1_830_dziok.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 4, 2011, 322-329

ISSN 1307-5543 – www.ejpam.com

Inequalities Involving Certain Integral Operator

Jacek Dziok1,∗, Mohamed Kamal Aouf2, Janusz Sokół3

1 Institue of Mathematics, University of Rzeszów, Rzeszów, Poland
2 Department of Mathematics Faculty of Science, Mansoura University, Mansoura, Egypt
3 Department of Mathematics, Rzeszów University of Technology, Rzeszów, Poland

Abstract. The object of this paper is to give several strict inequalities associated with the operator

Iλ
p,n
(a, b; c) (a, b ∈ R \ Z−0 , p, n ∈ N = {1,2, . . .}, λ > −p) defined by X.-L. Fu and M.-S. Liu, Some

subclasses of analytic functions involving the generalized Noor integral operator [see 3].

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, Integral operator, Hadamard product.

1. Introduction

LetAn(p) denote the class of functions of the form:

f (z) = zp +

∞
∑

k=n

ak+pzk+p (p, n ∈ N = {1,2, . . . .}), (1)

which are analytic and p-valent in the open unit disc U = {z : z ∈ C, |z| < 1}. For functions f

given by (1) and g ∈An(p) given by

g(z) = zp +

∞
∑

k=n

bk+pzk+p (z ∈ U) (2)

the Hadamard product (or convolution) of f and g is defined by

( f ∗ g)(z) = zp +

∞
∑

k=n

ak+p bk+pzk+p = (g ∗ f )(z) .

∗Corresponding author.

Email addresses: jdziok�univ.rzeszow.pl (J. Dziok), mkaouf127�yahoo.
om (M. Aouf),jsokol�prz.edu.pl (J. Sokół)

http://www.ejpam.com 322 c© 2011 EJPAM All rights reserved.



J. Dziok, M. Aouf, J. Sokół / Eur. J. Pure Appl. Math, 4 (2011), 322-329 323

For real or complex numbers a, b, c other than 0,−1,−2, . . ., the Gaussian hypergeometric

series is defined by

2F1(a, b; c; z) =

∞
∑

k=0

(a)k(b)k

(c)k(1)k
zk , (3)

where

(d)k =

¨

1 (k = 0; d ∈ C \ {0}),
d(d + 1) . . . (d + k− 1) (k ∈ N; d ∈ C),

we note that the series (3) converges absolutely for all z ∈ U so that it represents an analytic

function in U (see [8]).

With the aid of the Gaussian hypergeometric function 2F1(a, b; c; z), let us consider a

family of linear operators Iλp,n :An(p)→An(p) as follows:

Iλp,n(a, b; c) f (z) = zp +

∞
∑

k=n

(c)k(λ+ p)k

(a)k(b)k
ak+pzk+p

= zp
2F1(c, 1; a; z) ∗ zp

2F1(λ+ p, 1; b; z)

(a, b, c ∈ R \Z−0 ;λ > −p; z ∈ U) . (4)

The operator Iλp,n was introduced and studied by Fu and Liu [3].

We note that:

(i) In
1,1(a, n+1; a) f (z) = In f (z) (n> −1), where In is the Noor integral operator of n− th

order [see 6];

(ii) Iλ1,1(µ + 2,1; 1) f (z) = Iµ,λ f (z) (µ > −2,λ > −1), where Iµ,λ is the Choi–Saigo–

Srivastava operator [see 2];

(iii) Iλp,1(λ + p + 1, b; b) f (z) = Fλ,p( f )(z) (λ > −p), where Fλ,p( f )(z) is the generalized

Bernardi–Libera–Livingston operator [see 2];

(iv) Iλp,1(a, 1; c) f (z) = Iλp (a, c) f (z) (a, c ∈ R\Z−0 ,λ > −p), where Iλp (a, c) is the Cho–Kwon–

Srivastava operator [see 1];

(v) I1
p,1(n + p, c; c) f (z) = In,p f (z) (n > −p), where In,p is the Noor integral operator of

(n+ p− 1)− th order (see Liu and Noor [4] and Patel and Cho [7]).

Also it is easily to show that [see 3]:

Iλp,n(a,λ+ p; a) f (z) = I1
p,n(p+ 1, b; b) f (z) = f (z) and I1

p,n(a, p; a) f (z) =
z f ′(z)

p
,

z
�

Iλp,n(a, b; c) f (z)
�′
= (λ+ p)Iλ+1

p,n (a, b; c) f (z)−λIλp,n(a, b; c) f (z) (5)

and

z
�

Iλp,n(a+ 1, b; c) f (z)
�′
= aIλp,n(a, b; c) f (z)− (a− p)Iλp,n(a+ 1, b; c) f (z). (6)

By using the operator Iλp,n(a, b; c), we define the following classes of functions:



J. Dziok, M. Aouf, J. Sokół / Eur. J. Pure Appl. Math, 4 (2011), 322-329 324

Definition 1. Let Φ be the set of complex-valued functions ϕ(r, s, t),

ϕ(r, s, t) : C3→ C (C is the complex plane)

such that

1. ϕ(r, s, t) is continuous in a domain D ⊂ C3;

2. (0,0,0) ∈ D and
�

�ϕ(0,0,0)
�

�< 1;

3.

�

�

�ϕ
�

eiθ , f (λ,ζ,θ , p), g(λ,ζ,θ , p, M)
�

�

�

� > 1

whenever
�

eiθ , f (λ,ζ,θ , p), g(λ,ζ,θ , p, M)
�

∈ D,

with Re

¦

e−iθM
©

≥ ζ(ζ− 1), for all θ ∈ R, and for all ζ ≥ p ≥ 1, where

f (λ,ζ,θ , p) =

�

ζ+λ

λ+ p

�

eiθ

and

g(λ,ζ,θ , p, M) =
(λ+ 1)(λ+ 2ζ)eiθ +M

(λ+ p)(λ+ p+ 1)
.

Definition 2. LetH be the set of complex-valued functions h(r, s, t);

h(r, s, t) : C3→ C

such that

1. h(r, s, t) is continuous in a domain D ⊂ C3;

2. (1,1,1) ∈ D and
�

�g(1,1,1)
�

�< J (J > 1);

3.

�

�

�h
�

Jeiθ , f (λ,ζ,θ , p, J), g(λ,ζ,θ , p, J)
�

�

�

� ≥ J

whenever
�

Jeiθ , f (λ,ζ,θ , p, J), g(λ,ζ,θ , p, J , L)
�

∈ D,

with Re{L} ≥ ζ(ζ− 1) for all θ ∈ R and for all ζ ≥
J − 1

J + 1
, where

f (λ,ζ,θ , p, J) =
1+ ζ+ (λ+ p− 1)Jeiθ

(λ+ p)

and

g(λ,ζ,θ , p, J , L) =
1

(λ+ p+ 1)

¨

2+ ζ+ (λ+ p− 1)Jeiθ +
ζ− ζ2 + (λ+ p+ 1)ζJeiθ + L

ζ+ (λ+ p− 1)Jeiθ

«

.



J. Dziok, M. Aouf, J. Sokół / Eur. J. Pure Appl. Math, 4 (2011), 322-329 325

2. Main Results

We recall the following lemma due to Miller and Mocanu [5].

Lemma 1. Let w(z) = a+ wνz
ν + . . . be regular in U with ν ∈ N. If z0 = r0eiθ (0 < r0 < 1)

and
�

�w(z0)
�

� = max
|z|≤|z0|

|w(z)|, then

z0w′(z0) = ζw(z0) (7)

and

Re

¨

1+
z0w′′(z0)

w′(z0)

«

≥ ζ, (8)

where ζ is a real number and

ζ≥ ν

�

�w(z0)− a
�

�

2

�

�w(z0)
�

�

2
− |a|2

≥ ν

�

�w(z0)
�

�− |a|
�

�w(z0)
�

�+ |a|
. (9)

Note that if ν = 0 then the condition (9) becomes ζ ≥ ν ≥ 1.

Theorem 1. Let ϕ(r, s, t) ∈ Φ and let f in the classAn(p) satisfy

�

Iλp,n(a, b; c) f (z), Iλ+1
p,n (a, b; c) f (z), Iλ+2

p,n (a, b; c) f (z)
�

∈ D ⊂ C3 (10)

and �

�

�ϕ
�

Iλp,n(a, b; c) f (z), Iλ+1
p,n (a, b; c) f (z), Iλ+2

p,n (a, b; c) f (z)
�
�

�

�< 1 (11)

for a, b, c ∈ R \Z−0 , λ > −p, p ∈ N and z ∈ U. Then we have

�

�

�Iλp,n(a, b; c) f (z)

�

�

�< 1 (z ∈ U) . (12)

Proof. We define the function w by

w(z) = Iλp,n(a, b; c) f (z)
�

a, b, c ∈ R \Z−0 ;λ > −p; p ∈ N
�

(13)

for f belonging to the class An(p). Then, it follows that w ∈ An(p) and w(z) 6= 0 for

z ∈ U \ {0}. With the aid of (5), we have

Iλ+1
p,n (a, b; c) f (z) =

1

(λ+ p)

�

zw′(z) +λw(z)
�

(14)

and

Iλ+2
p,n (a, b; c) f (z) =

z2w′(z) + 2(λ+ 1)zw′(z) +λ(λ+ 1)w(z)

(λ+ p)(λ+ p+ 1)
. (15)

Suppose that z0 = r0eiθ (0< r0 < 1;θ ∈ R) and

w(z0) = max
|z|≤|z0|

|w(z)| = 1. (16)



J. Dziok, M. Aouf, J. Sokół / Eur. J. Pure Appl. Math, 4 (2011), 322-329 326

Then, letting w(z0) = eiθ and using (7) of Lemma 1, we obtain

Iλp,n(a, b; c) f (z0) = eiθ , (17)

Iλ+1
p,n (a, b; c) f (z0) =

1

(λ+ p)

�

z0w′(z0) +λw(z0)
�

=
(ζ+λ)

(λ+ p)
eiθ (18)

and

Iλ+2
p,n (a, b; c) f (z0) =

1

(λ+ p)(λ+ p+ 1)

�

(λ+ 1)(λ+ 2ζ)eiθ + z2
0 w′(z0)

�

=
(λ+ 1)(λ+ 2ζ)eiθ +M

(λ+ p)(λ+ p+ 1)
, (19)

where M = z2
0 w′′(z0) and ζ≥ p ≥ 1. Further, an application of (8) in Lemma 1, gives

Re

¨

z0w′′(z0)

w′(z0)

«

=Re

¨

z2
0 w′′(z0)

ζeiθ

«

≥ ζ− 1, (20)

or

Re

¦

e−iθM
©

≥ ζ(ζ− 1) (θ ∈ R;ζ ≥ 1). (21)

Since ϕ(r, s, t) ∈ Φ, we also have

�

�

�ϕ
�

Iλp,n(a, b; c) f (z), Iλ+1
p,n (a, b; c) f (z), Iλ+2

p,n (a, b; c) f (z)
�
�

�

�

=

�

�

�

�

ϕ

�

eiθ ,
ζ+λ

λ+ p
eiθ ,

1

(λ+ p)(λ+ p+ 1)

�

(λ+ 1)(λ+ 2ζ)eiθ +M
�

�
�

�

�

�

> 1 (22)

which contradicts the condition (11) of Theorem 1. Therefore, we conclude that

|w(z)| =
�

�

�Iλp,n(a, b; c) f (z)

�

�

� < 1 (z ∈ U) . (23)

This completes the proof of Theorem 1.

Corollary 1. Let ϕ1(r, s, t) = s and let f ∈ An(p) satisfy the conditions in Theorem 1 for

a, b, c ∈ R \Z−
0

, λ > −p, p ∈ N and z ∈ U. Then

�

�

�Iλ+i
p,n (a, b; c) f (z)

�

�

� < 1 (i = 0,1,2, . . . ; a, b, c ∈ R \Z−0 ;λ > −p; p ∈ N; z ∈ U).

Note that ϕ1(r, s, t) = s is in Φ, with the aid of Theorem 1, we have

�

�

�Iλp,n(a, b; c) f (z)

�

�

� < 1⇒
�

�

�Iλ+1
p,n (a, b; c) f (z)

�

�

� < 1

⇒
�

�

�Iλ+i
p,n (a, b; c) f (z)

�

�

� < 1 (i = 0,1,2, . . . ; a, b, c ∈ R \Z−0 ;λ > −p; p ∈ N; z ∈ U).



J. Dziok, M. Aouf, J. Sokół / Eur. J. Pure Appl. Math, 4 (2011), 322-329 327

Theorem 2. Let h(r, s, t) ∈ H , and let f ∈An(p) satisfy

 

Iλp,n(a, b; c) f (z)

Iλ−1
p,n (a, b; c) f (z)

,
Iλ+1
p,n (a, b; c) f (z)

Iλp,n(a, b; c) f (z)
,

Iλ+2
p,n (a, b; c) f (z)

Iλ+1
p,n (a, b; c) f (z)

!

∈ D⊂ C3 (24)

and
�

�

�

�

�

h

 

Iλp,n(a, b; c) f (z)

Iλ−1
p,n (a, b; c) f (z)

,
Iλ+1
p,n (a, b; c) f (z)

Iλp,n(a, b; c) f (z)
,

Iλ+2
p,n (a, b; c) f (z)

Iλ+1
p,n (a, b; c) f (z)

!
�

�

�

�

�

< J (25)

for some a, b, c, λ, p, n, J
�

a, b, c ∈ R \Z−
0

;λ > 1; p, n ∈ N; J > 1
�

and for all z ∈ U. Then we

have
�

�

�

�

�

Iλp,n(a, b; c) f (z)

Iλ−1
p,n (a, b; c) f (z)

�

�

�

�

�

< J (z ∈ U). (26)

Proof. We define the function w by

w(z) =
Iλp,n(a, b; c) f (z)

Iλ−1
p,n (a, b; c) f (z)

(27)

for f belonging to the class An(p). Then, it follows that w is either analytic or meromorphic

in U , w(0) = 1, and w(z) 6= 1. With the aid of the identity (5), we have

Iλ+1
p,n (a, b; c) f (z)

Iλp,n(a, b; c) f (z)
=

1

(λ+ p)

�

1+ (λ+ p− 1)w(z) +
zw′(z)

w(z)

�

(28)

and

Iλ+2
p,n (a, b; c) f (z)

Iλ+1
p,n (a, b; c) f (z)

=
1

(λ+ p+ 1)

¨

2+ (λ+ p− 1)w(z) +
zw′(z)

w(z)

+
(λ+ p− 1)zw′(z) +

zw′z)

w(z)
+

z2w′(z)

w(z)
−
�

zw′(z)

w(z)

�2

1+ (λ+ p− 1)w(z) +
zw′(z)

w(z)







. (29)

Suppose that z0 = r0eiθ (0< r0 < 1;θ ∈ R) and
�

�w(z0)
�

�= max
|z|≤|z0|

|w(z)| = J . Letting

w(z0) = Jeiθ and using Lemma 1 with a = ν = 1, we see that

Iλ+1
p,n (a, b; c) f (z0)

Iλp,n(a, b; c) f (z0)
=

1

(λ+ p)

�

1+ ζ+ (λ+ p− 1)Jeiθ
�

(30)

and

Iλ+2
p,n (a, b; c) f (z)

Iλ+1
p,n (a, b; c) f (z)

=
1

(λ+ p+ 1)

¨

2+ ζ+ (λ+ p− 1)Jeiθ +
ζ− ζ2 + (λ+ p− 1)ζJeiθ + L

ζ+ (λ+ p− 1)Jeiθ

«

,

(31)



REFERENCES 328

where L =
z2

0 w′′(z0)

w′(z0)
and ζ ≥

J − 1

J + 1
. Further, an application of (16) in Lemma 1 gives

Re {L} ≥ ζ(ζ− 1).

Since h(r, s, t) ∈H , we have

�

�

�

�

�

h

 

Iλp,n(a, b; c) f (z0)

Iλ−1
p,n (a, b; c) f (z0)

,
Iλ+1
p,n (a, b; c) f (z0)

Iλp,n(a, b; c) f (z0)
,

Iλ+2
p,n (a, b; c) f (z0)

Iλ+1
p,n (a, b; c) f (z0)

!
�

�

�

�

�

=

�

�

�

�

�

h

�

Jeiθ ,
1+ ζ+ (λ+ p− 1)Jeiθ

(λ+ p
+

1

(λ+ p+ 1)

¦

2+ ζ+ (λ+ p− 1)Jeiθ

+
ζ− ζ2 + (λ+ p− 1)ζJeiθ + L

ζ+ (λ+ p− 1)Jeiθ

«�

�

�

�

�

�

≥ J , (32)

which contradicts condition (25) of Theorem 2. Therefore, we conclude that

|w(z)| =

�

�

�

�

�

Iλp,n(a, b; c) f (z)

Iλ−1
p,n (a, b; c) f (z)

�

�

�

�

�

< J (33)

for some a, b, c ∈ R \Z−0 ,λ > 1, p, n ∈ N, J > 1 and for all z ∈ U . This completes the proof of

Theorem 2.

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[2] J Choi, M Saigo and H Srivastava. Some inclusion properties of a certain family of integral

operators. J. Math. Anal. Appl., 276:432–445, 2002.

[3] X Fu and M Liu. Some subclasses of analytic functions involving the generalized Noor

integral operator. J. Math. Anal. Appl., 323:190–208, 2006.

[4] J Liu and K Noor, Some properties of Noor integral operator. J. Natural Geometry, 21:81–

90, 2002.

[5] S Miller and P Mocanu. Second order differential inequalities in the complex plane. J.

Math. Anal. Appl., 65:289–305, 1978.

[6] K Noor and M Noor. On integral operators. J. Math. Anal. Appl., 238:341–352, 1999.

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