1.

A

function

f(z) ^A

is said to be an a-convex function if

f(z)

satisfiesthat

/,-. f’(z) #

^0and

.f’(z)

+ (1 a) >

0

e u, (13)

Re 1+

y,(,) y(,)

forsomerealnumber

We denote by

M(a)

theclass ofa-convexfunctions and weoften callit

Mocanu’s

class Miller,

Mocanu

and Reade

[4]

showedthefollowingTheorem

B

7 7 0 SFUKUI

THEOREM B. Itholds that

M(a)

C

S"

foranyreal numberaand

M(a)

C

K

C S"fora

>_

Wewillgivetworemarksabout Definition

REMARK

1. Ifthe condition

(13)

issatisfied,then the condition

f’(z) #

^{0 is}alwaystrue, so thislatter condition isnotneeded This fact isfoundedin

[4]

REMARK

2. The equality in

(13)

does not appear, because we have to consider the mimmal principle ofharmonic function intheopenunit disc

U

Wewilldefinenewsubclasses of

A

DEFINITION

2. Afunction

f(z)

E

A

issaidtobe an a-convex function oforder of type if

f ^(z)

^satisfies

Re a 1+

f’(z) +(l-a)

f(z) >

^zE

^U,

(14)

forsome reala

and/5

DEFINITION

3.

A

function

f(z) ^A

is saidtobe an a-convex function of

order/5

oftype 2 if

f ^(z)

^satisfies

Re a 1+

f’(z) J ^+(l-a) f(z) ^< B

^z

^U,

(15)

’for some reala

and/

Wedenoteby

M(a, 5)

^and

N(a, 5)

the classes of a-convex functionsoforder

5

^{of type andof}

type2, respectively Wecan put

M(a, 0) M(a).

^{Wenotethat itmustbe 1}

>/

in Definition 2 and 1

< 5

^{inDefinition3}

2.

LEMMA

AND

THEOREMS

To proveourresults we need thefollowinglemma which isaspecialcaseofCorollary in

[5]

LEMMA

Let

p(z)

1

+

PlZ

+

Pa^z

+

^{beanalyticinU If}there exists apointz0

U

such that Re

p(z) >

3’for

z01 ^< zl

^andP.e

^p(o)

^{"r,thenwehave}

Re

(2 1)

Nowwewillshowtwotheorems After thatusingthe abovelemma,

we,

^{willgive onlythe}

^proof

^of

Theorem1,because we caneasily givethe

proof

ofTheorem2inalmost the same was as in Theorem

THEOREM

1. It holdsthat

M(a, )

^C

^S" (3")

if it is satisfied that

( 1)

3’-1 <, 1>3’> ⁽²³⁾

23’

3"

+

^a

2(3’- 1) ->

^3’

->

⁰

⁽²⁴⁾

fora

>

0and1

>/

THEOREM

2. It holdsthat

N(a, 5)

^C

^S" (3’)

if it is satisfied that

>/5 ( >3’>0) ⁽²⁶⁾

3’+o2(3’_1)

ON a-CONVEXFUNCTIONSOFORDER 771 fora

<

⁰and 1

</

PROOF OF THEOREM1. Letus put

P(z) _f-TW

and

zf’(z)

W(z)

o. 1+

f’(z) ] + ^{(1- ) zf’(z) p()} + , ^zv’(z)

f(z) p(.)

Then

f ^(z)

^E

^{M (a, 5)}

^implies

ReW(z) Reap(z)+ ^{p’(z) p(z)} } ^> ’

^z

^u.

⁽²⁷⁾

Now

wemustshow

Rep(z) >

^-yforz E

U

Itistrueintheneighborhood ofz 0

So

if there exists a pointz0E

U

suchthat

Rep(z) >

,yfor

[z[ < [zo[

^and

Rep(z0)

7,then, bytheresult of our lemma and a

>

0,wehave

zop’ zo

7-

Re

W(zo) Rep(z0) +

Reo

_<

7

+

^o

_< ,

p(0) 2-

for

>

7

>

This contradictstheassumption

(2 7)

In

the same way we have

ReW(z0)<_7+o ^<_5,

^{for $}

^>7

>0 which leads to the contradiction Thus there exists no such apointz0in

U

Thiscompletestheproofof Theorem

IZI

3.

APPLICATIONS

We^willshow someapplications ofTheorem and Theorem 2

Firstofall,weput/5 0 in

(2 3)

^and

(2 4)

of Theorem Then we have

COROLLARY

1. Itholds that

1 o

+ V/(o + 80)

M(o) cS*(7)

^for

<7 < (0>1) (3 1)

0 1

M(o) cS’(7)

^for

I-

^<7_<

(2_>o_> 1). (3

2)

COROLLARY

2. Theorem

A

is true, thatis,

K

C

S’(})

Puttingc 1 in Corollary above we have_3’

1/2,

^{so we caneasily seethat}

^K

^C

^S" ()

^from

M(1) K

COROLLARY

3.

It

holdsthat

M(o, 15)

C

S"

for any realnumbers0

dnd

¹

> 5 ^>

⁰

When we put -y 0 in

(2 2)

ofthe lemma, we haveRe^z0rc(z0)_p(o)

<

0 So wehave Re

W(zo)

Re

p(zo) + are

^zov’(z,o)_p(so)

<

0 Since theassumptionofTheorem is

ReW(z)

>/, ^z

U,

itcontradmts thecondition

f(z) M(o, )

^forI

>/ >

⁰

COROLLARY

4. Theorem

B

is true, thatis,

M(a)

C

S"

for anyrealnumber

(3 3)

M(a) cKcS"

for 0> 1.

(34)

PROOF.

The fact

(3.3)

is a direct result of Corollary ³ We will show

(3 4)

^Since

S’(71)

^C

S’(72)

^for0

^<

^q,2

<

")’1

<

1, wehave

M(o)

C

S*(7) c S*(}) ^{c S"}

^for0

^>

ⁱⁿ

^{(3 1)}

^of

Corollary Therefore,

7 7 2 SFUKUI

Re a- 1+

f’(z) +(l-a)

f(z)

^>0,

^{zeU (3}

⁵⁾

is writtensuch as

zf’(z)) ^{zf’(z) >}

^O.

c.Re 1+

f’(z) > (a-1)Re

f(zi

Thisshows that

M (a)

^C

^K

^fora

>

1 l-I

Asanapplicationof Theorem 2 wehave thefollowing:

COROLLARY

5. Itholds that

N(a,/3)

C

K

C

S’()

^for

^> >

^{1 anda}

<

¹

PROOF. WeputT=

1/2

ⁱⁿ

^{(2 5)}

^or

^{(2 6)}

^{Then weobtain}

q >>

^{landa< -1 Firstthis}

shows

N(a, )

^C

^S" () ^Next

the inequality

(1 5)

Re(a(1+ zf’(z)) f’() f(z) zEU

impliesthat

zf’(z)

</3 (1 a)Re < ( a) Re <

^O.

.

^Re

+

f’(z) f(z) f(z)

z_/

_>0,

zEU

Thus we obtainRe 1

+

_f,(z)]

REFERENCES

1] MOCANU, P.,

Une propri6de convexit6g6n6ralis6edans la th6oric de larepr6sentation conforme, Mathemattca(Cluj), 11

(34)(1969),

127-133

[2] MARX, A.,

Untersuchungenuber schlichteAbbildungen,Math.

Ann. (1932),

^40-67

[3] STROHHACKER, E.,

BeitragezurTheorieder schlichitenFunctionen,Math.

Z.,

³⁷

(1933),

^356- 380

[4] MILLER,

S

S, MOCANU, P

and

READE, M.O,

All a-convex functions are univalent and starlike, Proc. Amer.Math.

Soc.,

37

(2) (1973),

^553-554

[5] FUKUI, S,

OnJack’s lemma and Miller-Mocanu’s lemma, Bull.

Fac., of

^{Education, Wakayama}

University

Nat.

Sci,45

(1995),

1-7