International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 1 (2016), 1-8
http://www.etamaths.com

APPLICATIONS OF EXTREMAL THEOREM TO A CLASS OF P-VALENT

ANALYTIC FUNCTIONS

LIANGPENG XIONG∗ AND XIAOLI LIU

Abstract. A subclass J
m,l
p,λ

(ξ,α) of p-valent analytic functions with a generalized multiplier trans-

formation operator is introduced. We discuss the compactness as well as the extreme points of

J
m,l
p,λ

(ξ,α) under the topology of uniform convergence. Finally, as one of the applications of ex-

tremal theorem, we solve the sharp distortion inequalities problem as

max
f∈J m,l

p,λ
(ξ,α)

|f(χ)(z)|,χ = 0, 1, 2, ...

Several related basic results and remarks about the old or new classes are also presented.

1. Introduction

Let Ap denote the class of functions of the form

(1) f(z) = zp +

∞∑
k=p+1

akz
k,p ∈ N = {1, 2, 3, ...},

which are analytic in 4 = {z : z ∈ C, |z| < 1}.
In fact, Ap is a vector space over C with the usual definitions of addition and scalar multiplication

for functions. Let the topology on Ap be given by a metric ρ which is equivalent to the topological of
uniform convergence on compact subsets, where the ρ is determined as

ρ(f,g) =

∞∑
n=1

1

2n
‖f −g‖n

1 + ‖f −g‖n

whenever f and g belong to Ap, and ‖f‖n = max{|f(z)| : |z| = rn, 0 < rn < 1, lim
n→∞

rn = 1}. It follows
from theorems of Weierstrass and Montel that this topology space is complete(see [11], P38).

If F ⊂ Ap then F is called locally uniformly bounded if there is a constant M such that |f(z)|6 M
whenever f ∈ F. Moreover, Montel’s theorem implies that F ⊂ Ap is compact if and only if F is closed
and locally uniformly bounded(see [11], P39).

We use the notation HF for the closed convex hull of F, where

HF =
{ ∞∑
k=1

tkfk,fk ∈ F, tk > 0,
∞∑
k=1

tk = 1
}
.

Let V be a subclass of a linear topological space. If v ∈ V and if v = tf1 + (1 − t)f2, 0 < t < 1,f1 ∈
V,f2 ∈ V can make sure that f1 = f2, then we say v ∈ EV, where EV denotes the set of extreme
points of V. Again, suppose that M is a convex subset of Ap and J : M → R, if

J(tf + (1 − t)g) 6 tJ(f) + (1 − t)J(g)

whenever f,g ∈ M and 0 6 t 6 1, then linear functional J is called convex on M.

2010 Mathematics Subject Classification. 30C35, 30C45, 35Q30.
Key words and phrases. P-valent functions; extreme points; linear topological space; distortion inequalities; multiplier

transformation operator.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

1



2 XIONG AND LIU

Recently, Prajapat [18] and Sharma et al. [20] studied a generalized multiplier transformation
operator Jmp (λ,l) : Ap → Ap as

(2) Jmp (λ,l)f(z) =



p + l

λ
zp−

p+l
λ

∫ z
0

t
p+l
λ
−p−1Jm+1p (λ,l)f(t)dt, m ∈ Z−,

λ

p + l
z1+p−

p+l
λ

(
z
p+l
λ
−pJm−1p (λ,l)f(z)

)′
, m ∈ Z+,

f(z), m = 0.

where m ∈ Z = {..,−2,−1, 0, 1, 2, ..},λ > 0, l > −p. It easily follows from the above definition of the
operator that the series expansion of Jmp (λ,l)f(z) for f(z) of the form (1.1) is given by

(3) Jmp (λ,l)f(z) = z
p +

∞∑
k=p+1

(
1 +

λ(k −p)
p + l

)m
akz

k.

We note that operator Jmp (λ,l) contains kinds of operators introduced and studied by different
mathematicians (for details, see [3, 4, 9, 13]). Now, we reproduce here briefly some of these special
cases as follows.

Remark 1.1. (i) For m ∈ Z+ ∪{0}, Jmp (λ,l) ≡ Imp (λ,l), (see Cătaş[5]).
(ii) For m ∈ Z+ ∪{0}, Jmp (1, l) ≡ Ip(m,l), (see Kumar et al. [12]).
(iii) For m ∈ Z+ ∪{0}, Jmp (1, 0) ≡ Dmp , (see Orhan et al. [16]).
(iv) For m ∈ Z− ∪{0}, Jmp (λ,l) operator, (see El-Ashwah and Aouf [10]).
(v) For m ∈ Z− ∪{0}, Jmp (1, 1) operator, (see Patel and Sahoo [17]).

For various other special cases studied earlier of the operator Jmp (λ,l), one can refer to [8, 14, 15, 18].
Let Vp be the subclass of Ap consisting functions of the form

(4) f∇(z) = z
p −

∞∑
k=p+1

akz
k,ak > 0,p ∈ N = {1, 2, 3, ...},

It is easy to see that

(5) Jmp (λ,l)f∇(z) = z
p −

∞∑
k=p+1

(
1 +

λ(k −p)
p + l

)m
akz

k.

We introduce the class J
m,l
p,λ (ξ,α) as a subclass of Vp consisting of functions f which satisfy

(6) <

{
z
(
Jmp (λ,l)f∇(z)

)′
+ ξz2

(
Jmp (λ,l)f∇(z)

)′′
ξz
(
Jmp (λ,l)f∇(z)

)′
+ (1 − ξ)Jmp (λ,l)f∇(z)

}
> α,z ∈4,

where 0 6 α < p, 0 6 ξ 6 1,m ∈ Z+ ∪{0},λ > 0, l > −p.
Here, for reader’s convenience, we depict some of subclasses related the above J

m,l
p,λ (ξ,α).

Remark 1.2. (i) J
0,l
p,λ(ξ,α) was studied by Altintaş et al.[1] and Xiong [21].

(ii) J
0,l
1,λ(ξ,α) was studied by Altintaş [2].

(iii) J
0,l
p,λ(0,α) ≡ S

∗
p(α) was studied by Darwish and Aouf [6].

(iv) J
0,l
p,λ(1,α) ≡ Kp(α) was studied by Darwish and Aouf [6].

(v) J
0,l
1,λ(0,α) ≡ S

∗(α) and J
0,l
1,λ(1,α) ≡ K(α) were studied by Srivastava et al. [19] and Domokos

[7], respectively.

In this paper we obtain the extreme points for class J
m,l
p,λ (ξ,α). Furthermore, the sharp distortion

inequalities are given by using the extreme points theory.



P-VALENT ANALYTIC FUNCTIONS 3

2. Preliminary results

In this section, we give a sufficient and necessary condition for the functions f∇(z) ∈ Vp to be in
J

m,l
p,λ (ξ,α). As a Lemma, we find that the class J

m,l
p,λ (ξ,α) is the closed convex hull of a set M .

Lemma 2.1. A function f(z) = zp −
∞∑

k=p+1

akz
k belongs to J

m,l
p,λ (ξ,α) if and only if

(7)

∞∑
k=p+1

ψ(k)ak 6 (p−α)[ξ(p− 1) + 1] (p ∈ N = {1, 2, ..}).

where

(8) ψ(k) = (k −α)[ξ(k − 1) + 1]
(

1 +
λ(k −p)
p + l

)m
.

Proof. If we taking n = 1 and

bk =
(

1 +
λ(k −p)
p + l

)m
ak,

then this Lemma is an immediate consequence of Altintas et al. [1] (Also see Xiong [21], Lemma 1])

in function f∇(z) = z
p −

∞∑
k=p+1

bkz
k. �

Lemma 2.2. If M = {fk(z) : k = p,p + 1,p + 2, ...}, then J
m,l
p,λ (ξ,α) = HM , where fk(z) are

determined by 

fp(z) = z

p, k = p,

fk(z) = z
p −

(p−α)[ξ(p− 1) + 1]
ψ(k)

zk, k > p + 1.

Proof. Suppose that f(z) ∈ HM and

f(z) = λpfp(z) +

∞∑
k=p+1

λkfk(z),λk > 0,λp +
∞∑

k=p+1

λk = 1.

Making use of the elements in M , we can express

f(z) = λpz
p +

∞∑
k=p+1

λk

[
zp −

(p−α)[ξ(p− 1) + 1]
ψ(k)

zk
]

= λpz
p +

∞∑
k=p+1

λkz
p −

∞∑
k=p+1

λk
(p−α)[ξ(p− 1) + 1]

ψ(k)
zk

=
(
λp +

∞∑
k=p+1

λk

)
zp −

∞∑
k=p+1

λk
(p−α)[ξ(p− 1) + 1]

ψ(k)
zk

= zp −
∞∑

k=p+1

λk
(p−α)[ξ(p− 1) + 1]

ψ(k)
zk = zp −

∞∑
k=p+1

bkz
k,

where

(9) bk = λk
(p−α)[ξ(p− 1) + 1]

ψ(k)
.

Consequently, by using (2.3) and the Lemma 2.1, we are lead to

∞∑
k=p+1

ψ(k)

(p−α)[ξ(p− 1) + 1]
bk =

∞∑
k=p+1

λk = 1 −λp 6 1,

thus, it follows that f(z) ∈ J m,lp,λ (ξ,α). This implies HM ⊂ J
m,l
p,λ (ξ,α).



4 XIONG AND LIU

We next consider J
m,l
p,λ (ξ,α) ⊂ HM . If a function f(z) ∈ J

m,l
p,λ (ξ,α), recalling Lemma 2.1 we see

that

ak 6
(p−α)[ξ(p− 1) + 1]

ψ(k)
(k > p + 1).

Here, taking

λk =
ψ(k)

(p−α)[ξ(p− 1) + 1]
ak,

where k > p + 1 and λp = 1−
∞∑

k=p+1

λk, then we can know 0 6 λk 6 1,k > p. Hence, we conclude that

f(z) = zp −
∞∑

k=p+1

akz
k = zp −

∞∑
k=p+1

(p−α)[ξ(p− 1) + 1]
ψ(k)

λkz
k

= zp −
∞∑

k=p+1

λk

[
zp −

(
zp −

(p−α)[ξ(p− 1) + 1]
ψ(k)

zk
)]

= zp −
∞∑

k=p+1

λkz
p +

∞∑
k=p+1

λk

(
zp −

(p−α)[ξ(p− 1) + 1]
ψ(k)

zk
)

=
(

1 −
∞∑

k=p+1

λk

)
zp +

∞∑
k=p+1

λkfk(z) =

∞∑
k=p

λkfk(z).

This proves that J
m,l
p,λ (ξ,α) ⊂ HM and completes the proof of Lemma 2.2. �

Lemma 2.3. ([11],P44) Let X be a locally convex linear topological space and let U be a compact subset
of X. If HU be a compact then EHU ⊂ U.

3. Compactness and Extreme points

In this section, we prove that the class J
m,l
p,λ (ξ,α) is compact subset of Ap, which can help us to

obtain the extreme points of J
m,l
p,λ (ξ,α) and to complete the works in section 4.

Theorem 3.1. The class J
m,l
p,λ (ξ,α) is a compact subset of Ap.

Proof. We need to prove J
m,l
p,λ (ξ,α) is closed and locally uniformly bounded. Suppose that

f(z) = zp −
∞∑

k=p+1

akz
k ∈ J m,lp,λ (ξ,α)

and |z|6 r, 0 < r < 1, then

|f(z)|6 rp +
∞∑

k=p+1

ψ(k)|ak|
rk

ψ(k)
6 rp + (p−α)[ξ(p− 1) + 1]

∞∑
k=p+1

rk

ψ(k)
.

Moreover, it is easy to see that

lim
k−→∞

sup
( rk
ψ(k)

)1
k

= r lim
k−→∞

sup(ψ(k))−
1
k = r < 1,

this shows that the series
∞∑

k=p+1

rk

ψ(k)
is convergent, so the results above assert that J

m,l
p,λ (ξ,α) is locally

uniformly bounded.

We next prove that the J
m,l
p,λ (ξ,α) is sequentially closed and suppose that

{fJ(z)}⊂ J m,lp,λ (ξ,α) ⊂ Vp ⊂ Ap



P-VALENT ANALYTIC FUNCTIONS 5

and {fJ(z)} → f(z) as J → ∞, where fJ(z) = zp −
∞∑

k=p+1

aJkz
k. In fact, Weierstrass’ theorem asserts

f(z) ∈ Vp([11], P38), so we can set f(z) = zp −
∞∑

k=p+1

akz
k, thus, it is easy to know that aJk → ak as

J →∞. Because of Lemma 2.1 and fJ(z) ∈ J m,lp,λ (ξ,α), for a positive integer H, then we have

H∑
k=p+1

ψ(k)aJk 6
∞∑

k=p+1

ψ(k)aJk 6 (p−α)[ξ(p− 1) + 1].

Choosing J →∞, it follows that

H∑
k=p+1

ψ(k)ak 6 (p−α)[ξ(p− 1) + 1].

Furthermore, taking H → +∞, this leads to
∞∑

k=p+1

ψ(k)ak 6 (p−α)[ξ(p− 1) + 1],

which implies that f(z) ∈ J m,lp,λ (ξ,α). This completes the proof of Theorem 3.1. �

Theorem 3.2. EJ
m,l
p,λ (ξ,α) = M , where M is defined as Lemma 2.2.

Proof. Suppose that

zp −
(p−α)[ξ(p− 1) + 1]

ψ(k)
zk = tg1(z) + (1 − t)g2(z),

where

gi(z) = z
p −

∞∑
k=p+1

ak,iz
k ∈ J m,lp,λ (ξ,α), 0 < t < 1, i = 1, 2,

then we have

(10)
(p−α)[ξ(p− 1) + 1]

ψ(k)
= tak,1 + (1 − t)ak,2.

Since gi(z) ∈ J
m,l
p,λ (ξ,α), Lemma 2.1 gives

(11) ak,i 6
(p−α)[ξ(p− 1) + 1]

ψ(k)
, i = 1, 2.

The (10) and (11) imply that

ak,1 = ak,2 =
(p−α)[ξ(p− 1) + 1]

ψ(k)
.

Thus, we can know that g1(z) = g2(z), which gives us

fk(z) = z
p −

(p−α)[ξ(p− 1) + 1]
ψ(k)

zk ∈ EJ m,lp,λ (ξ,α), k > p + 1.

Likewise, we also can obtain fp(z) = z
p ∈ EJ m,lp,λ (ξ,α). In fact, we get M ⊂ EJ

m,l
p,λ (ξ,α). Again, us-

ing the Lemma 2.2, we know J
m,l
p,λ (ξ,α) = HM . Moreover, Theorem 3.1 shows that M ⊂ J

m,l
p,λ (ξ,α)

is a compact set, thus, using the Lemma 2.3, it suffices to verify that EJ
m,l
p,λ (ξ,α) = EHM ⊂ M .

This completes the proof of Theorem 3.2. �



6 XIONG AND LIU

4. Applications of extreme points theorem

We want to maximize the |f(X)(z)|(X = 0, 1, 2, ...) over J m,lp,λ (ξ,α) by making critical use of the
information about extreme points. For this point, let linear functional J : J

m,l
p,λ (ξ,α) → R be defined

as:

J(f) = |f(X)(z)|,f(z) ∈ J m,lp,λ (ξ,α),X = {0, 1, 2, ...}.
It is easy to see that J is a continuous and convex functional. By using the known Krein-Milman
theorem, ([11], P45, Theorem 4.6) gives a important result: let F be a compact subset of A and J is
a real-valued, continuous, convex functional on HF , then

max{J(f) : f ∈ HF} = max{J(f) : f ∈ F} = max{J(f) : f ∈ EHF}.

Thus, following the Lemma 2.2, Theorem 3.1 and Theorem 3.2, we can know that

max{|f(X)(z)| : f ∈ J m,lp,λ (ξ,α)} = max{|f
(X)(z)| : f ∈ HM}

= max{|f(X)(z)| : f ∈ M}.

Hence, in order to solve the extremal problems on |f(X)(z)| over J m,lp,λ (ξ,α), it needs only to solve
them over the smaller class M . We next turn to consider the |f(X)(z)| over M . In Lemma 2.2, taking
the function

f(z) = zp −
(p−α)[ξ(p− 1) + 1]

ψ(k)
zk, k > p + 1,

and after some simplifications, then we have

f(χ)(z) =

p!

(p−χ)!
zp−χ −

(p−α)[ξ(p− 1) + 1]k!
(k −χ)!ψ(k)

zk−χ, χ = 0, 1, 2, ...,p

f(χ)(z) = −
(p−α)[ξ(p− 1) + 1]k!

(k −χ)!ψ(k)
zk−χ, χ = p + 1,p + 2, ...,k

f(χ)(z) = 0, χ > k.

Setting χ ∈ N0 = {0, 1, 2, ...}, 0 < r < 1, we define the sequence {J
(χ)
k } as follows:

Case I: if χ = 0, then

J
(χ)
k =

{
0, k < p + 1,
(p−α)[ξ(p−1)+1]

ψ(k)
rk, k > p + 1.

Case II: if χ ∈ N = {1, 2, 3, ...}, then

J
(χ)
k =




0, k < max{χ,p + 1},
(p−α)[ξ(p− 1) + 1]k!

(k −χ)!ψ(k)
rk−χ, k > max{χ,p + 1}.

We can easily prove that J
(χ)
k → 0 as k →∞, this implies that there is a kχ ∈{p+1,p+2, ...}(χ ∈ N0),

such that

(12) J
(χ)
kχ

= max{J (χ)k : k = p + 1,p + 2, ...}.

Finally, we now present the deserved results according to the analysis above.

Theorem 4.1. Suppose that f(z) = zp −
∞∑

k=p+1

akz
k ∈ J m,lp,λ (ξ,α), where |z| = r < 1, then

(1) If χ = 0, 1, 2, ...,p, we have N1 6 |f(χ)(z)|6 N2, where

N1 =
p!

(p−χ)!
rp−χ −

(p−α)[ξ(p− 1) + 1]kχ!
(kχ −χ)!ψ(kχ)

rkχ−χ

and

N2 =
p!

(p−χ)!
rp−χ +

(p−α)[ξ(p− 1) + 1]kχ!
(kχ −χ)!ψ(kχ)

rkχ−χ.



P-VALENT ANALYTIC FUNCTIONS 7

(2)If χ = p + 1,p + 2, ..., we have

−
(p−α)[ξ(p− 1) + 1]kχ!

(kχ −χ)!ψ(kχ)
rkχ−χ 6 |f(χ)(z)|6

(p−α)[ξ(p− 1) + 1]kχ!
(kχ −χ)!ψ(kχ)

rkχ−χ.

All the above kχ is defined as (12), and ψ(kχ) are the values of ψ(k) in (2.2) whenever k = kχ. The
results are sharp and the extremal functions are given by the M of Lemma 2.2.

Two special cases of Theorem 4.1 when m = 0,ξ = 0 and m = 0,ξ = 1 yield, respectively,

Corollary 4.1. Suppose that f(z) = zp −
∞∑

k=p+1

akz
k ∈ S∗p(α), where |z| = r < 1, then

(1) If χ = 0, 1, 2, ...,p, we have N1 6 |f(χ)(z)|6 N2, where

N1 =
p!

(p−χ)!
rp−χ −

(p−α)kχ!
(kχ −χ)!(kχ −α)

rkχ−χ

and

N2 =
p!

(p−χ)!
rp−χ +

(p−α)kχ!
(kχ −χ)!(kχ −α)

rkχ−χ

(2)If χ = p + 1,p + 2, ..., we have

−
(p−α)kχ!

(kχ −χ)!(kχ −α)
rkχ−χ 6 |f(χ)(z)|6

(p−α)kχ!
(kχ −χ)!(kχ −α)

rkχ−χ,

All the above kχ is defined as (12). The results are sharp and the extremal functions are given by the
M of Lemma 2.2.

Corollary 4.2. Suppose that f(z) = zp −
∞∑

k=p+1

akz
k ∈ Kp(α), where |z| = r < 1, then

(1) If χ = 0, 1, 2, ...,p, we have N1 6 |f(χ)(z)|6 N2, where

N1 =
p!

(p−χ)!
rp−χ −

(p−α)p ·kχ!
(kχ −χ)!(kχ −α)kχ

rkχ−χ

and

N2 =
p!

(p−χ)!
rp−χ −

(p−α)p ·kχ!
(kχ −χ)!(kχ −α)kχ

rkχ−χ

(2)If χ = p + 1,p + 2, ..., we have

−
(p−α)p ·kχ!

(kχ −χ)!(kχ −α)kχ
rkχ−χ 6 |f(χ)(z)|6

(p−α)p ·kχ!
(kχ −χ)!(kχ −α)kχ

rkχ−χ.

All the above kχ is defined as (12). The results are sharp and the extremal functions are given by the
M of Lemma 2.2.

Remark 4.1. In fact, various other interesting consequences of our main results (which are asserted by
Theorems 3.1, 3.2 and 4.1 and Corollaries 4.1 to 4.2 above) can be derived by appropriately choosing
special parameters as the Remark 1.1 and Remark 1.2. The details may be left as an exercise for the
interested reader.

Acknowledgement: The authors are thankful to the referees for reading this paper and this work
was supported by Scientific Research Fund of SiChuan Provincial Education Department of China,
Grant No.14ZB0364.



8 XIONG AND LIU

References

[1] O.Altintaş, H.Irmak and H.M.Srivastava, Fractional calculus and certain starlike func- tions with negative coeffi-

cients, Comput. Math. Appl., 30(2)(1995), 9-16.
[2] O.Altintaş, On a subclass of certain starlike functions with negative coefficients, Math. Japonica, 36(3)(1991), 1-7.

[3] Y.Abu Muhanna, L.L.Li and S.Ponnusamy, Extremal problems on the class of convex functions of order −1/2,
Archiv der mathematik, 103(6)(2014), 461-471.

[4] R.M.Ali, S.R.Mondal and V.Ravichandran, On the Janowski convexity and starlikeness of the confluent hypergeo-

metric function, Bulletin of the belgian mathematical society-simon stevin, 22(2)(2015), 227-250.

[5] A.Cătaş, On certain classes of p-valent functions defined by multiplier transformations, in Proceedings of the
international Symposium on Geometric Function Theory and Applications: GFTA 2007 (Eds. S. Owa, Y. Polatoglu),

TC Istanbul University Publications, Turkey, 2008, pp. 241-250.
[6] H.E.Darwish and M.K.Aouf, Generalizations of modified-Hadamard products of p-valent functions with negative

coefficients, Math. Comput. Modelling 49(1-2)(2009), 38-45.

[7] T.Domokos, On a subclass of certain starlike functions with negative coefficients, Stud. Univ. Babeş-Bolyai Math.,
36(1991), 29-36.

[8] J.Dziok, Applications of extreme points to distortion estimates, Appl.Math.Comput. 215(2009), 71-77.

[9] E.Deniz and H.Orhan, Loewner chains and univalence criteria related with ruscheweyh and salagean derivatives,
Journal of applied analysis and computation, 5(3)(2015), 465-478.

[10] R.M.El-Ashwah and M.K.Aouf, Some properties of new integral operator, Acta Univ.Apulensis Math.Inform.,

24(2010), 51-61.
[11] D.Hallenbeck and T.H.MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory,

39-46. Pitman Advanced Publishing Program, Boston, Pitman, (1984)

[12] S.S.Kumar, H.C.Taneja and V.Ravichandran, Classes of multivalent functions defined by Dzoik-Srivastava linear
operator and multiplier transformations, Kyungpook Math. J., 46(2006), 97-109.

[13] S.Kanas and D.Raducanu, Some class of analytic functions related to conic domains, Mathematica slovaca,
64(5)(2014), 1183-1196.

[14] J.L.Liu, Certain sufficient conditions for strongly starlike functions associated with an integral operator, Bul-

l.Malays.Math.Sci.Soc., 34(1)(2011), 21-30.
[15] G.Murugusundaramoorthy and K.Vijaya, Second hankel determinant for bi-univalent analytic functions associated

with hohlov operator, International journal of analysis and applications, 8(1)(2015), 22-29.

[16] H.Orhan and H.Kiziltunc, A generalization on subfamily of p-valent functions with negative coefficients, Appl.
Math.Comput., 155(2004), 521-530.

[17] J.Patel, P.Sahoo, Certain subclasses of multivalent analytic functions, Indian J. Pure Appl. Math., 34(3)(2003),

487-500.
[18] J.K.Prajapat, Subordination and superordination preserving properties for generalized multiplier transformation

operator, Math. Comput. Modelling, 55(2012), 1456-1465.

[19] H.M.Srivastava, S.Owa and S.K.Chatterjea, A note on certain classes of starlike functions,
Rend.Sem.Mat.Univ.Padova, 77(1987), 115-124.

[20] Poonam Sharma, J.K.Prajapat and R.K.Raina, Certain subordination results involving a generalized multiplier
transformation operator, Journal of Classical Analysis, 2(1)(2013), 85-106.

[21] L.P.Xiong, Some general results and extreme points of p-valent functions with negative coefficients, Demonstratio

mathematica, 44(2)(2011), 261-272.

Department of Mathematics, Engineering and Technical College of Chengdu University of Technology,
Leshan, Sichuan, 614007, P.R.China

∗Corresponding author: xlpwxf@163.com