Paper-6-Acta24.2010. 127KB Jun 04 2011 12:03:14 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 24/2010
pp. 51-61

SOME PROPERTIES OF NEW INTEGRAL OPERATOR

R. M. El-Ashwah and M. K. Aouf
Abstract. Certain integral operator Jpm (λ, ℓ)(λ ≥ 0; ℓ ≥ 0; p ∈ N ; m ∈ N0 =
N ∪ {0}), where N = {1, 2, ...} is introduced for functions of the form f (z) = z p +
∞
X
ak z k which are analytic and p-valent in the open unit disc U = {z : |z| < 1}.

k=p+1

The opject of the present paper is to give an applications of the above operator to
the differential inequalities.
2000 Mathematics Subject Classification: 30C45.
1.Introduction

Let A(p) denote the class of functions of the form:
p

f (z) = z +

∞
X

ak z k

(p ∈ N = {1, 2, ....})

(1.1)

k=p+1

which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. In [3] Catas
extended the multiplier transformation and defined the operator Ipm (λ, ℓ)f (z) on
A(p) by the following infinite series


∞ 
X
p + ℓ + λ(k − p) m
m
p
Ip (λ, ℓ)f (z) = z +
an z n
p+ℓ
k=p+1

(λ ≥ 0; ℓ ≥ 0; p ∈ N, m ∈ N0 ; z ∈ U ).

(1.2)

We note that:
′

Ip0 (1, 0)f (z)

= f (z)

and

Ip1 (1, 0)f (z)

zf (z)
=
.
p

By specializing the parameters m, λ, ℓ and p, we obtain the following operators
studied by various authors :
51

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

(i) Ipm (1, ℓ) = Ip (m, ℓ)f (z) (see Kumar et al. [9] and Srivastava et al. [16]);
(ii) Ipm (1, 0)f (z) = Dpm f (z) (see [2], [8] and [11]);
(iii) I1m (1, ℓ)f (z) = Iℓm f (z) (see Cho and Srivastava [4] and Cho and Kim [5]);
(iv) I1m (1, 0) = Dm f (z) (m ∈ N0 ) (see Salagean [14]);

(v) I1m (λ, 0) = Dλm (see Al-Aboudi [1]);
(vi) I1m (1, 1) = I m f (z) (see Uralegaddi and Somanatha [17]);
m f (z), where D m f (z) is defined by
(vii) Ipm (λ, 0) = Dλ,p
λ,p
m
Dλ,p
f (z)


∞ 
X
p + λ(k − p) m
=z +
ak z k .
p
p

k=p+1

Furthermore we define the integral operator Jpm (λ, ℓ)f (z) as follows:
Jp0 (λ, ℓ)f (z) = f (z),
Jp1 (λ, ℓ)f (z)

Jp2 (λ, ℓ)f (z) =

=





p+ℓ
λ

p+ℓ
λ





z

z p−(

p−( p+ℓ
)
λ

Zz

t(

p+ℓ
)−p−1
λ

f (t)dt

(f (z) ∈ A(p); z ∈ U ),

0

p+ℓ
)
λ

Zz

t(

p+ℓ
)−p−1
λ

Jp1 (λ, ℓ)f (t)dt

(f (z) ∈ A(p); z ∈ U ),

0

and, in general,
Jpm (λ, ℓ)f (z)

=

Jp1 (λ, ℓ)
x

=





p+ℓ
λ



z

p−(

p + ℓ Zz p + ℓ
)
)−p−1
(
λ
Jpm−1 (λ, ℓ)f (t)dt
t λ
0


 p 
 p 
zp
z
z

1
1
∗ Jp (λ, ℓ)
∗ ....Jp (λ, ℓ)
∗ f (z)
1−z
1−z
1−z
m − times
y
(f (z) ∈ A(p); m ∈ N ; z ∈ U ).

(1.3)

We note that if f (z) ∈ A(p), then from (1.1) and (1.3), we have
m
∞ 
X
p+ℓ
m

p
Jp (λ, ℓ)f (z) = z +
ak z k (m ∈ N0 = N ∪ {0}). (1.4)
p + ℓ + λ(k − p)
k=p+1

From (1.4), it is easy to verify that
′

λz(Jpm+1 (λ, ℓ)f (z)) = (ℓ + p)Jpm (λ, ℓ)f (z) − [ℓ + p(1 − λ)] Jpm+1 (λ, ℓ)f (z) (λ > 0).
(1.5)
52

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

We note that:
(i)J1m (λ, 0)f (z) = Iλ−m f (z) (see Patel [12])

=

(

f (z) ∈ A(1) : Iλ−m f (z) = z +

∞
X

[1 + λ(k − 1)]−m ak z k , m ∈ N0

k=2

)

;

(ii) J1α (1, 1)f (z) = I α f (z) (see Jung et al. [7]);
)
(
α
∞ 
X
2
ak z k ; α > 0; z ∈ U ;
= f (z) ∈ A(1) : I α f (z) = z +
k+1
k=2

(iii) Jpα (1, 1)f (z) = Ipα f (z) (see Shams et al. [15]);


α
∞ 


X
p+1
= f (z) ∈ A(p) : Ipα f (z) = z p +
ak z k ; α > 0; z ∈ U ;


k+1
k=p+1

(iv) Jpm (1, 1)f (z) = Dm f (z) (see Patel and Sahoo [13]);



m
∞ 


X
p+1
ak z k ; m is any integer; z ∈ U .
= f (z) ∈ A(p) : Dm f (z) = z p +


k+1
k=p+1

(v) J1m (1, 1)f (z) = I m f (z) (see Flett [6]);

=

m
∞ 
X
2
ak z k ; m ∈ N0 ; z ∈ U
f (z) ∈ A(1) : I f (z) = z +
k+1

(

m

k=2

)

;

(iv) J1m (1, 0)f (z) = I m f (z) (see Salagean [14])
(
)
∞
X
m
−m
k
= f (z) ∈ A(1) : I f (z) = z +
k ak z ; m ∈ N0 ; z ∈ U .
k=2

Also we note that:
(i)Jpm (1, 0)f (z) = Jpm f (z)


∞  


X
m
p
k
m
p
ak z ; m ∈ N0 ; z ∈ U ;
= f (z) ∈ A(p) : Jp f (z) = z +


k
k=p+1

53

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

(ii) Jpm (1, ℓ)f (z) = Jpm (ℓ)f (z)


m
∞ 


X
p+ℓ
ak z k ; m ∈ N0 ; ℓ ≥ 0; z ∈ U ;
= f (z) ∈ A(p) : Jpm (ℓ)f (z) = z p +


k+ℓ
k=p+1

m f (z)
(iii) Jpm (λ, 0)f (z) = Jλ,p


m
∞ 


X
p
m
f (z) = z p +
= f (z) ∈ A(p) : Jλ,p
ak z k ; m ∈ N0 ; λ ≥ 0; z ∈ U .


p + λ(k − p)
k=p+1

By using the operator Jpm (λ, ℓ), we define the following classes of functions.

Definition 1. Let Φ be the set of complex-valued functions ϕ(r, s, t);
ϕ(r, s, t) : C 3 → C (C is complex plane)
such that
(i) ϕ(r, s, t) is continuous in a domain D ⊂ C 3 ;
(ii) (0,
0, 0) ∈ D and |ϕ(0, 0, 0)| < 1;

hn
h

h
i
i

p(1−λ)+ℓ
1 
iθ ,  1 
1
+
2
ζ + p(1−λ)+ℓ
e
ζ+
(iii) ϕeiθ ,  p+ℓ
2
λ
λ
p+ℓ

λ
λ
 !
i2 
h

p(1−λ)+ℓ
eiθ + M
 > 1,
λ



h

h
i
i
p(1−λ)+ℓ
p(1−λ)+ℓ
1
1
iθ
iθ
2 

whenever e ,  p+ℓ  ζ +
1+
2
e
,
ζ+
λ
λ
p+ℓ

λ
λ
!
i2 
h


p(1−λ)+ℓ
eiθ + M
∈ D, λ > 0, ℓ ≥ 0, with Re e−iθ M ≥ ζ(ζ − 1), for
λ

all θ ∈ R, and for all ζ ≥ p ≥ 1.

Definition 2. Let H be a set of complex-valued functions h(r, s, t);
h(r, s, t) : C 3 → C
(i) h(r, s, t) is continuous in a domain D ⊂ C 3 ;
(ii) (1,
1, 1) ∈ D and |h(1,
 1, 1)| < M (M" > 1);
 
p+ℓ



ζ + λ M eiθ

1 
iθ


(iii) h M e ,
ζ + p+ℓ
M eiθ +
,  p+ℓ
λ
p+ℓ

λ
λ
54

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

ζ − ζ2 +
ζ+
whenever


iθ

ζ+

M e ,







p+ℓ
λ



p+ℓ
λ



p+ℓ
λ

ζM eiθ + L



M eiθ



p+ℓ
λ




 ≥ M,



"


M eiθ
1

 ζ + p+ℓ
M eiθ +
,
λ

ζ − ζ2 +

+
ζ+

p+ℓ
λ



p+ℓ
λ





p+ℓ
λ



ζM eiθ + L



 ∈ D,

M eiθ

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

M −1
.
M +1

2.Main Results
We recall the following lemmas due to Miller and Mocenu [10].
Lemma 1 [10]. Let w(z) = bp z p + bp+1 z p+1 + ....(p ∈ N ) be regular in the unit disc
U with w(0) 6= 0 (z ∈ U ).
If z0 = r0 eiθ0 (0 < r0 < 1) and |w(z0 )| = max |w(z)| , then
|z|≤z0

′

and

z0 w (z0 ) = ζw(z0 )

(2.1)

(

(2.2)

′′

z0 w (z0 )
Re 1 +
w′ (z0 )

)

≥ζ

and
where ζ is real and ζ ≥ p ≥ 1.
Lemma 2 [10]. Let w(z) = a + wν z ν + .... be regular in U with w(z) 6= a and
ν ≥ 1 If z0 = r0 eiθ0 (0 < r0 < 1) and |w(z0 )| = max |w(z)| , then
|z|≤z0

′

z0 w (z0 ) = ζw(z0 )
and

(

′′

z0 w (z0 )
Re 1 +
w′ (z0 )

55

)

≥ ζ,

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

where ζ is a real number and
ζ≥ν

|w(z0 )| − |a|
|w(z0 ) − a|2
2
2 ≥ ν |w(z )| + |a| .
|w(z0 )| − |a|
0

(2.3)

Theorem 1. Let ϕ(r, s, t) ∈ Φ and let f (z) belonging to the class A(p) satisfy
(i) (Jpm (λ, ℓ)f (z) , Jpm−1 (λ, ℓ)f (z) , Jpm−2 (λ, ℓ)f (z)) ∈ D ⊂ C 3
and


(ii) ϕ(Jpm (λ, ℓ)f (z) , Ipm−1 (λ, ℓ)f (z) , Ipm−2 (λ, ℓ)f (z)) < 1

for m > 2, p ∈ N, λ > 0, ℓ ≥ 0 and z ∈ U. Then we have

 m
Jp (λ, ℓ)f (z) < 1 (z ∈ U ).

(2.4)

Proof. We define the analytic function w(z) by
Jpm (λ, ℓ)f (z) = w(z)

(m > 2; p ∈ N ; λ > 0; ℓ ≥ 0)

(2.5)

for f (z) belonging to the class A(p). Then, it folllows that w(z) ∈ A(p) and
w(z) 6= 0 (z ∈ U ). With the aid of the idenlitty (1.5), we have



1
p(1 − λ) + ℓ
′
m−1

Jp (λ, ℓ)f (z) = 
w(z) + zw (z)
(2.6)
p+ℓ
λ
λ

and

Jpm−2 (λ, ℓ)f (z)

=

1


p+ℓ
λ



2

(

p(1 − λ) + ℓ
λ



p(1 − λ) + ℓ
1+2
λ

2



w(z) +


zw (z) + z w (z) .
′

2

′′

(2.7)

Suppose that z0 = r0 eiθ0 (0 < r0 < 1; θ0 ∈ R) and
w(z0 ) = max |w(z)| = 1.
|z|≤|z0 |

(2.8)

Then, letting w(z0 ) = eiθ0 and using (2.1) of Lemma 1, we obtain
Jpm (λ, ℓ)f (z0 ) = w(z0 ) = eiθo ,
56

(2.9)

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator




p(1 − λ) + ℓ
′
= p+ℓ
w(z0 ) + z0 w (z0 )
λ
( λ )



p(1 − λ) + ℓ
1
ζ+
eiθ0
= p+ℓ
λ
)
(
1

Jpm−1 (λ, ℓ)f (z0 )

λ

and
Jpm−2 (λ, ℓ)f (z0 ) =

=

1
2
( p+ℓ
λ )

"(

1
2
( p+ℓ
λ )





1+2

p(1 − λ) + ℓ
1+2
λ



h

p(1−λ)+ℓ
λ

i

ζ+

h

p(1−λ)+ℓ
λ

i
′′
z02 w (z0 )


p(1 − λ) + ℓ
ζ+
λ

2 )

iθ0

e

i2 

eiθ0 +

#

+M ,

(2.11)

′′

where M = z02 w (z0 ) and ζ ≥ p ≥ 1.
Further, an application of (2.2) in Lemma 1, gives
)
)
(
(
′′
′′
z02 w (z0 )
z0 w (z0 )
≥ ζ − 1,
= Re
Re
ζeiθ0
w′ (z0 )
or

n
o
Re e−iθ0 M ≥ ζ(ζ − 1)

(θ0 ∈ R; ζ ≥ 1).

(2.12)

(2.13)

Since ϕ(r, s, t) ∈ Φ, we also have



ϕ(Jpm (λ, ℓ)f (z0 ), J m−1 (λ, ℓ)f (z0 ), J m−2 (λ, ℓ)f (z0 ))









1
p(1 − λ) + ℓ
p(1 − λ) + ℓ
1

iθ0
iθ0
e , p+ℓ
ζ
1+2
ζ+
= ϕ e , p+ℓ

λ
λ
( λ )
( λ )2
#!

 )

p(1 − λ) + ℓ 2 iθ0

+
e +M >1
(2.14)

λ

which contradicts the condition (ii) of Theorem 1 Therefore, we conclude that


|w(z)| = Jpm (λ, ℓ)f (z) < 1 (z ∈ U ; m > 2).

Corollary 1. Let ϕ1 (r, s, t) = s and let f (z) ∈ A(p) satisfy the conditions in
Theorem 1 for m > 2; p ∈ N ; λ > 0; ℓ ≥ 0 and z ∈ U. Then

 m+i
(Jp (λ, ℓ)f (z) < 1 (i = 0, 1, ...; m > 2; p ∈ N ; λ > 0; ℓ ≥ 0)
57

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

Proof. Note that ϕ1 (r, s, t) = s in Φ, with the aid of Theorem 1, we have



 m−1
Jp (λ, ℓ)f (z) < 1 ⇒ Jpm (λ, ℓ)f (z) < 1 (m > 2; p ∈ N ; λ > 0; ℓ ≥ 0)


⇒ Jpm+i (λ, ℓ)f (z) < 1

(i = 0, 1, 2, ...; m > 2; p ∈ N ; λ > 0; ℓ ≥ 0; z ∈ U ).

Theorem 2. Let h(r, s, t) ∈ H, and let f (z) belonging to A(p) satisfying
!
Jpm−1 (λ, ℓ)f (z) Jpm−2 (λ, ℓ)f (z) Jpm−3 (λ, ℓ)f (z)
,
∈ D ⊂ C3
,
(i)
Jpm (λ, ℓ)f (z) Jpm−1 (λ, ℓ)f (z) Jpm−2 (λ, ℓ)f (z)
and




(ii) h


Jpm−1 (λ, ℓ)f (z) Jpm−2 (λ, ℓ)f (z) Jpm−3 (λ, ℓ)f (z)
,
,
Jpm (λ, ℓ)f (z) Jpm−1 (λ, ℓ)f (z) Jpm−2 (λ, ℓ)f (z)

!


 0; ℓ ≥ 0; m > 3; p ∈ N ; M > 1) and for all z ∈ U. Then
we have


 J m−1 (λ, ℓ)f (z) 

 p
(2.16)
 < M (z ∈ U )
 m
 Jp (λ, ℓ)f (z) 
Proof. We define the function w(z) by


 J m−1 (λ, ℓ)f (z) 

 p
 = w(z) (λ > 0, ℓ ≥ 0; m > 3; p ∈ N )
 m
 Jp (λ, ℓ)f (z) 

(2.17)

for f (z) belonging to the class A(p). Then, it follows that w(z) is either analytic
or meromorphic in U , w(0) = 1, and w(z) 6= 1. With the aid of the identity (1.5),
we have


#
"
′
 J m−2 (λ, ℓ)f (z) 


zw
(z)
1
 p
 
p+ℓ

w(z) +
(2.18)
 m−1
=
λ
p+ℓ
 Jp (λ, ℓ)f (z) 
w(z)
λ

and



#
"
′
 J m−3 (λ, ℓ)f (z) 


zw (z)
  1 
 p
p+ℓ
+
w(z) +
=
 m−2
λ
p+ℓ
 Jp (λ, ℓ)f (z) 
w(z)
λ



p+ℓ
λ




′′
′
′
′
z 2 w (z)
zw (z) 2
(z)
+
−
(
)
zw (z) + zw
w(z)
w(z)
w(z)



.
′
zw (z)
p+ℓ
w(z)
+
λ
w(z)
58

(2.19)

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

Suppose that z0 = r0 eiθ0 (0 < r0 < 1; θ0 ∈ R) and |w(z0 )| = max |w(z)| = M.
|z|≤|z0 |

Letting w(z0 ) = M eiθ0 and applying Lemma 2 with a = ν = 1, we see that
Jpm−2 (λ, ℓ)f (z)
Jpm−1 (λ, ℓ)f (z)
and
Jpm−3 (λ, ℓ)f (z0 )
Jpm−2 (λ, ℓ)f (z0 )
′′

z 2 w (z )

=

1
p+ℓ
λ

h


i
iθ0
 ζ + p+ℓ
M
e
λ

(2.20)





p+ℓ
2
iθ0 + L 



ζM
e
ζ
−
ζ
+
λ
1

 ζ + p+ℓ

=
,
M eiθ0 +
λ
p+ℓ
p+ℓ 

ζ+
M eiθ0
λ

λ

M −1
where L = 0w(z0 )0 and ζ ≥ M
+1 .
Further, an application of (ii) in Lemma 2 gives

Re {L} ≥ ζ(ζ − 1).
Since h(r, s, t) ∈ H, we have

!

Jpm−1 (λ, ℓ)f (z0 ) Jpm−2 (λ, ℓ)f (z0 ) Jpm−3 (λ, ℓ)f (z0 ) 

,
,
h


Jpm (λ, ℓ)f (z0 ) Jpm−1 (λ, ℓ)f (z0 ) Jpm−2 (λ, ℓ)f (z0 ) 



 

 

M eiθ0
ζ + p+ℓ

λ
1
iθ0






ζ+ p+ℓ
,
= h M e ,
M eiθ0 +
λ
p+ℓ
p+ℓ

λ
λ



iθ0 + L  
ζ − ζ 2 + p+ℓ
ζM
e

λ
 ≥ M,


p+ℓ
 
ζ + λ M eiθ0

which contradicts condition (ii) of Theorem 2. Therefore, we conclude that


 J m−1 (λ, ℓ)f (z) 
 p

|w(z)| =  m
(2.23)
 0, ℓ ≥ 0, m > 3, p ∈ N and z ∈ U. This completes the proof of Theorem
2.
Remarks. (i)Putting λ = 1 in the above results, we obtain the correspording
results for the operatr Jpm (ℓ)f (z);
(ii) Putting ℓ = 0 in the above results, we obtain the correspording results for
m f (z);
the operatr Jλ,p
59

R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

(iii) Putting λ = ℓ = 1 in the above results, we obtain the correspording results
for the operatr Dm f (z).
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R. M. El-Ashwah, M. K. Aouf - Some Properties of New Integral Operator

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R. M. El-Ashwah
Department of Mathematics
Faculty of Science (Damietta Branch)
University of Mansoura University
New Damietta 34517, Egypt
email:r elashwah@yahoo.com
M. K. Aouf
Department of Mathematics
Faculty of Science
University of Mansoura University
Mansoura 35516, Egypt
email:mkaouf127@yuhoo.com

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