CUBO A Mathematical Journal
Vol.14, No¯ 01, (119–125). March 2012

Majorization for certain classes of analytic functions defined
by a new operator

E. A. Eljamal and M. Darus

School of Mathematical Sciences,

Faculty of Science and Technology,

Universiti Kebangsaan Malaysia,

Bangi 43600 Selangor D. Ehsan, Malaysia.

email: n-ebtisam@yahoo.com , maslina@ukm.my

ABSTRACT

In the present paper, we investigate the majorization properties for certain classes of

multivalent analytic functions defined by a new operator. Moreover, we pointed out

some new and known consequences of our main result.

RESUMEN

En el presente art́ıculo, investigamos las propiedades de mayorización para ciertas clases

de funciones anaĺıticas multivalentes definidas por un nuevo operador. Además, resalta-

mos algunas consecuencias -nuevas y conocidas- de nuestro resultado princresultado.

Keywords and Phrases: Majorization properties, multivalent functions, Ruscheweyh derivative

operator, Hadamard product.

2010 AMS Mathematics Subject Classification: 30C45.



120 E. A. Eljamal and M. Darus CUBO
14, 1 (2012)

1 Introduction

Let f and g be analytic in the open unit disk U = {z : z ∈ C, |z| < 1}. We say that f is majorized
by g in U and write

f(z) � g(z) (z ∈ U) (1.1)

if there exists a function ϕ, analytic in U such that

|ϕ(z)| ≤ 1 and f(z) = ϕ(z)g(z) (z ∈ U). (1.2)

It maybe noted here that (1.1) is closely related to the concept of quasi-subordination between

analytic functions. Let Ap denote the class of functions of the form

f(z) = zp +

∞∑
k=p+1

akz
k,(p ∈ N = {1,2, ...}), (1.3)

which are analytic and multivalent in the open unit disk U. In particular, if p = 1, thenA1 = A.

For functions fj ∈ Ap given by

fj(z) = z
p +

∞∑
k=p+1

ak,jz
k,(j = 1,2;p ∈ N), (1.4)

we define the Hadamard product or convolution of two functions f1 and f2 by

f1 ∗ f2(z) = zp +
∞∑

k=p+1

ak1,ak2z
k = (f2 ∗ f1)(z). (1.5)

.

Definition 1.1. Let the function f be in the class Ap. Ruscheweyh derivative operator is given by

Rn = zp +

∞∑
k=p+1

C(k,n)akz
k. (1.6)

Next we define the following differential operator,

D0 = f(z) = zp +

∞∑
k=p+1

akz
k

D1n,λ1,λ2,p = D
0f(z)

p − pλ1 + λ2(k − p)

p + λ2(k − p)
+ (D0f(z))′

zλ1

p + λ2(k − p)

= zp + Σ∞k=p+1
[
p + (λ1 + λ2)(k − p)

p + λ2(k − p)

]
akz

k,

and

D2n,λ1,λ2,p = D
1
n,λ1,λ2,p

f(z)
p − pλ1 + λ2(k − p)

p + λ2(k − p)
+ (D1n,λ1,λ2,pf(z))

′ zλ1

p + λ2(k − p)



CUBO
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Majorization for certain classes of analytic functions . . . 121

= zp + Σ∞k=p+1
[
p + (λ1 + λ2)(k − p)

p + λ2(k − p)

]2
akzk.

In general,

Dmn,λ1,λ2,pf(z) = D(D
n−1f(z) = zp + Σ∞k=p+1

[
p + (λ1 + λ2)(k − p)

p + λ2(k − p)

]m
akz

k (1.7)

where (m,n ∈ N0 = N ∪ {0},λ2 ≥ λ1 ≥ 0). By applying convolution product on (1.6) and (1.7) we
have the following operator

Dmn,λ1,λ2,pf(z) = z
p + Σ∞k=p+1

[
p + (λ1 + λ2)(k − p)

p + λ2(k − p)

]m
C(k,n)akz

k, (1.8)

where C(k,n) =
Γ(k+n)
Γ(k)

.

Moreover, for m,n ∈ N0, λ2 ≥ λ1 ≥ 0

(p + λ2(k − p))D
m,n
λ1,λ2,p

f(z) = (p + λ2(k − p) − pλ1)D
m,n
λ1,λ2,p

f(z) + λ1z(D
m,n
λ1,λ2,p

f(z))′ (1.9)

Special cases of this operator include:

• the Ruscheweyh derivative operator in the case D0,n0,0,1f(z) ≡ R
n [6],

• the Salagean derivative operator in the case Dm,01,0,1f(z) ≡ D
m ≡ Sn [2],

• the generalized Salagean derivative operator introduced by Al-Oboudi in the case Dm,0λ1,0,1f(z) ≡
Dmλ1[1],

• the generalized Ruscheweyh derivative operator in the case D1,nλ1,0,1f(z) ≡ D
λ1
n [3], and

• the generalized Al-Shaqsi and Darus derivative operator in the caseDm,nλ1,0,1f(z) ≡ D
m,λ1
n [4].

To further our work, we need to define a class of functions as follows:

Definition 1.2. A function f ∈ Ap is said to be in the class S
m,p,j
λ1,λ2,n

[A,B,γ] of p-valent functions

of complex order γ 6= 0 in U if and only if{
1 +

1

γ

(
z(Dm,nλ1,λ2,pf(z))

(j+1)

(Dm,n
λ1,λ2,p

f(z))(j)
− p + j

)}
≺
1 + Az

1 + Bz
. (1.10)

(z ∈ U,p ∈ N, j ∈ N0 = N ∪ {0},γ ∈ C − {0},λ2 ≥ λ1 ≥ 0).

Clearly, we have the following relationships:

(i) S0,1,00,0,0[1,−1,γ] = S(γ)



122 E. A. Eljamal and M. Darus CUBO
14, 1 (2012)

(ii) S0,1,10,0,0[1,−1,γ] = K(γ)

(iii) S0,1,00,0,0[1,−1,1 − α] = S
∗ for 0 < α < 1. The classes S(γ) and K(γ) are said to be classes

of starlike and convex of complex order γ 6= 0 in U and S∗(α) denote the class of starlike functions
of order α in U.

A majorization problem for the class S(γ) has been investigated by Altintas e.tal [5] and for

the class S∗=S∗(0) has been investigated by MacGregor [7]. In the present paper, we investigate a

majorization problem for the class S
m,p,j
λ1,λ2,α

[A,B,γ].

2 Majorization problem for the class S
m,p,j
λ1,λ2,n

[A,B,γ]

Theorem 2.1. Let the function f ∈ Ap and suppose that g ∈ S
m,p,j
λ1,λ2,n

[A,B,γ]. If (Dm,nλ1,λ2,pf(z))
(j)

is majorized by (Dm,nλ1,λ2,pg(z))
(j) in U, then∣∣∣(Dm+1,nλ1,λ2,pf(z))(j)∣∣∣ ≤ ∣∣∣(Dm,nλ1,λ2,pg(z))(j)∣∣∣ for |z| ≤ r0, (2.1)

where r0 = r0(p,γ,λ1,λ2,A,B) is the smallest positive root of the equation

r3
∣∣∣∣γ(A − B) −

(
p + λ2(k − p)

λ1

)
B

∣∣∣∣ −
[
p + λ2(k − p)

λ1
+ 2|B|

]
r2−

[∣∣∣∣γ(A − B) − (p + λ2(k − p)λ1 )B
∣∣∣∣ + 2

]
r +

(
p + λ2(k − p)

λ1

)
= 0, (2.2)

(−1 ≤ B < A ≤ 1;P ∈ N;γ ∈ C − {0}).

Proof. Since g ∈ Sm,p,j
λ1,λ2,n

[A,B,γ] we find from (1.10) that

1 +
1

γ

(
z(Dm,nλ1,λ2,pg(z))

(j+1)

(Dm,n
λ1,λ2,p

g(z))(j)
− p + j

)
=
1 + Aw(z)

1 + Bw(z)
(2.3)

(γ ∈ C − 0,j,p ∈ N and p > j), where w is analytic in U with

w(0) = 0 and |w(z)| < z (z ∈ U).

From (2.3) we get

z(Dm,nλ1,λ2,pg(z))
(j+1)

(Dm,n
λ1,λ2,p

g(z))(j)
=

(p − j) + [γ(A − B) + (p − j)B]w(z)

1 + Bw(z)
(2.4)

and

z(Dm,nλ1,λ2,pf(z))
(j+1) = (p +

λ2(k − p)

λ1
)(Dm+1,nλ1,λ2,pf(z))

(j)+



CUBO
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Majorization for certain classes of analytic functions . . . 123

(p − j −
λ2(k − p)

λ1
)(Dm,nλ1,λ2,pf(z))

(j). (2.5)

By virtue of (2.4) and (2.5) we get

∣∣∣(Dm,nλ1,λ2,pg(z))(j)∣∣∣ ≤
p+λ2(k−p)

λ1
[1 + |B|z|]

(
p+λ2(k−p)

λ1
)|γ(A − B) − (

p+λ2(k−p)
λ1

)|B|z|
|(Dm+1,nλ1,λ2,pg(z))

(j)|. (2.6)

Next, since (Dm,nλ1,λ2,pf(z))
(j) is majorized by (Dm,nλ1,λ2,pg(z))

(j) in the unit disk U, we have from

(1.2) that

(Dm,nλ1,λ2,pf(z))
(j) = ϕ(z)(Dm,nλ1,λ2,pg(z))

(j).

Differentiating it with respect to z and multiplying by z we get

z(Dm,nλ1,λ2,pf(z))
(j+1) = zϕ′(z)(Dm,nλ1,λ2,pg(z))

(j) + zϕ(z)(Dm,nλ1,λ2,pg(z))
(j+1).

Now by using (2.5) in the above equation, it yields

(Dm,nλ1,λ2,pf(z))
(j) =

zϕ′(z)(Dm,nλ1,λ2,pg(z))
(j)

p+λ2(k−p)
λ1

+ ϕ(z)(Dm,nλ1,λ2,pg(z))
(j) (2.7)

Thus, by noting that ϕ ∈ Ω satisfies the inequality (see, e.g. Nehari [8])

|ϕ′(z)| ≤
1 − |ϕ(z)|2

1 − |z|2
(z ∈ U) (2.8)

and using (2.6) and (2.8) in (2.7), we get∣∣∣(Dm+1,nλ1,λ2,pf(z))(j)∣∣∣ ≤[
|ϕ(z)| +

1 − |ϕ(z)|2

1 − |z|2
|z|(1 + |B||z|)

p+λ2(k−p)
λ1

− |γ(A − B) − (
p+λ2(k−p)

λ1
)|B||z

]∣∣∣(Dm,nλ1,λ2,pg(z))(j+1)∣∣∣ (2.9)
which upon setting

|z| = r and |ϕ(z)| = ρ (0 ≤ ρ ≤ 1)

leads us to the inequality ∣∣∣(Dm+1,nλ1,λ2,pf(z))(j)∣∣∣ ≤
φ(ρ)

(1 − r2)(
p+λ2(k−p)

λ1
) − |γ(A − B) − (

p+λ2(k−p)
λ1

)B|r

∣∣∣(Dm+1,nλ1,λ2,pg(z))(j)∣∣∣ (2.10)
where

φ(ρ) = −r(1 + |B|)ρ2 + (1 − r2)[
(
p + λ2(k − p)

λ1
) − |γ(A − B) + (

p + λ2(k − p)

λ1
)B|r)

]
ρ + r(1 + |B|r) (2.11)



124 E. A. Eljamal and M. Darus CUBO
14, 1 (2012)

takes its maximum value at ρ = 1 with r1 = r1(p,γ,λ1,λ2,A,B) for r1(p,γ,λ1,λ2,A,B) is the

smallest positive root of equation (2.2). Furthermore, if 0 ≤ ρ ≤ r1(p,γ,λ1,λ2,A,B), then function
ψ(ρ) defined by

ψ(ρ) = −σ(1 + |B|σ)ρ2 + (1 − σ2)[
(
p + λ2(k − 1)

λ1
) − |γ(A − B) + (

p + λ2(k − p)

λ1
)B|σ)

]
ρ + σ(1 + |B|σ) (2.12)

is seen to be an increasing function on the interval 0 ≤ ρ ≤ 1 so that

ψ(ρ) ≤ ψ(1) = (1 − σ2)(
p + λ2(k − p)

λ1
) − |γ(A − B) + (

p + λ2(k − p)

λ1
)B|σ) (2.13)

0 ≤ ρ ≤ 1; (0 ≤ σ ≤ r1(p,γ,λ1,λ2,A,B)).

Hence upon setting ρ = 1 in (2.13) we conclude that (2.1) of Theorem 2.1 holds true for |z| ≤
r1(p,γ,λ1,λ2,A,B) where r1(p,γ,λ1,λ2,A,B) is the smallest positive root of equation (2.2). This

completes the proof of the Theorem 2.1.

Setting p = 1, m = 0, A = 1,B = −1 and j = 0 in Theorem 2.1 we get

Corollary 2.1. Let the function f ∈ A be analytic in the open unit disk Uand suppose that g ∈
S0,1,00,0,0[1,−1,γ] = S(γ). If f(z) is majorized by g(z) in U, then

|f′(z)| ≤ |g′(z)| (|z| < r3)

where

r3 = r3(γ) =
3 + |2γ − 1| −

√
9 + 2|2γ − 1| + |2γ − 1|2

2|2γ − 1|
.

This is a known result obtained by Altintas[5].

For γ = 1, the above corollary reduces to the following result:

Corollary 2.2. Let the function f(z) ∈ A be analytic univalent in the open unit disk U and suppose
that g ∈ S∗ = S∗(0). If f is majorized by g in U, then

|f′(z)| ≤ |g′(z)| (|z| ≤ 2 −
√
3)

which is a known result obtained by MacGregor [7].

Some other work related to the class defined by (1.3) can be seen in [9] and of course elsewhere. In

fact, recently Ibrahim [10] used the concept of majorization to find solutions of fractional differential

equations in the unit disk.

Acknowledgement The work presented here was supported by UKM-ST-06-FRGS0244-2010.

Received: April 2011. Revised: June 2011.



CUBO
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Majorization for certain classes of analytic functions . . . 125

References

[1] F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Internat.

J. Math. Math. Sci., 27(2004), 1429-1436.

[2] G. Salagean, Subclasses of univalent functions, Lecture in Math. Springer Verlag, Berlin,

1013(1983), 362-372.

[3] K. Al-Shaqsi and M. Darus, On univalent functions with respect to k-symmetric points defined

by a generalization Ruscheweyh derivative operators, Jour. Anal. Appl., 7(2009), 53-61.

[4] M. Darus and K. Al-Shaqsi, Differential Sandwich Theorems with Generalised Derivative

Operator, Int. J. Comput. Math. Sci., (22)(2008), 75-78.

[5] O. Altintas, Ö.Özkan and H. M. Srivastava, Majorization by starlike functions of complex

order, Complex Var. 46(2001), 207-218.

[6] St. Ruscheweyh, New certain for univalent functions, Proc. Amer.Math. soc. 49(1975),109-115.

[7] T. H. MacGregor, Majorization by univalent functions, Duke Math. J. 34(1967), 95-102.

[8] Z. Nehari, Confformal mapping, MacGraw-Hill Book Company, New York,Toronto and Lon-

don (1955).

[9] M. Darus and R. W. Ibrahim, Multivalent functions based on a linear operator, Miskolc

Mathematical Notes, 11(1) (2010), 43-52.

[10] R. W. Ibrahim, Existence and uniqueness of holomorphic solutions for fractional Cauchy

problem, J. Math. Anal. Appl., 380 (2011), 232-240.