J.
Indones.
Math.
Soc.
(MIHMI)
Vol.
No.
, pp.
27Ae38.
ON CERTAIN CLASSES OF P -VALENT
FUNCTIONS DEFINED BY
MULTIPLIER TRANSFORMATION
AND DIFFERENTIAL OPERATOR
Tehranchi .
Kulkarni and G.
Murugusundaramoorthy Abstract.
In this paper, we discuss the p-valent functions that satisfy the differential subordinations z(Ip .
,)f .
Oe.
(Ip .
,)f .
O a .
B (AOeB))z a.
We also obtain coefficient inequalities, extreme points, integral representation and arithmetic mean.
Further we investigate some interesting properties of operators defined on Ap .
, j, , a.
B).
INTRODUCTION
Denote by A the class of functions f .
= z Oe n=2 an z n analytic in D = .
OO C : .
< .
Also denote Ap the class of all analytic functions of the form f .
= kz p 2pOe1 tnOep 1 z nOep 1 Oe 2 F1 .
, b.
, .
< 1
n=pOe1 2 F1 .
, b.
, .
, .
, .
tnOep 1 .
, .
n! e.
= a.
1, n Oe .
, c > b > 0, c > a b and e.
, n Oe p .
, n Oe p .
, k > 0.
, n Oe p .
Oe p .
! Received 14 August 2005, revised 28 July 2006, accepted 2 August 2006 .
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words and Phrases: Multivalent function.
Differential subordinations, multiplier transformation, differential operator.
Tehranchi .
Kulkarni and G.
Murugusundaramoorthy These functions are analytic in the unit disk D (For details see .
, .
Definition 1.
A function f OO Ap is said to be in the class SpO () , p-valently n 0 o .
starlike functions of order , if it satisfies Re zff .
> , .
O < p, z OO D).
We note that Sp .
= Sp , the class of p-valently starlike functions in D.
A function f OOnAp is said to o be in the class Cp () of p-valently convex of order , if it satisfies .
Re 1 zff 0 .
> , .
O < p, z OO D).
Let h.
be analytic and h.
= 1.
A function f OO Ap is in the class SpO .
if zf 0 .
O h.
, f .
z OO OI.
The class SpO .
and a corresponding convex class Cp .
are defined by Ma and Minda .
But results about the convex class can be obtained easily from the corresponding result of functions in SpO .
= 1Oez then the classes reduce to the usual classes of starlike and convex functions.
= 1 .
Oe.
z , 0 O < p, then the classes reduce to the usual classes of starlike 1Oez 1 Az and convex functions of order .
If h.
= 1 Bz Oe 1 O B < A O 1, then the classes reduce to the class of Janowski starlike function SpO [A.
B] defined by 1 Az f OO Ap :
1 Bz SpO [A.
B] = Oe 1 O B < A O 1, z OO D .
, 0 < O 1, then the classes reduce to the classes of strongly If h.
= 1Oez starlike and convex function of order that consists of univalent functions f OO A AA 0 AA Aarg zf .
A < A , 0 < O 1, z OO OI f .
A or equivalently we have SS O () = f OO Ap :
1Oez , 0 < O 1, z OO OI .
ObradovicU and Owa .
Silverman .
ObradovicU and Tuneski .
and Tuneski .
have studied the properties of classes of functions defined in terms of the ratio of .
1 zff 0 .
and zff .
On certain classes of p-valent functions Definition 1.
A function f OO Ap is said to be p-valent Bazilevic of type and order if there exists a function g OO SpO such that zf 0 .
1Oe .
> , .
OO OI) .
for some ( Ou .
O < .
We denote by Bp (, ), the subclass of Ap consisting of all such functions.
In particular, a function in Bp .
, ) = Bp () is said to be p-valently close-to-convex of order in OI.
Definition 1.
For two functions f and g, analytic in OI we say f is subordinate to g denoted by f O g if there exists a Schwarz function w.
, analytic in OI with w.
= 1 and .
| < 1, such that f .
= g.
), z OO OI.
In particular, if the function g is univalent in OI, the above subordination is equivalent to f .
= g.
, f (OI) OC g(OI).
Also, we say that g is superordinate to f .
Using the techniques of Cho and Srivastava.
,Cho and Kim.
and Uralegadi and Somanatha.
we define the following transformation.
Definition 1.
We define the multiplier transformation operator Ip .
, ) on the O infinite series f .
= z p Oe
an z n as n=p 1
O AA
n Ip .
, )f .
= z Oe an z n , ( Ou .
n=1 p We note that SaUlaUgean derivative operators .
is closely related to the operators Ip .
, ) when the coefficient of f .
is positive.
Also note that the class I1 .
, .
= Ir .
,I1 .
, ) = Ir the classes studied in .
Definition 1.
For each f .
= z p Oe n=p 1 = an z n we have O z pOej Oe an z nOej Oe .
! Oe n=p 1 where n, p OO N, p > j, and j OO N0 = .
O N .
For j = 0 we have f .
= f .
Definition 1.
A function f OO Ap is said to be in the class Ap .
, j.
if it satisfies z(Ip .
, )f .
O h.
Oe .
(Ip .
, )f .
Tehranchi .
Kulkarni and G.
Murugusundaramoorthy h.
= 1 A Oe B z a 1 Bz z OO OI, and Oe1 O B < A O 1, 0 < < p, a > 0 , we denote Ap .
, j.
= Ap .
, j, , a.
B).
We say that f .
is superordinate to h.
if f .
satisfies the following h.
O z(Ip .
, )f .
Oe .
(Ip .
, )f .
where h.
is analytic in OI and h.
= 1.
We note that if z(Ip .
, )f .
Oe .
B (A Oe B)).
O .
(Ip .
, )f .
) so h.
= 1 by choosing j = r = 0, p = 1, then f .
OO SpO .
Also a = A = = 1.
B = Oe1, f .
OO SpO .
But if a = = 1 and Oe1 O B < A O 1 then f .
OO S O [A.
B], class of Janowski starlike function.
If we put p = a = = A = 1.
B = Oe1 then f .
OO SS O .
classes of strongly starlike.
By Definition 1.
, if g.
starlike and j = r = 0 and p = a = n OO S 0 , univalent zf .
A = = 1.
B = Oe1 and if Re f .
Oe1 g2 .
> 1, then f .
OO B.
, .
class Bazilevic function of type = 2 and order = 1.
MAIN RESULTS
In this section we obtain sharp coefficient estimates for functions in Ap .
, j, , a.
B).
Theorem 2.
Let f .
be of the form .
Then f OO Ap .
, j, , a.
B) if and only if a.
B)(.
, .
Oe .
Oe .
, j .
) .
, .
Oe .
, .
km < 1 (A Oe B).
, j .
, j .
m=p 1 Ar r .
, .
= m p .
Oe .
! Oe 1 O B < A O 1, 0 < < p, j < p.
, .
= .
Ou 0, p, r OO N On certain classes of p-valent functions Proof.
The function f .
of the form .
can be expressed as f .
= z p Oe knOep 1 z nOep 1 , or n=2p f .
= z p Oe km z m m=p 1 where m = n Oe p 1 and km = .
m! , and also we have for all r, j OO N0 (Ip .
, )f .
) Ar p! m pOej km z mOej Oe .
! Oe m=p 1 , .
z pOej Oe , .
, .
km z mOej .
m=p 1 Let f .
OO Ap .
, j, a, .
B) then A az(Ip .
, )f .
Oe a.
Oe .
(Ip .
, )f .
A A R.
Oe .
(I .
, )f .
Oe Baz(I .
, )f .
A < 1
where R = aB (A Oe B).
Now, we can write , .
, .
Oe .
Oe .
, j .
)km z mOej m=p 1 < 1.
, .
, .
Oe .
R Oe .
, j .
km z mOej (A Oe B).
, j .
z pOej Oe m=p 1 We choose the values of z on the real axis and letting z Ie 1Oe then we have O , .
, .
Oe .
Oe .
, j .
)km m=p 1 (A Oe B).
, j .
, .
B(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
)km m=p 1 , .
B)(.
, .
Oe .
m=p 1 Oe.
, j .
) Oe .
, .
(A Oe B).
Oe .
]km < (A Oe B).
, j .
Conversely, we assume that the condition .
holds true.
Hence it is sufficient to show that f OO Ap .
, j, , a.
B), that is to prove that A az(Ip .
, )f .
Oe a.
Oe .
(Ip .
, )f .
A A .
Oe .
R(I .
, )f .
Oe Baz(I .
, )f .
A < 1.
< 1,
Tehranchi .
Kulkarni and G.
Murugusundaramoorthy But we have A az(Ip .
, )f .
Oe a.
Oe .
(Ip .
, )f .
A A .
Oe .
R(I .
, )f .
Oe Baz(I .
, )f .
A O
= |.
, .
, .
Oe .
Oe .
, j .
)km z mOej ]/ m=p 1 [(A Oe B).
, j .
z pOej Oe r .
, .
B(.
, .
Oe .
Oe .
, j .
) Oe m=p 1 .
, .
(A Oe B).
Oe .
)km ]| O < {.
, .
, .
Oe .
Oe .
, j .
)km ]/ m=p 1 [(A Oe B).
, j .
O Oe r .
, .
B(.
, .
Oe .
Oe .
, j .
) Oe m=p 1 .
, .
(A Oe B).
Oe .
)km ]} < 1 and so proof is complete.
The inequality .
is sharp for the function f .
= z p Oe (A Oe B).
, j .
, .
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
] with q Ou 1 p.
Corollary 2.
Let f OO Ap .
, j, , a.
B) then we have (AOeB).
,j .
km < r .
B)(.
Oe.
Oe.
,j .
)Oe.
(AOeB).
Oe.
] .
with m Ou p 1 In the next theorem we prove that the class Ap .
, j, , .
B) is closed under linear combination.
Theorem 2.
Let fq .
= z p Oe Then the function F .
= O m=p 1 km,q z m .
= 1, 2.
A A A , .
be in Ap .
, j, , a.
B).
dq fq .
where dq = 1, is also in Ap .
, j, , a.
B).
On certain classes of p-valent functions Proof.
We have F .
= z Oe km,q z m=1 p yE t m=1 p = z Oe = z Oe .
, .
O km,q z m=1 p dq km,q (A Oe B).
, j .
, .
Oe .
Oe .
, j .
yE t dq dm,q O a.
B)(.
, .
Oe .
Oe .
, j .
) .
, .
Oe .
, .
Oe (A Oe B).
, j .
, j .
Hence we obtain O a.
B)(.
, .
Oe .
Oe .
, j .
m=1 p m=1 p dq = 1.
Now we prove that the class Ap .
, j, , a.
B) is closed under arithmetic mean.
Theorem 2.
Let fj .
= z p Oe m=1 p km,q z m .
= 1, 2.
A A A , .
are in Ap .
, j, , a.
B).
Then the function F .
= z p Oe m=1 p bm z m where bm = 1s km,q , is also in Ap .
, j, , a.
B).
Proof.
Since fj .
OO Ap .
, j, , a.
B), then by .
we have O a.
B)(.
, .
Oe .
Oe .
, j .
) .
, .
Oe .
Oe r .
, .
(A Oe B).
, j .
, j .
m=1 p km,q < 1, q = 1, 2, 3.
A A A s.
Therefore .
, .
B)(.
Oe.
Oe.
,j .
) Oe .
Oe.
(AOeB).
,j .
,j .
Oe.
= m=1 p r .
, .
B)(.
Oe.
Oe.
,j .
) Oe (AOeB).
,j .
,j .
Ps APO B)(.
Oe.
Oe.
,j .
) .
Oe.
O s q=1 Oe .
,j .
km,q m=1 p .
, .
(AOeB).
,j .
O q=1 1s = 1 m=1 p and this completes the proof.
Theorem 2.
(Extreme Point.
Let fp .
= z p and for m Ou 1 p fm .
z p Oe (A Oe B).
, j .
, .
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
] Tehranchi .
Kulkarni and G.
Murugusundaramoorthy Then the function f .
OO Ap .
, j, , a.
B) if and only if it can be expressed in the O AAm fm .
= where AAm Ou 0 and O AAm = 1.
Proof.
Suppose that f can be expressed in the form .
then we have O AAm fm .
AAp fp .
AAm fm .
m=p 1 AAp z p AAm .
p Oe m=p 1 (A Oe B).
, j .
, .
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
] O z p Oe m=1 p Consequently O (A Oe B).
, j .
z m AAm r .
, .
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
] r .
, .
m=p 1 a.
B)(.
, .
Oe .
Oe .
, j .
) .
, .
Oe .
Oe (A Oe B).
, j .
, j .
(A Oe B).
, j .
AAn r .
, .
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
] O AAm = 1 Oe AAp < 1.
m=1 p Therefore we conclude the result.
Conversely, let f OO Ap .
, j, , a.
B) since by .
we may set AAm = r .
, .
km a.
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
(A Oe B).
, j .
with m Ou 1 p.
Therefore AAm Ou 0 and if we set AAp = 1 Oe m=1 p AAn then we can f .
z p Oe km z m m=1 p z p Oe m=1 p (A Oe B).
, j .
AAm z m , .
B)(.
, .
Oe .
Oe .
, j .
) Oe .
, .
(A Oe B).
Oe .
] On certain classes of p-valent functions z p Oe AAm .
p Oe fm .
) m=1 p z p .
AAm ) Oe m=1 p AAm fm .
m=1 p AAm fm .
Remark 2.
The extreme points of the class Ap .
, j, , a.
B) are the function fp .
, fm p .
, m Ou 1 p as in Theorem 2.
In the following theorem, we obtain the integral representation for Ap .
, j, , a.
B).
Theorem 2.
Let f .
OO Ap .
, j, , a.
B) then AZ z p(O.
R .
= exp 0 t.
BO.
) where |O.
| < 1, z OO U and R = aB (A Oe B).
Also AZ p(AOeB) f .
= z exp log.
Oe Bx.
dAA.
where AA.
is the probability measure on X = .
: .
= .
Proof.
Set z(Ip .
, )f .
= Q.
Oe .
(Ip .
, )f .
A a(Q.
Oe.
A Since f .
OO Ap .
, j, , a.
B) so A ROeaBQ.
A < 1 where R = aB (A Oe B).
a(Q.
Oe.
= O.
, |O.
| < 1.
Consequently we put ROeaBQ.
R a Finally we can write Q.
= a aBO.
(Ip .
, )f .
Oe .
(O.
R .
BO.
) (Ip .
, )f .
Then we have log(Ip .
, )f .
) .
Z z .
(Ip .
, )f .
) AZ z = exp .
Oe .
(O.
R .
BO.
) .
Oe .
(O.
R .
BO.
) Tehranchi .
Kulkarni and G.
Murugusundaramoorthy for r = j = 0 we have AZ z p(O.
R .
BO.
) f .
= exp For obtaining the second representation let X = .
: .
= .
then we have a(Q.
Oe.
ROeaBQ.
= xz, z OO OI and then we conclude that (Ip .
, ).
(Ip .
, )f .
Oe .
Oe .
(A Oe B) .
Oe .
(Rxz .
Bx.
Bx.
x(A Oe B) .
Oe .
Bx.
log(Ip .
, )f .
) (A Oe B) log.
Bx.
= .
Oe .
Oe .
(A Oe B) (Ip .
, )f .
Bx.
pOej Oe.
(AOeB) .
pOej dAA.
(Ip .
, )f .
) = z log.
Oe Bx.
where AA.
is probability measure on X.
For j = r = 0 we have AZ f .
= z exp log.
Oe Bx.
p(AOeB) dAA.
Now, we introduce an integral operator due to Bernardi .
Lc .
) = Z z f .
tcOe1 dt, .
> Oe.
and we study the effect of this operator on class Ap .
, j, , a.
B).
Theorem 2.
If f OO Ap .
, j, , a.
B) then Lc .
) is also in Ap .
, j, , a.
B).
Proof.
If f .
= z p Oe m=1 p Lc .
) = km z m then z p Oe Z zyE t Oe m=1 p km z m .
m=1 p tcOe1 dt On certain classes of p-valent functions Since m > p then m c O 1 so we have a.
B)(.
Oe.
Oe.
,j .
) .
Oe.
Oe m=1 p (AOeB).
,j .
,j .
m c km PO Oe .
Oe.
km < 1.
O m=1 p r .
, .
B)(.
Oe.
Oe.
,j .
) (AOeB).
,j .
,j .
PO Thus Lc .
) OO Ap .
, j, , a.
B).
REFERENCES