8_856_selvakumaran.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1048-1054
ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Majorization for Certain Classes of Analytic Functions Defined

by a Generalized Operator

C. Selvaraj, K.A. Selvakumaran∗

Dept. of Mathematics, Presidency College (Autonomous), Chennai-600 005, India

Abstract. In this paper, we investigate majorization properties for certain classes of multivalent an-
alytic functions defined by a generalized operator. Also, we point out some new and known conse-
quences of our main result.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, Starlike functions, Hadamard product, Subordination,
Majorization

1. Introduction and preliminaries

LetAp denote the class of functions f (z) of the form

f (z) = zp +

∞
∑

n=1

anzp+n, (p ∈ N := {1,2,3, . . .}), (1)

which are analytic and p-valent in the open unit disk U = {z : z ∈ C and |z| < 1}. Also let
A1 =:A . For functions f j ∈Ap given by

f j(z) = zp +

∞
∑

n=1

an, jz
p+n, ( j = 1,2; p ∈ N), (2)

∗Corresponding author.

Email addresses: pam
9439�yahoo.
o.in (C. Selvaraj), selvaa1826�gmail.
om (K. Selvakumaran)

http://www.ejpam.com 1048 c© 2010 EJPAM All rights reserved.



C. Selvaraj, K. Selvakumaran / Eur. J. Pure Appl. Math, 3 (2010), 1048-1054 1049

we define the Hadamard product (or convolution) of f1 and f2 by

( f1 ∗ f2)(z) = zp +

∞
∑

n=1

an,1an,2zp+n = ( f2 ∗ f1)(z).

Let f (z) and g(z) be analytic in U . Then we say that the function f (z) is subordinate to
g(z) in U , if there exists an analytic function w(z) in U with

w(0) = 0, |w(z)| < 1 (z ∈ U ),
such that

f (z) = g(w(z)) (z ∈ U ).
We denote this subordination by f (z) ≺ g(z). Furthermore, if the function g(z) is univalent
in U , then f (z)≺ g(z) (z ∈ U )⇐⇒ f (0) = g(0) and f (U )⊂ g(U ).

Suppose that the functions f (z) and g(z) are analytic in the open unit disk U . Then we
say that the function f (z) is majorized by g(z) in U (see [5]) and write

f (z)≪ g(z) (z ∈ U ), (3)

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

|ϕ(z)| ≤ 1 and f (z) = ϕ(z)g(z) (z ∈ U ).
The majorization (3) is closely related to the concept of quasi-subordination between analytic
functions in U .

Let α1,α2, . . . ,αq and β1,β2, . . . ,βs (q, s ∈ N ∪ {0},q ≤ s + 1) be complex numbers such
that βl 6= 0,−1,−2, . . . for l ∈ {1,2, . . . , s}. The generalized hypergeometric function qFs is
given by

qFs(α1,α2, . . . ,αq;β1,β2, . . . ,βs; z) =

∞
∑

n=0

(α1)n(α2)n . . . (αq)n

(β1)n(β2)n . . . (βs)n

zn

n!
, (z ∈ U ),

where (x)n denotes the Pochhammer symbol defined by

(x)n = x(x + 1)(x + 2) · · · (x + n− 1) for n ∈ N and (x)0 = 1.

Corresponding to a function G p
q,s(α1;β1; z) defined by

Gq,s(α1,β1; z) := zp
qFs(α1,α2, . . . ,αq;β1,β2, . . . ,βs; z), (4)

C.Selvaraj and K.R.Karthikeyan recently defined the following generalized differential opera-
tor D

p,m
λ
(α1,β1) f :Ap −→Ap by

D
p,0
λ
(α1,β1) f (z) = f (z) ∗ G p

q,s(α1,β1; z),

D
p,1
λ
(α1,β1) f (z) = (1−λ)( f (z) ∗ G p

q,s(α1,β1; z)) +
λ

p
z( f (z) ∗ G p

q,s(α1,β1; z))′,

D
p,m
λ
(α1,β1) f (z) = D

p,1
λ
(D

p,m−1
λ

(α1,β1) f (z)),

(5)



C. Selvaraj, K. Selvakumaran / Eur. J. Pure Appl. Math, 3 (2010), 1048-1054 1050

where m ∈ N0 = N∪ {0} and λ≥ 0.
If f (z) ∈Ap, then we have

D
p,m
λ
(α1,β1) f (z) = zp +

∞
∑

n=1

�

p+λn

p

�m (α1)n(α2)n . . . (αq)n

(β1)n(β2)n . . . (βs)n
an

zp+n

n!
. (6)

It can be seen that, by specializing the parameters the operator D
p,m
λ
(α1,β1) f (z) reduces to

many known and new integral and differential operators. In particular, when m = 0 and p = 1
the operator D

p,m
λ
(α1,β1) f (z) reduces to the well known Dziok- Srivastava operator [3] and

for p = 1, q = 2, s = 1, α1 = β1, and α2 = 1, it reduces to the operator introduced by F.
AL-Oboudi [1]. Further we remark that, when p = 1, q = 2, s = 1, α1 = β1, α2 = 1, and
λ= 1 the operator D

p,m
λ
(α1,β1) f (z) reduces to the operator introduced by G. S. Sălăgean [8].

It can be easily verified from (6) that

λz(D
p,m
λ
(α1,β1) f (z))

′ = pD
p,m+1
λ

(α1,β1) f (z)− p(1−λ)Dp,m
λ
(α1,β1) f (z). (7)

Using the operator D
p,m
λ
(α1,β1) f (z) we now define the following class of p-valent analytic

functions.

Definition 1. A function f (z) ∈ Ap is said to be in the class S
p, j
λ,m(A, B;γ) of p-valent functions

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

Re

�

1+
1

γ

�

z
�

D
p,m
λ
(α1,β1) f (z)
�( j+1)

�

D
p,m
λ
(α1,β1) f (z)
�( j)

− p+ j

��

≺ 1+ Az

1+ Bz
(8)

(z ∈ U ; −1≤ B < A≤ 1; p ∈ N; m, j ∈ N0; γ ∈ C− {0}; |γλ(A− B) + pB| ≤ p).

It can be seen that, by specializing the parameters the class S
p, j
λ,m(A, B;γ) reduces to many

known subclasses of analytic functions. In particular, when A= 1 and B = −1 the class reduces
to the class S

p, j
λ,m(γ) which has recently been introduced by C.Selvaraj and K.A.Selvakumaran

[9]. Further, when q = 2, s = 1, α1 = β1, and α2 = 1, we have the following relationships:

(1) S
1,0
λ,0(1,−1;γ) = S (γ) (γ ∈ C− {0}).

(2) S
1,1
λ,0(1,−1;γ) =K (γ) (γ ∈ C− {0}).

(3) S
1,0
λ,0(1,−1; 1−α) = S ∗(α) for 0≤ α < 1.

The classes S (γ) and K (γ) are said to be the classes of starlike and convex functions of
complex order γ 6= 0 in U which were studied by M. A. Nasr and M. K. Aouf [6] and P.
Wiatrowski [10] and S ∗(α) is the class of starlike functions of order α in U .



C. Selvaraj, K. Selvakumaran / Eur. J. Pure Appl. Math, 3 (2010), 1048-1054 1051

2. Majorization Problem for the Class S
p, j
λ,m(A, B;γ)

Theorem 1. Let the function f (z) be in the class Ap and suppose that g(z) ∈ S
p, j
λ,m(A, B;γ). If

�

D
p,m
λ
(α1,β1) f (z)
�( j)

is majorized by
�

D
p,m
λ
(α1,β1)g(z)
�( j)

in U for j ∈ N0, then

�

�

�

D
p,m+1
λ

(α1,β1) f (z)
�( j)
�

� ≤
�

�

�

D
p,m+1
λ

(α1,β1)g(z)
�( j)
�

� for |z| ≤ r1, (9)

where r1 = r1(p,γ,λ,A, B) is the smallest positive root of the equation

|γλ(A− B) + pB|r3 − (p+ 2λ|B|)r2− (|γλ(A− B) + pB|+ 2λ)r + p = 0 (10)

(−1≤ B < A≤ 1; p ∈ N; γ ∈ C− {0}; λ≥ 0).

Proof. Let

h(z) = 1+
1

γ

�

z
�

D
p,m
λ
(α1,β1)g(z)
�( j+1)

�

D
p,m
λ
(α1,β1)g(z)
�( j)

− p+ j

�

(11)

(p ∈ N; m, j ∈ N0; γ ∈ C− {0}; p > j).

Since g(z) ∈ S
p, j
λ,m(γ), we find from (8) that

h(z) =
1+ Aw(z)

1+ Bw(z)
, (12)

where w(z) is analytic in U , which satisfies the conditions

w(0) = 0 and |w(z)| < 1 (z ∈ U ).
It follows from (11) and (12)that

z
�

D
p,m
λ
(α1,β1)g(z)
�( j+1)

�

D
p,m
λ
(α1,β1)g(z)
�( j)

=
(p− j)+ [γ(A− B) + (p− j)B]w(z)

1+ Bw(z)
(13)

In view of

λz(D
p,m
λ
(α1,β1) f (z))

( j+1) = p(D
p,m+1
λ

(α1,β1) f (z))
( j) − (p− pλ+λ j)(D

p,m
λ
(α1,β1) f (z))

( j),
(14)

(13) immediately yields the following inequality:
�

�

�

�

�

D
p,m
λ
(α1,β1)g(z)
�( j)

�

�

�

�

≤ p(1+ |B||z|)
p− |γλ(A− B) + pB||z|

�

�

�

�

�

D
p,m+1
λ

(α1,β1)g(z)
�( j)

�

�

�

�

. (15)

Since
�

D
p,m
λ
(α1,β1) f (z)
�( j)

is majorized by
�

D
p,m
λ
(α1,β1)g(z)
�( j)

inU , there exist an analytic
function ϕ(z) such that

�

D
p,m
λ
(α1,β1) f (z)
�( j)
= ϕ(z)
�

D
p,m
λ
(α1,β1)g(z)
�( j)

(16)



C. Selvaraj, K. Selvakumaran / Eur. J. Pure Appl. Math, 3 (2010), 1048-1054 1052

and |ϕ(z)| ≤ 1 (z ∈ U ). Thus we have

z
�

D
p,m
λ
(α1,β1) f (z)
�( j+1)

= zϕ′(z)
�

D
p,m
λ
(α1,β1)g(z)
�( j)
+ zϕ(z)
�

D
p,m
λ
(α1,β1)g(z)
�( j+1).

(17)
Using (14), in the above equation, we get

�

D
p,m+1
λ

(α1,β1) f (z)
�( j)
=
λz

p
ϕ′(z)
�

D
p,m
λ
(α1,β1)g(z)
�( j)
+ϕ(z)
�

D
p,m+1
λ

(α1,β1)g(z)
�( j).

(18)
Noting that ϕ(z) satisfies (cf. [4, 7])

|ϕ′(z)| ≤ 1− |ϕ(z)|2
1− |z|2 (z ∈ U ), (19)

we see that
�

�

�

�

�

D
p,m+1
λ

(α1,β1) f (z)
�( j)

�

�

�

�

≤
�

ϕ(z) +
1− |ϕ(z)|2

1− |z|2
λ|z|(1+ |B||z|)

p− |γλ(A− B) + pB||z|
�
�

�

�

�

�

D
p,m+1
λ

(α1,β1)g(z)
�( j)

�

�

�

�

(20)

which, upon setting
|z| = r, and |ϕ(z)| = ρ (0≤ ρ ≤ 1)

leads us to the following inequality:
�

�

�

�

�

D
p,m+1
λ

(α1,β1) f (z)
�( j)

�

�

�

�

≤ Θ(ρ)

(1− r2)(p− |γλ(A− B) + pB|r)

�

�

�

�

�

D
p,m+1
λ

(α1,β1)g(z)
�( j)

�

�

�

�

, (21)

where the function Θ(ρ) defined by

Θ(ρ) := −λr(1+ |B|r)ρ2+ (1− r2)(p− |γλ(A− B) + pB|r)ρ +λr(1+ |B|r) (0≤ ρ ≤ 1)

takes its maximum value at ρ = 1 with r = r1(p,γ,λ,A, B), the smallest positive root of the
equation (10). Furthermore, if 0≤ σ ≤ r1(p,γ,λ,A, B), then the function

Φ(ρ) := −λσ(1+ |B|σ)ρ2 + (1−σ2)(p− |γλ(A− B) + pB|σ)ρ +λσ(1+ |B|σ)

increases in the interval 0≤ ρ ≤ 1, so that Φ(ρ) does not exceed

Φ(1) = (1−σ2)(p− |γλ(A− B) + pB|σ) (0≤ σ ≤ r1(p,γ,λ,A, B)).

Therefore, from this fact, (21) gives the inequality (9).

As a special case of Theorem 1, when A= 1 and B = −1, we have



C. Selvaraj, K. Selvakumaran / Eur. J. Pure Appl. Math, 3 (2010), 1048-1054 1053

Corollary 1. [9] Let the function f (z) be in the class Ap and suppose that g(z) ∈ S
p, j
λ,m(γ). If

�

D
p,m
λ
(α1,β1) f (z)
�( j)

is majorized by
�

D
p,m
λ
(α1,β1)g(z)
�( j)

in U for j ∈ N0, then

�

�

�

D
p,m+1
λ

(α1,β1) f (z)
�( j)
�

� ≤
�

�

�

D
p,m+1
λ

(α1,β1)g(z)
�( j)
�

� for |z| ≤ r1, (22)

where

r1 = r1(p,γ,λ) :=
k−
p

k2 − 4p|2γλ− p|
2|2γλ− p| (23)

(k := 2λ+ p+ |2γλ− p|; p ∈ N; γ ∈ C− {0}; λ≥ 0).

Setting A= 1, B = −1, p = 1 and j = 0 in Theorem 1, we have

Corollary 2. Let the function f (z) ∈ A be analytic and univalent in the open unit disk U and

suppose that g(z) ∈ S
1,0
λ,m(γ). If
�

D
1,m
λ
(α1,β1) f (z)
�

is majorized by
�

D
1,m
λ
(α1,β1)g(z)
�

in U ,

then
�

�

�

D
1,m+1
λ

(α1,β1) f (z)
�

�

� ≤
�

�

�

D
1,m+1
λ

(α1,β1)g(z)
�

�

� for |z| ≤ r2, (24)

where

r2 :=
k−
p

k2 − 4|2γλ− 1|
2|2γλ− 1| (25)

(k := 2λ+ 1+ |2γλ− 1|; γ ∈ C− {0}; λ ≥ 0).

Further putting λ= 1, m = 0, q = 2, s = 1, α1 = β1, and α2 = 1 in Corollary 2, we get

Corollary 3. [2] Let the function f (z) ∈ A be analytic and univalent in the open unit disk U
and suppose that g(z) ∈ S (γ). If f (z) is majorized by g(z) in U , then

�

� f ′(z)
�

� ≤
�

�g′(z)
�

� for |z| ≤ r3, (26)

where

r3 :=
3+ |2γ− 1| −
p

9+ 2|2γ− 1|+ |2γ− 1|2
2|2γ− 1| . (27)

For γ= 1, Corollary 3 reduces to the following result:

Corollary 4. [5] Let the function f (z) ∈ A be analytic and univalent in the open unit disk U
and suppose that g(z) ∈ S ∗ = S ∗(0). If f (z) is majorized by g(z) in U , then

�

� f ′(z)
�

� ≤
�

�g′(z)
�

� for |z| ≤ 2−p3. (28)



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