Certain Subclasses of Analysic P-Valent Functions With Respect To Other Points 1

IOSR Journal of Mathematics (IOSR-JM)

e-ISSN: 2278-5728, p-ISSN:2319-765X. Volume 10, Issue 4 Ver. I (Jul-Aug. 2014), PP 61-77 www.iosrjournals.org

www.iosrjournals.org 61 | Page

Certain Subclasses of Analysic P-Valent Functions With Respect

To Other Points

1Hamzat J. O,

2Sangoniyi S. O.

1Department of Pure and Applied Mathematics,Ladoke Akintola University, of Technology, Ogbomoso

P.M.B. 4000, Ogbomoso, Oyo State, Nigeria. 2Department of Mathematics,Emmanuel Alayande College of Education, OyoP.M.B. 1010, Oyo State, Nigeria.

Abstract: Let T be the class of analytic functions of the form:

kk

k zazzf

2

defined in the open unit disk 1: zzD normalized with 0f and 01/ f where is an

arbitrary fixed point in D. The authors study certain

subclasses lpS ns ,,,,,*

, , lpS nc ,,,,,*

, and lpS nsc ,,,,,*

, of analytic p-valent

function with negative coefficients in the unit disk. The results presented in this paper include coefficient bounds

and distortion properties for functions belonging to these subclasses. Further results include linear combination of several functions and radii of starlikeness and convexity for functions belonging to the aforemention

subclasses.

Mathematics Subject Classification: Primary 30C45

Keywords and phrases: Analytic function, Univalent function, Starlikeness, Convexity, Coefficient

inequalities.

I. Introduction

Let T denote the class of functions of the form

kk

k zazzf

2 (1)

which are analytic and univalent in the open unit disk 1: zzD and normalized with 0f

and 01/ f , where is an arbitrary fixed point in D. TS denotes the class of analytic and

univalent functions (see[1, 7, 8]).

Also, let pT be the class of analytic p-valent functions zf of the form

kp

k

kp

pzazzf

1

. (2)

Now, supposing we pose index alpha on (2) such hat

kp

k

kp

pzazzf

1

.

This is,

...2

2

1

1

p

p

p

p

pzazazzf

(3)

Expanding (3) binomially, we have

j

pp

j

j

pzazazzf ...1

2

21

1

where ,1

,..!2

12

.

Finally, we have

Certain Subclasses of Analysic P-Valent Functions With Respect To Other Points


)(zf kp

k

kp

pzaz

1

(4)

Our motivation for this work is that several authors and researchers (see[1,2,3,4,5,6,7,8,9,10]) to

mention but few have worked extensively on coefficient bounds for functions of the type (2), hardly will you

find coefficient inequalities for the functions of the type (4) perhaps due to the difficulties the index α always

pose. However, in the present paper, the authors aim at generalizing 0.. ei . Here, we let ),( pT be

the class of functions of the form (4) where α is real.

Using Aouf et al derivative operator (see[2,3]), we can write for function

,pTzf that

kp

k

kP

nn

n

p zal

lkpz

l

lpzflI

1

,1

11

1

11,

,0Nn ,Np

,0l

,0

0 and

.1k

At this junction, the authors wish to define the following:

Definition A:

(i.) Let the function zf be defined by (2). Then lpSzf n ,,,* if and only if

,0

,

,

,

/

,

zflI

zflIzn

p

n

p

,0Nn Dz (6)

and lpwSn ,,,* denote the class of starliken functions.

(ii.) Let function zf be defined by (2). Then zf

nsS , lp,,, if and only if

DzNnzflIzflI

zflIzn

p

n

p

n

p

,,0

,,

,0

,,

/

,

. (7)

and lpS ns ,,,*

, denotes the class of starlike functions with respect to symmetric points and is an

arbitrary fixed point in D.

(iii) Let function zf be defined by (2). Then zf ,,,,,,

nsS if and only if

pzflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

p

n

p

n

p

n

p

,,

,

,,

,

,,

/

,

,,

/

,

(8)

for some 0 10,1 and z D where is an arbitrary fixed point in D. We

denote the class of starliken with respect to symmetric points by

nsS , ...,,, lp

Definition B: Let the function zf be defined by (4). Then zf is said to be -n-starlike with

respect to symmetric point if it satisfies the condition:

pzflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

p

n

p

n

p

n

p

,,

,

,,

,

,,

/

,

,,

/

,(9)

where ,0 Nn 10 , 10 ,

11

10

p and z D. We denote the class of

starliken with respect to symmetric points by lpS ns ,,,,,,*

, and is an arbitrary fixed

point in D.

Definition C: Let the function zf be defined by (4). Then zf is said to be starliken with

respect to conjugate point if it satisfies the condition:



p

zflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

pw

n

p

n

p

n

pw

,,

,

,,

,

,,

/

,

,,

/

, (10)

where NNn }0{0 , 10 , 10 ,

11

10

p, z D and is arbitrarily fixed in

D. We denote the class of starliken with respect to symmetric points by

ncS . .,,,,,, lp

Definition D: Let the functions zf be defined by (4). Then zf is said to be starliken with

respect to symmetric conjugate points if satisfies the condition:

p

zflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

pw

n

p

n

p

n

pw

,,

,

,,

,

,,

/

,

,,

/

,

(11)

where NNn }0{0 , 10 , 10 , 0

1

1p < 1, z D and is arbitrarily fixed in

D. We denote the class of -n-starlike with respect to symmetric points by

nscS ,. .,,,,,, lp

II. Coefficient Inequalities

Theorem 2.1: Let the function zf be defined by (4) and 0,, ,,

zflIzflI n

p

n

p

for z . Then, zf

nsS ,. l,,,,,, if and only if

kp

n

pp

k

k al

lkppkdr

1

111111)(

1

n

pp

l

lpp

1

11111

(12)

Proof: using (4), (5) and (9), that is

pzflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

p

n

p

n

p

n

p

,,

,

,,

,

,,

/

,

,,

/

,

1k

k

zkapzzf

and

zflI n

p ,,

=

kp

kp

k

n

p

n

zl

lkpz

l

lp

1 1

11

1

11.

For convenience, we let

kp

kp

k

kpppzazzf

1

11

and

zflI n

p ,,



kp

kp

n

k

kpp

n

pza

l

lkpz

l

lp

1

111

1

111

1

This implies that

kp

kp

n

k

kpp

n

pza

l

lkpkpz

l

lpp

1

111

1

111

1

n

p

l

lpp

1

)1(1)1(1

kp

kp

n

k

pza

l

lkppk

)(1

1111

1

.

That is

pn

ppz

l

lpp

1

11111

.0

1

11111

1

kp

kp

n

k

kpkpza

l

lkppkpk

Letting ,drz we have

1 1

11111

k

kp

kp

n

kpkpdra

l

lkppkpk

.0

1

11111

n

pp

l

lpp

Therefore, by the maximum modulus theorem, we have .,,,,,,*

, lpSzf ns

For the converse, let us suppose that

.

,,

,

,,

,

,,

/

,

,,

/

,

zflIzfl

zflz

zflIzflI

zflIz

n

p

n

p

n

p

n

p

n

p

n

p

This implies that,

1

1

1111

)1()()1(

k

kp

kp

kppp

p

kp

k

kppp

zapkzp

zapkzp

where

n

l

lp

1

11 and

.

1

11n

l

lkp

Since zz for all z , then we have

1

1

1111

11

k

k

kp

kpp

k

kpk

kp

drapkp

drapkp (13)



If we choose values of z on the real axis so that

zflIzflI

zflIzn

p

n

p

in

pw

,,

,

,,

,is real,

0,, ,,

zflIzflI n

p

n

p , zfor and upon clearing the denominator in (13).

Also letting drz through the real values, then we obtain

kkp

k

kp

k

k

kp

kpdrapkdrapk

1 1

111

pp

pp 111

and this completes the proof of theorem 2.1.

Corollary 2.1: Let the function zf defined by (4) be in the class lpS ns ,,,,,,*

, .

Then, we have

DzandNnkdrpkpk

pa

kkpkp

p

kp

0,1111

111

(14)

where and are as earlier defined.

The equality in (14) is attained for function zf given by

kp

kkpp

ppp

zdrpkpk

pzzf

11)1(

111. (15)

Theorem 2.2: Let the function zf be defined by (4). Then lpSzf nc ,,,,,,*

, if and only

if

1

12121k

k

kp pdrapk (16)

Proof: Using (4), (5) and (10), then

p

zflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

pw

n

p

n

p

n

pw

,,

,

,,

,

,,

/

,

,,

/

,

1k

kp

kp

pzazzf

and

zflI n

p ,,

1 1

11

1

11

k

kp

kp

n

p

n

zal

lkpz

l

lp

Let

1k

kp

kp

pzazzf

and

zflI n

p ,,

=

kp

kp

n

k

p

n

zal

lkpz

l

lp

1 1

1

1

11.



From (10), we have

1 1

11

1

11

k

kp

kp

n

p

n

zal

lkppkz

l

lpp

p

n

zl

lpp

1

11)2(

kp

kp

n

k

zal

lkpppk

)(

1

112

1

.

This implies that

p

n

zl

lpp

1

1121

1

.01

1112

k

kp

kp

n

zal

lkppkpk

Hence,

1 1

1112

k

kp

kp

n

zal

lkppkpk

n

l

lpp

1

1112

Letting ,drz we obtain

1 1

1112

k

k

kp

n

dral

lkppkpk

n

l

lpp

1

1112

and the complete the proof of Theorem 2.2.

Corollary 2.2: Let the function zf defined by (4) be in the class .,,,,,,*

, lpS nc

Then, we have

k

kp

n

n

kp

dral

lkpk

l

lpp

a

1

11121

1

1112

(17)

Nnk ,1 and z D.

The equality in (17) is attained for the function zf given by

kp

k

kp

n

n

pz

dral

lppk

l

lpp

zzf

1

1112)1(

1

111)2(

Nnk ,1 and zD. (18)

Theorem 2.3: Let the function zf defined by (4) be in the class .,,,,,,*

, lpS nsc

Then, we have



1 1

111111

k

k

kp

n

kpdra

l

lkppk

n

pp

l

lpp

1

11)1()1(1

. (19)

Proof 2.3: Using (4),(5) and (11), we have

p

zflIzflI

zflIzp

zflIzflI

zflIz

n

p

n

p

n

pw

n

p

n

p

n

pw

,,

,

,,

,

,,

/

,

,,

/

,

and

zfpI n

p ,,

kp

kp

nkp

k

p

n

pza

l

lkpz

l

lp

1

111

1

111

1

From (11),we have

p

n

pz

l

lpp

1

111

kp

kp

n

k

kpza

l

lkppk

1 1

111

p

n

pz

l

lpp

1

1111

0

1

11)1(1)(

1

kp

kp

n

k

kp zal

lkppkp

.

This implies that

1 1

1111)1(

k

kp

kp

n

kpza

l

lkppkpk

.0

1

11111

pn

ppz

l

lpp

Letting ,drz then we have

1 1

111111

k

k

kp

n

kpdra

l

lkppk

n

pp

l

lpp

1

11)1()1(1

.

and this ends the proof of theorem 2.3.

Corollary2.3: Let the function zf defined by (4) be in the class lpS nsc ,,,,,,*

, . Then, we have



k

kp

n

kp

n

pp

kp

dral

lkppkpk

l

lpp

a

1

11111

1

11111

(20)

k 1 and oNn .

The equality in (20) is attained for the function zf given by

zf

kp

k

kp

n

kp

n

pp

pz

dral

lkppkpk

l

lpp

z

1

11111

1

11111

(21)

k 1 and oNn

III. Distortion Theorem

Theorem 3.1: Let the function zf be defined by (4) be in the class .,,,,,,*

, lpS ns .

Then, we have

in

pp

n

pppi

p

l

lpp

l

lpdrp

l

lpdr

1

11111

1

1111̀1

1

11

11

zflI i

p ,,

in

pp

n

pppi

p

l

lpp

l

lpdrp

l

lpdr

1

11111

1

1111̀1

1

11

11

(22)

for Dz , where ni 0 and is arbitrarily fixed in D. The result is best possible.

Proof 2.3: We note that .,,,,,,*

, lpSzf ns if and only if

lpSzflI ins

i

p ,,,,,,, *

,,

and

kp

kp

k

i

p

i

i

p zal

lpz

l

lpzflI

1

,1

1

1

11, . (23)

Now, using theorem 2.1 we have

1 1

11

k

kp

i

al

lkp



drl

lpp

l

lp

l

lpp

n

pp

in

pp

1

11111

1

1

1

11111

11

. (24)

That is,

kp

k

i

al

lk

1 1

11

drl

lpp

l

lpp

in

pp

n

pp

1

11111

1

11111

11

. (25)

It follow from (23) and (25) that

zflI i

p ,,

kp

k

kp

ip

i

zal

lkpz

l

lp

1 1

11

1

11

1

1

1

11

1

11

k

kp

i

p

i

pa

l

lkpdr

l

lpdr

in

pp

n

pppi

p

l

lpp

l

lpdrp

l

lpdr

1

11111

1

11111

1

11

11

(26)

nd also

zflI i

p ,,

in

pp

n

ppp

i

p

l

lpp

l

lpdrp

l

lpdr

1

11111

1

11111

1

11

11

(27)

Finally, we see that the inequality in (22) is attained by the function

zflI i

p ,,

p

i

zl

lp

1

11

kp

in

pp

n

pp

z

drl

lpp

l

lpp

1

11111

1

11111

11

(28)

or by



kp

n

pp

n

pp

pz

drl

lpp

l

lpp

zzf

1

11111

1

11111

11

(29)

and this complete the proof of theorem 3.1.

Corollary 3.1: Let the function zf be defined by (4) be in the class .,,,,,,*

, lS ns . Then, we

have

zf

l

lpp

l

lpp

drn

pp

n

pp

p

1

11111

1

11111

11

n

pp

n

pp

p

l

lpp

l

lpp

dr

1

11111

1

11111

11

(30)

for Dz , where is arbitrarily fixed in D .The result is best possible for the function zf given by (29).

Proof: For 0i in theorem 3.1, we can easily show (30).

Theorem3.2: Let the function zf defined by (4) be in the class lpS inc ,,,,,,*

, .

Then, we have

in

np

l

lp

l

lpdrp

p

l

lpdr

1

1121

1

1112

1

11

zflI i

p ,,

in

n

pi

p

l

lpp

l

lpdrp

l

lpdr

1

1121

1

1112

1

11

(31)

for Dz , where ni 0 and is arbitrarily fixed in D. The result is best possible for the function

zf given by

zflI i

p ,,

kp

in

n

p

i

z

drl

lpp

l

lpp

zl

lp

1

1121

1

1112

1

11 (32)

or by

kp

n

n

pz

drl

lpp

l

lpp

zzf

1

1121

1

1112

. (33)



Corollary 3.2: Let the function )(zf defined by (4) be in the class .,,,,,,*

, lpS nc

Then, we have

)(

1

11)2(1

1

)1(11)2(

zf

l

lpp

l

lpdrp

drn

n

p

p

n

n

p

p

l

lpp

l

lpdrp

dr

1

11)2(1

1

)1(11)2(

(34)

for Dz , where is arbitrarily fixed in D .The result is best possible for the function )(zf given by (33).

Theorem 3.3: Let the function )(zf defined by (4) be in the class

lpS insc ,,,,,,*

, .

Then, we have

in

pp

n

pppi

p

l

lpp

l

lpdrp

l

lpdr

1

1)1()1(11

1

)1(1)1()1(1)(

1

)1(1

11

)(),(, zflI i

p

in

pp

n

pppi

p

l

lpp

l

lpdrp

l

lpdr

1

1)1()1(11

1

)1(1)1()1(1)(

1

)1(1

11

(35)

for Dz , where ni 0 and is arbitrarily fixed in D. This result is sharp. Our next results shall include

linear combinations of several functions of the type (4).

IV. 4. Convex linear Combination

Theorem 4.1: Let

Npandzzf p

p ,0,)()( (36)

and

)(zf kp

.)(

)(1

)1(1)1()1(1)1(

1

)1(1)1()1(1

kp

k

n

kpkp

n

pp

pz

drl

lkppk

l

lpp

z

(37)

Then, lpSzf ns ,,,,,,)( *

, if and only if it can be expressed in the form:



0

)()(k

kpkp zfzf

, (38)

where

100

k

kpkp and . (39)

Proof: Let

0

)()(k

kpkp zfzf

1

)()(k

kpkppp zfzf

. (40)

Using (36), (37), (38) and (39), it is easily seen that

)(zf

.)(

)(1

)1(1)1()1(1)1(

1

)1(1)1()1(1

kp

k

n

kpkp

kp

n

pp

pz

drl

lkppk

l

lpp

z

(41)

since

kp

kn

pp

k

n

kpkp

l

lpp

drl

lkppk

1

1

)1(1)1()1(1

)(1

)1(1)1()1(1)1(

1

.

)(1

)1(1)1()1(1)1(

1

)1(1)1()1(1

k

kp

k

n

kpkp

n

pp

drl

lkppk

l

lpp

11 p . (42)

It follows from theorem 2.1 that the function lpSzf ns ,,,,,,)( *

, . Conversely,

let us suppose that lpSzf ns ,,,,,,)( *

, . Since

k

n

kpkp

n

pp

kp

drl

lkppk

l

lpp

a

)(1

)1(1)1()1(1)1(

1

)1(1)1()1(1

DzandNnNpk 10,10,1,,,1 0 .

Setting

n

pp

k

n

kpkp

kp

l

lpp

drl

lkppk

1

)1(1)1()1(1

)(1

)1(1)1()1(1)1(

and



1

1k

kpp .

It follows that

0

)()(k

kpkp zfzf

and this completes the proof of the theorem.

Corollary 4.1: The extreme points of the class lpS ns ,,,,,,*

, are functions given by (36) and (37).

Theorem 4.2: Let

Npandzzf p

p ,0,)()( (43)

and

)(zf kp

kp

n

pp

n

p z

drl

lpp

l

lpp

z

)(1

1)1()1(11

1

)1(112

)(11

(44)

Then, lpSzf nc ,,,,,,)( *

, if and only if it can be expressed in the form

0

)()(k

kpkp zfzf

, (45)

where

100

k

kpkp and . (46)

The proof is similar to that of theorem 4.1.

Corollary 4.2: The extreme points of the class lpS nc ,,,,,,*


Theorem 4.3. Let

Npandzzf p

p ,0,)()( (47) and

)(zf kp pz )(

.

)(1

)1(1)1()1(1)1(

1

)1(1)1()1(1

kp

k

n

kpkp

n

pp

z

drl

lkppk

l

lpp

48)

Then, lpSzf nsc ,,,,,,)( *

, if and only if it can be expressed in the form:

0

)()(k

kpkp zfzf

, (49)

where

100

k

kpkp and .

The proof is also similar to that of theorem 4.1.

Corollary 4.3: The extreme points of the class lpS nsc ,,,,,,*


Theorem 4.4: The class lpS ns ,,,,,,*

, is closed under convex linear combination.

Proof: Let us suppose that the function )(1 zf and

)(2 zf defined by



);2,1()()()(1

,

zjazzfk

jkp

p

j (50)

are in the class lpS ns ,,,,,,*

, .

Setting

)10(,)()1()()( 21 zfzfzf . (51)

Then from (50), we can write

);10()()()1()()()(1

2,1, Dzzaazzf kp

k

kpkp

p

(52)

Thus, in view of theorem 2.1, we can have that

)()1()()()1()1(1)1( 2,1,

1

kpkp

k

k

kpkp aadrpk

)()()1()1(1)1( 1,

1

kp

k

k

kpkp adrpk

)()()1()1(1)1()1( 2,

1

kp

k

k

kpkp adrpk

pppp pp )1()1(1)1()1()1(1

ppp )1()1(1 ,

nn

l

lpand

l

lkp

1

)1(1

1

)1(1

which show that lpSzf ns ,,,,,,)( *

, and this complete the proof of theorem 4.4.

Theorem 4.5: the class lpS nc ,,,,,,*

, is closed under convex linear combination.

Proof: Let us suppose that the function )(1 zf and

)(2 zf defined by

);2,1()()()(1

,

zjazzfk

jkp

p

j (53)

are in the class lpS nc ,,,,,,*

, .

Setting

)10(,)()1()()( 21 zfzfzf . (54)


).;10()()()1()()()(1

2,1, Dzzaazzf kp

k

kpkp

p

(55)


5. Radii of starlikeness and convexity

Theorem 5.1: Let the function)(zf defined by (4) in the class lpS ns ,,,,,,*

, .

Then )(zf is starlike of order )10( in 1|| rz , where

.

1

)1(1)1()1(1)1(

)(1

)1(1)1()1(1)1()(

inf

1

1

k

n

pp

k

n

kpkp

k

l

lppkp

drl

lkppkp

r

(56)

The result is best possible with the extremal function given by (15) and 1r attains its infimum for 1k .



Proof: It is ample to show that

)(

))()((

zf

zfz

for 1rz . This implies that

p

zaz

zak

kp

k kp

p

kp

kpk

1

11

or

pza

zak

k

k kp

k

kpk

1

1

1

1.

That is

1

11

p

zapkk

kpk. (57)

From theorem 2.1, we have

.1

1

111111

1

1111111

1

n

pp

kp

n

k

kpkpk

l

kpp

al

kppkdr

(58)

Hence, (57) is proven true if

p

zpkk

1

n

pp

n

kpk

l

kpp

l

kppkdr

1

111111

1

1111111

.

That is,

k

n

pp

k

n

kpkp

l

lppkp

drl

lkppkp

z

1

1

)1(1)1()1(1)1(

)(1

)1(1)1()1(1)1()(

1k

and this ends the proof of theorem 5.1.

Remark: It is clear that 1r attains its infimum at 1k for the function zf given by

kp

n

pp

n

pp

pz

l

lpp

l

lpp

zzf

1

11111

1

11111

11

.

Theorem 5.2: Let the function zf defined by (4) be in the class lpS nc ,,,,,,*

, . Then zf is

starlike of order 10 in 2rz , where



k

n

k

n

k

l

kpp

drl

kppkp

r

1

1

11112

1

111121

inf2

. (59)

The result is best possible with the extremal function given by (18) and 2r attains its infimum for 1k . The

proof is similar to that of Theorem 5.1.

Theorem 5.3: Let the function zf defined by (4) be in the class lpS nsc ,,,,,,*

, . Then zf is

starlike order 10 in 3rz , where

k

n

pp

k

n

kpkp

k

l

ppkpk

drl

kppkp

r

1

1

111111

1

1111111

inf3

(60)

Proof: It is ample to show that

ppf

fzi

//

1

for 1rz . This implies that

p

zakpzp

zakpk

kp

kpk

p

kp

kpk

1

1

1

1

1

or

p

zap

kp

zap

kpk

k

kpk

k

kpk

1

1

1

.

That is,

11

pp

zapkpkk

kpk. (61)

Now, (61) is proven true if

pp

zapkpkk

kp

n

pp

n

kpkpk

l

pp

l

kppkdr

1

111111

1

1111111

. (62)

Solving for z , we have that



.

1

111111

1

1111111

1

k

n

pp

k

n

kpkp

l

ppkpk

drl

kppkp

z

This completes the proof of theorem 5.3.

References [1]. M. Acu, S. Owa. On some subclass of univalent functions, J. Inequalitity in Pure and Appl. Math., 6 (3) 70(2005), 1-14.

[2]. M. K. Aouf, R.M. El-Ashwali, S.M. El-Deeb, Certain classes of Univalent functions with negative coefficient and n-starlike with

respect to certain points, Malemankn Bectink, 62(3) (2010), 215-226.

[3]. M.K Aouf, A.Shamandy, A.O Mostafa, and S.M Madian, A subclass of starlikem functions, Acta Universitation

Apulensis No 21 (2010), 135-142.

[4]. J. Dziok, On the convex combination of the Dziok-Srivastava operator, Appl. Math. Comput., 188(2006), 1214-1220.

[5]. R.M. El-Ashwah, D.K. Thomas, Some subclass of close-to-convex functions, J. Ramaunjan. Math. Soc., 2(1987), 86-100.

[6]. S.A. Halim, A. Janteng, M. Darus, Coefficient properties for classes with negative coefficient and starlike with respect to other

points, proceeding of the 13th Math. Sci. Nat. Symposium, UEM., 2(2005), 658-663.

[7]. S.Kanas, F. Ronning, Uniformly starlike and Convex functions and related classes of univalent functions, Ann. Univ. Mariae Curie

– Sklodowska Section A, 53 (1999), 95-105.

[8]. A.T. Oladipo, On subclasses of analytic and univalent functions, Advances in Applied Mathematical Analysis functions, 4 (1)

(2009), 87-93.

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