International Journal of Analysis and Applications
ISSN 2291-8639
Volume 1, Number 2 (2013), 100-105
http://www.etamaths.com

A SUBORDINATION THEOREM INVOLVING A MULTIPLIER

TRANSFORMATION

SUKHWINDER SINGH BILLING

Abstract. We, here, study a certain differential subordination involving a

multiplier transformation which unifies some known differential operators. As

a special case to our main result, we find some new results providing the best
dominant for zp/f(z), z/f(z) and zp−1/f′(z), 1/f′(z).

1. Introduction

Let A be the class of all functions f analytic in the open unit disk E = {z ∈ C :
|z| < 1} and normalized by the conditions that f(0) = f′(0) − 1 = 0. Thus, f ∈A
has the Taylor series expansion

f(z) = z +

∞∑
k=2

akz
k.

Let Ap denote the class of functions of the form f(z) = zp +
∞∑

k=p+1

akz
k,p ∈ N =

{1, 2, 3, · · ·}, which are analytic and multivalent in the open unit disk E. Note
A1 = A. For f ∈Ap, define the multiplier transformation Ip(n,λ) as

Ip(n,λ)f(z) = z
p +

∞∑
k=p+1

(
k + λ

p + λ

)n
akz

k, (λ ≥ 0,n ∈ N0 = N∪{0}).

The operator Ip(n,λ) has been recently studied by Aghalary et al. [3]. I1(n, 0) is
the well-known Sălăgean [1] derivative operator Dn, defined for f ∈A as under:

Dnf(z) = z +

∞∑
k=2

knakz
k.

For two analytic functions f and g in the unit disk E, we say that f is subordinate
to g in E and write as f ≺ g if there exists a Schwarz function w analytic in E
with w(0) = 0 and |w(z)| < 1, z ∈ E such that f(z) = g(w(z)), z ∈ E. In case the
function g is univalent, the above subordination is equivalent to: f(0) = g(0) and
f(E) ⊂ g(E).

2010 Mathematics Subject Classification. 30C80, 30C45.
Key words and phrases. Differential subordination, Multiplier transformation, Analytic func-

tion; Univalent function, p-valent function.

c©2013 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

100



A SUBORDINATION THEOREM 101

Let Φ : C2 ×E → C be an analytic function, p be an analytic function in E such
that (p(z),zp′(z); z) ∈ C2 × E for all z ∈ E and h be univalent in E. Then the
function p is said to satisfy first order differential subordination if

(1) Φ(p(z),zp′(z); z) ≺ h(z), Φ(p(0), 0; 0) = h(0).

A univalent function q is called a dominant of the differential subordination (1) if
p(0) = q(0) and p ≺ q for all p satisfying (1). A dominant q̃ that satisfies q̃ ≺ q for
each dominant q of (1), is said to be the best dominant of (1).

Obradovic̆ [2], introduced and studied the class N(α), 0 < α < 1 of functions
f ∈A satisfying the following inequality

<

{
f′(z)

(
z

f(z)

)1+α}
> 0, z ∈ E.

He called it, the class of non-Bazilevic̆ functions.
In 2005, Wang et al. [6] introduced the generalized class N(λ,α,A,B) of non-

Bazilevic̆ functions which is analytically defined as:

N(λ,α,A,B) =
{
f ∈A : (1 + λ)

(
z

f(z)

)α
−λ

zf′(z)

f(z)

(
z

f(z)

)α
≺

1 + Az

1 + Bz
,

}
where 0 < α < 1, λ ∈ C, −1 ≤ B ≤ 1, A 6= B, A ∈ R.

Wang et al. [6] studied the class N(λ,α,A,B) and made some estimates on(
z

f(z)

)α
.

Using the concept of differential subordination, Shanmugam et al. [5] studied

the differential operator (1+λ)

(
z

f(z)

)α
−λ

zf′(z)

f(z)

(
z

f(z)

)α
and obtained the best

dominant for

(
z

f(z)

)α
.

The main objective of this paper is to unify the above mentioned differential
operators. For this, we establish a differential subordination involving the multiplier
transformation Ip(n,λ), defined above. As special cases of main theorem, we obtain

best dominant for zp/f(z), z/f(z) and zp−1/f′(z), 1/f′(z) and some known results
also appear as special cases to our main result.

To prove our main result, we shall make use of the following lemma of Miller
and Macanu [4].

Lemma 1.1. Let q be univalent in E and let θ and φ be analytic in a domain
D containing q(E), with φ(w) 6= 0, when w ∈ q(E). Set Q(z) = zq′(z)φ[q(z)],
h(z) = θ[q(z)] + Q(z) and suppose that either

(i) h is convex, or
(ii) Q is starlike.

In addition, assume that

(iii) < zh
′(z)

Q(z)
> 0, z ∈ E.

If p is analytic in E, with p(0) = q(0),p(E) ⊂ D and

θ[p(z)] + zp′(z)φ[p(z)] ≺ θ[q(z)] + zq′(z)φ[q(z)],

then p(z) ≺ q(z) and q is the best dominant.



102 BILLING

2. Main Results

In what follows, all the powers taken are the principal ones.

Theorem 2.1. Let α and β be non-zero complex numbers such that <(β/α) > 0

and let f ∈Ap,
(

zp

Ip(n,λ)f(z)

)β
6= 0, z ∈ E, satisfy the differential subordination

(2)(
zp

Ip(n,λ)f(z)

)β [
1 + α−α

Ip(n + 1,λ)f(z)

Ip(n,λ)f(z)

]
≺

1 + Az

1 + Bz
+

α

β(p + λ)

(A−B)z
(1 + Bz)2

,

then (
zp

Ip(n,λ)f(z)

)β
≺

1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E,

and
1 + Az

1 + Bz
is the best dominant.

Proof: On writing u(z) =

(
zp

Ip(n,λ)f(z)

)β
, a little calculation yields that

(3)

(
zp

Ip(n,λ)f(z)

)β [
1 + α−α

Ip(n + 1,λ)f(z)

Ip(n,λ)f(z)

]
= u(z) +

α

β(p + λ)
zu′(z),

Define the functions θ and φ as follows:

θ(w) = w and φ(w) =
α

β(p + λ)
.

Clearly, the functions θ and φ are analytic in domain D = C and φ(w) 6= 0, w ∈ D.

Select q(z) =
1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E and define the functions Q and h

as follows:

Q(z) = zq′(z)φ(q(z)) =
α

β(p + λ)
zq′(z) =

α

β(p + λ)

(A−B)z
(1 + Bz)2

,

and

(4) h(z) = θ(q(z))+Q(z) = q(z)+
α

β(p + λ)
zq′(z) =

1 + Az

1 + Bz
+

α

β(p + λ)

(A−B)z
(1 + Bz)2

.

A little calculation yields

<
(
zQ′(z)

Q(z)

)
= <

(
1 +

zq′′(z)

q′(z)

)
= <

(
1 −Bz
1 + Bz

)
> 0, z ∈ E,

i.e. Q is starlike in E and

<
(
zh′(z)

Q(z)

)
= <

(
1 +

zq′′(z)

q′(z)
+ (p + λ)

β

α

)
= <

(
1 −Bz
1 + Bz

)
+(p+λ)<

(
β

α

)
> 0, z ∈ E.

Thus conditions (ii) and (iii) of Lemma 1.1, are satisfied. In view of (2), (3) and
(4), we have

θ[u(z)] + zu′(z)φ[u(z)] ≺ θ[q(z)] + zq′(z)φ[q(z)].
Therefore, the proof follows from Lemma 1.1.

For p = 1 and λ = 0 in above theorem, we get the following result involving
Sălăgean operator.



A SUBORDINATION THEOREM 103

Theorem 2.2. If α, β are non-zero complex numbers such that <(β/α) > 0. If

f ∈A,
(

z

Dnf(z)

)β
6= 0, z ∈ E, satisfies

(
z

Dnf(z)

)β [
1 + α−α

Dn+1f(z)

Dnf(z)

]
≺

1 + Az

1 + Bz
+
α

β

(A−B)z
(1 + Bz)2

, −1 ≤ B < A ≤ 1, z ∈ E,

then (
z

Dnf(z)

)β
≺

1 + Az

1 + Bz
, z ∈ E.

3. Dominant for zp/f(z), z/f(z)

This section is concerned with the results giving the best dominant for zp/f(z)
and z/f(z). Select λ = n = 0 in Theorem 2.1, we obtain the following result.

Corollary 3.1. Let α, β be non-zero complex numbers such that <(β/α) > 0

and let f ∈Ap,
(

zp

f(z)

)β
6= 0, z ∈ E, satisfy

(1 + α)

(
zp

f(z)

)β
−α

zf′(z)

pf(z)

(
zp

f(z)

)β
≺

1 + Az

1 + Bz
+

α

pβ

(A−B)z
(1 + Bz)2

, z ∈ E,

then (
zp

f(z)

)β
≺

1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E.

Taking β = 1 in above theorem, we obtain:

Corollary 3.2. Suppose that α is a non-zero complex number such that <(1/α) >
0 and suppose that f ∈Ap,

zp

f(z)
6= 0, z ∈ E, satisfies

(1 + α)
zp

f(z)
−α

zp+1f′(z)

p(f(z))2
≺

1 + Az

1 + Bz
+
α

p

(A−B)z
(1 + Bz)2

, z ∈ E,

then
zp

f(z)
≺

1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E.

On writing α = −1 in Corollary 3.1, we get:

Corollary 3.3. Let β be a complex number with <(β) < 0 and let f ∈Ap,
(

zp

f(z)

)β
6=

0, z ∈ E, satisfy

zf′(z)

pf(z)

(
zp

f(z)

)β
≺

1 + Az

1 + Bz
−

1

pβ

(A−B)z
(1 + Bz)2

, −1 ≤ B < A ≤ 1, z ∈ E,

then (
zp

f(z)

)β
≺

1 + Az

1 + Bz
, z ∈ E.

Selecting α = β = 1/2 in Corollary 3.1, we get:



104 BILLING

Corollary 3.4. If f ∈Ap,
√

zp

f(z)
6= 0, z ∈ E, satisfies

√
zp

f(z)

(
3 −

zf′(z)

pf(z)

)
≺

2(1 + Az)

1 + Bz
+

2

p

(A−B)z
(1 + Bz)2

, z ∈ E,

then √
zp

f(z)
≺

1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E.

Taking p = 1 in Corollary 3.2, we have the following result.

Corollary 3.5. If α is a non-zero complex number such that <(1/α) > 0 and
if f ∈A,

z

f(z)
6= 0, z ∈ E, satisfies

(1 + α)
z

f(z)
−α

z2f′(z)

(f(z))2
≺

1 + Az

1 + Bz
+ α

(A−B)z
(1 + Bz)2

, −1 ≤ B < A ≤ 1, z ∈ E,

then
z

f(z)
≺

1 + Az

1 + Bz
, z ∈ E.

Setting p = 1 in Corollary 3.3, we have the following result.

Corollary 3.6. If β is a complex number with <(β) < 0 and if f ∈A,
(

z

f(z)

)β
6=

0, z ∈ E, satisfies

zβ+1f′(z)

(f(z))β+1
≺

1 + Az

1 + Bz
−

1

β

(A−B)z
(1 + Bz)2

, −1 ≤ B < A ≤ 1, z ∈ E,

then (
z

f(z)

)β
≺

1 + Az

1 + Bz
, z ∈ E.

Setting p = 1 in Corollary 3.1, we obtain, below, the result of Shanmugam et al.
[5].

Corollary 3.7. If α, β are non-zero complex numbers such that <(β/α) > 0.

If f ∈A,
(

z

f(z)

)β
6= 0, z ∈ E, satisfies

(1 + α)

(
z

f(z)

)β
−αf′(z)

(
z

f(z)

)1+β
≺

1 + Az

1 + Bz
+
α

β

(A−B)z
(1 + Bz)2

, z ∈ E,

then (
z

f(z)

)β
≺

1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E.

4. Dominant for zp−1/f′(z), 1/f′(z)

We obtain here, the best dominant for zp−1/f′(z) and 1/f′(z) as special cases
to our main result. Select λ = 0 and n = 1 in Theorem 2.1, we obtain:



A SUBORDINATION THEOREM 105

Corollary 4.1. Let α, β be non-zero complex numbers such that <(β/α) > 0

and let f ∈Ap,
(
pzp−1

f′(z)

)β
6= 0, z ∈ E, satisfy

(1 +α)

(
pzp−1

f′(z)

)β
−
α

p

(
1 +

zf′′(z)

f′(z)

)(
pzp−1

f′(z)

)β
≺

1 + Az

1 + Bz
+
α

pβ

(A−B)z
(1 + Bz)2

, z ∈ E,

then (
pzp−1

f′(z)

)β
≺

1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1, z ∈ E.

Taking β = 1 in above theorem, we obtain:

Corollary 4.2. Suppose that α is a non-zero complex number such that <(1/α) >

0 and suppose that f ∈Ap,
pzp−1

f′(z)
6= 0, z ∈ E, satisfies

(1 + α)
pzp−1

f′(z)
−α

zp−1

f′(z)

(
1 +

zf′′(z)

f′(z)

)
≺

1 + Az

1 + Bz
+
α

p

(A−B)z
(1 + Bz)2

, z ∈ E,

then
zp−1

f′(z)
≺

1 + Az

p(1 + Bz)
, −1 ≤ B < A ≤ 1, z ∈ E.

Taking p = 1 in Corollary 4.2, we have the following result.

Corollary 4.3. If α is a non-zero complex number such that <(1/α) > 0 and
if f ∈A,

1

f′(z)
6= 0, z ∈ E, satisfies

1

f′(z)

(
1 −α

zf′′(z)

f′(z)

)
≺

1 + Az

1 + Bz
+ α

(A−B)z
(1 + Bz)2

, −1 ≤ B < A ≤ 1, z ∈ E,

then
1

f′(z)
≺

1 + Az

1 + Bz
, z ∈ E.

References

[1] G. S. Sălăgean, Subclasses of univalent functions, Lecture Notes in Math., 1013 362–372,
Springer-Verlag, Heideberg,1983.

[2] M. Obradovic̆, A class of univalent functions, Hokkaido Mathematical Journal, 27(2)(1998)

329–335.
[3] R. Aghalary, R. M. Ali, S. B. Joshi and V. Ravichandran, Inequalities for analytic functions

defined by certain linear operators, Int. J. Math. Sci., 4(2005) 267–274.

[4] S. S. Miller and P. T. Mocanu, Differential Suordinations : Theory and Applications, (No.
225), Marcel Dekker, New York and Basel, 2000.

[5] T. N. Shanmugam, S. Sivasubramanian and H. Silverman, On sandwich theorems for some

classes of analytic functions, International J. Math. and Math. Sci., Article ID 29684(2006)
pp.1–13.

[6] Z. Wang, C. Gao and M. Liao, On certain generalized class of non-Bazilevic̆ functions, Acta

Mathematica Academiae Paedagogicae Nýıregyháziensis, New Series, 21(2)(2005) 147–154.

Department of Applied Sciences, Baba Banda Singh Bahadur Engineering College,

Fatehgarh Sahib-140 407, Punjab, India