International Journal of Analysis and Applications
ISSN 2291-8639
Volume 15, Number 2 (2017), 146-154
DOI: 10.28924/2291-8639-15-2017-146

FRACTIONAL OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE

FIRST DERIVATIVES ARE s-PREINVEX IN THE SECOND SENSE

BADREDDINE MEFTAH∗

Abstract. In this paper, we establish an fractional identity. Using this new identity we derives some

fractional Ostrowski’s inequalities for functions whose first derivatives are s-preinvex in the second

sense.

1. Introduction

In 1938, A.M. Ostrowski proved an interesting integral inequality, given by the following theorem

Theorem 1.1. [10] Let f : I → R, where I ⊆ R is an interval, be a mapping in the interior I◦of I,
and a,b ∈ I◦, with a < b. If |f′| ≤ M for all x ∈ [a,b], then∣∣∣∣∣∣f(x) − 1b−a

b∫
a

f (t) dt

∣∣∣∣∣∣ ≤ M (b−a)
[

1

4
+

(
x− a+b

2

)2
(b−a)2

]
, ∀x ∈ [a,b] . (1.1)

In the last decades, the inequality (1.1) has attracted much interest by many researchers, and
considerable papers have been published concerning the generalizations, variants, and extensions of
the inequality (1.1), for more detail we refer readers to [4, 7–9, 13, 16, 17] and references cited therein.

Recently, lot of efforts have been made by many mathematicians to generalize the classical convexity.
Hanson [3], introduced a new class of generalized convex functions, called invex functions. In [1], the
authors gave the concept of preinvex functions which is special case of invexity, and many authors have
study their basic properties, and their role in optimization, variational inequalities and equilibrium
problems, we refer readers to [11, 12, 15, 20, 21].

İşcan [5] established the following Ostrowski inequalities for functions whose derivatives are preinvex

Theorem 1.2. [5, Theorem 2.2] Let A ⊆ R be an open invex subset with respect to η : A×A → R
and a,b ∈ A with a < a + η(b,a). Suppose that f : A → R is a differentiable function and |f′| is
preinvex function on A. If f′ is integrable on [a,a + η(b,a)], then the following inequality∣∣∣∣∣∣∣f(x) − 1η(b,a)

a+η(b,a)∫
a

f(u)du

∣∣∣∣∣∣∣ ≤
η(b,a)

6

×
{(

3
(
x−a
η(b,a)

)2
− 2

(
x−a
η(b,a)

)3
+ 2

(
a+η(b,a)−x

η(b,a)

)3)
|f′(a)|

+

(
1 − 3

(
x−a
η(b,a)

)2
+ 4

(
x−a
η(b,a)

)3)
|f′(b)|

}
holds for each x ∈ [a,a + η(b,a)].

Theorem 1.3. [5, Theorem 2.8] Let A ⊆ R be an open invex subset with respect to η : A×A → R
and a,b ∈ A with a < a + η(b,a). Suppose that f : A → R is a differentiable function and |f′|q is

Received 21st July, 2017; accepted 30th September, 2017; published 1st November, 2017.
2010 Mathematics Subject Classification. 26D10, 26D15, 26A51.
Key words and phrases. Ostrowski inequality; midpoint inequality; Hölder inequality; power mean inequality; s-

preinvex functions.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

146



FRACTIONAL OSTROWSKI TYPE INEQUALITIES 147

preinvex function on [a,a + η(b,a)] for some fixed q ≥ 1. If f′ is integrable on [a,a + η(b,a)], then the
following inequality∣∣∣∣∣∣∣f(x) − 1η(b,a)

a+η(b,a)∫
a

f(u)du

∣∣∣∣∣∣∣ ≤ η (b,a)
(

1
2

)1−1
q

×

{(
x−a
η(b,a)

)2(1−1q ) ((x−a)2(3η(b,a)−2x+2a)
6η3(b,a)

|f′(a)|q +
1

3

(
x−a
η(b,a)

)3
|f′(b)|q

)1
q

+
(
a+η(b,a)−x

η(b,a)

)2(1−1q ) (1
3

(
a+η(b,a)−x

η(b,a)

)3
|f′(a)|q

+

(
1

6
+

(x−a)2(2x−3η(b,a)−2a)
6η3(b,a)

)
|f′(b)|q

)1
q

}
holds for each x ∈ [a,a + η(b,a)].

Kirmaci [7] established the following midpoint inequalities for differentiable convex functions

Theorem 1.4. [7, Theorem 2.2] Let f : I◦ ⊂ R → R be a differentiable mapping on I◦, a,b ∈ I◦ (I◦
is the interior of I) with a < b. If |f′| is convex on [a,b], then we have∣∣∣∣∣∣ 1b−a

b∫
a

f(x)dx−f
(
a+b

2

)∣∣∣∣∣∣ ≤ b−a8 (|f′(a)| + |f′(b)|) .
Theorem 1.5. [7, Theorem 2.3] Let f : I◦ ⊂ R → R be a differentiable mapping on I◦, a,b ∈ I◦ (I◦

is the interior of I) with a < b and let p > 1. If |f′|
p
p−1 is convex on [a,b], then we have∣∣∣∣∣∣ 1b−a

b∫
a

f(x)dx−f
(
a+b

2

)∣∣∣∣∣∣ ≤ b−a16
(

4
p+1

)1
p

((
3 |f′(a)|

p
p−1 + |f′(b)|

p
p−1
)p−1

p

+
(
|f′(a)|

p
p−1 + 3 |f′(b)|

p
p−1
)p−1

p

)
.

Wang et al. [18] established the following midpoint inequalities for functions whose the power of the
absolute value of the first derivatives are preinvex

Theorem 1.6. [18, Theorem 3.1] Let A ⊆ R be an open invex subset with respect to η : A×A → R
and let f : A → R be a differentiable function. If |f′|q is preinvex on A for q ≥ 1, then for every
a,b ∈ A with η (b,a) 6= 0 we have∣∣∣∣∣∣∣ 1η(b,a)

a+η(b,a)∫
a

f(u)du−f
(

2a+η(b,a)
2

)∣∣∣∣∣∣∣
≤ |η(b,a)|

8

((
|f′(a)|q+2|f′(b)|q

3

)1
q

+

(
2|f′(a)|q+|f′(b)|q

3

)1
q

)
.

Theorem 1.7. [18, Corollary 3.2] Let A ⊆ R be an open invex subset with respect to η : A×A → R
and let f : A → R be a differentiable function. If |f′| is preinvex on A, then for every a,b ∈ A with
η (b,a) 6= 0 we have∣∣∣∣∣∣∣ 1η(b,a)

a+η(b,a)∫
a

f(u)du−f
(

2a+η(b,a)
2

)∣∣∣∣∣∣∣ ≤
|η(b,a)|

8
(|f′(a)| + |f′(b)|) .

Motivated by these results, in this paper we establish an fractional identity, and then using this
equality we derive some Ostrowski’s inequalities for functions whose first derivatives in absolute value
are s-preinvex in the second sense.



148 B. MEFTAH

2. Preliminaries

In this section we recall some concepts of convexity that are well known in the literature. Throughout
this section I is an interval of R.

Definition 2.1. [14] A function f : I → R is said to be convex, if
f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f(y)

holds for all x,y ∈ I and all t ∈ [0, 1].
Definition 2.2. [2] A nonnegative function f : I ⊂ [0,∞) → R is said to be s-convex in the second
sense for some fixed s ∈ (0, 1], if the following inequality

f(tx + (1 − t)y) ≤ tsf(x) + (1 − t)sf(y)
holds for all x,y ∈ I and t ∈ [0, 1].

Let K be a subset in Rn and let f : K → R and η : K ×K → Rn be continuous functions.
Definition 2.3. [20] A set K is said to be invex at x with respect to η, if

x + tη (y,x) ∈ K
holds for all x,y ∈ K and t ∈ [0, 1].
K is said to be an invex set with respect to η if K is invex at each x ∈ K.

Definition 2.4. [20] A function f on the invex set K is said to be preinvex with respect to η, if

f (x + tη (y,x)) ≤ (1 − t) f (x) + tf(y)
holds for all x,y ∈ K and t ∈ [0, 1].
Definition 2.5. [19] A nonnegative function f on the invex set K ⊆ [0,∞) is said to be s-preinvex
in the second sense with respect to η, for some fixed s ∈ (0, 1], if

f (x + tη (y,x)) ≤ (1 − t)sf (x) + tsf(y)
holds for all x,y ∈ K and t ∈ [0, 1].
Definition 2.6. [6] Let f ∈ L1[a,b]. The Riemann-Liouville integrals Jαa+f and J

α
b−
f of order α > 0

with a ≥ 0 are defined by

Jαa+f(x) =
1

Γ (α)

x∫
a

(x− t)α−1 f(t)dt, x > a

Jαb−f(x) =
1

Γ (α)

b∫
x

(t−x)α−1 f(t)dt, b > x

respectively, where Γ(α) =
∞∫
0

e−ttα−1dt, is the Gamma function and J0
a+
f(x) = J0

b−
f(x) = f(x).

3. Main Results

In what follows η : K ×K → R, and K ⊂ R an invex subset with respect to η, and a,b ∈ K◦ the
interior of K such that [a,a + η(b,a)] ⊂ K. At first, we prove the following lemma.
Lemma 3.1. Let f : [a,a + η(b,a)] → R be a differentiable function with a < a + η(b,a). If f′ ∈ L
([a,a + η(b,a)]), then the following equality for fractional integrals((

x−a
η(b,a)

)α
+
(

1 − x−a
η(b,a)

)α)
f(x) − Γ(α+1)

(η(b,a))α
(Jαx−f(a) + J

α
x+f(a + η (b,a)))

= η (b,a)




x−a
η(b,a)∫

0

tαf′(a + tη (b,a))dt−
1∫
x−a
η(b,a)

(1 − t)α f′(a + tη (b,a))dt


 (3.1)

holds for all x ∈ [a,a + η(b,a)].



FRACTIONAL OSTROWSKI TYPE INEQUALITIES 149

Proof. Integrating by parts right hand side of (3.1), we get

η (b,a)




x−a
η(b,a)∫

0

tαf′(a + tη (b,a))dt−
1∫
x−a
η(b,a)

(1 − t)α f′(a + tη (b,a))dt




=


( x−aη(b,a))α f(x) −α

x−a
η(b,a)∫

0

tα−1f(a + tη (b,a))dt

+
(

1 − x−a
η(b,a)

)α
f(x) −α

1∫
x−a
η(b,a)

(1 − t)α−1 f(a + tη (b,a))dt




=
((

x−a
η(b,a)

)α
+
(

1 − x−a
η(b,a)

)α)
f(x) −α




x−a
η(b,a)∫

0

tα−1f(a + tη (b,a))dt

+

1∫
x−a
η(b,a)

(1 − t)α−1 f(a + tη (b,a))dt


 . (3.2)

Using the change of variable u = a + tη (b,a), (3.2) becomes

η (b,a)




x−a
η(b,a)∫

0

tαf′(a + tη (b,a))dt−
1∫
x−a
η(b,a)

(1 − t)α f′(a + tη (b,a))dt




=
((

x−a
η(b,a)

)α
+
(

1 − x−a
η(b,a)

)α)
f(x)

− α
(η(b,a))α


 x∫

a

(u−a)α−1 f(u)du +
a+η(b,a)∫

x

(η (b,a) + a−u)α−1 f(u)du




=
((

x−a
η(b,a)

)α
+
(

1 − x−a
η(b,a)

)α)
f(x) − Γ(α+1)

(η(b,a))α
(Jαx−f(a) + J

α
x+f(a + η (b,a))) ,

which is the desired result. �

Theorem 3.1. Let f : [a,a + η (b,a)] → R be a differentiable function such that η(b,a) > 0 and
f′ ∈ L ([a,a + η (b,a)]). If |f′| is s-preinvex in the second sense for some fixed s ∈ (0, 1], then the
following inequality for fractional integrals∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣

≤ η (b,a)
((

B x−a
η(b,a)

(α + 1,s + 1) + 1
α+s+1

(
1 − x−a

η(b,a)

)α+s+1)
|f′(a)|

+

(
1

α+s+1

(
x−a
η(b,a)

)α+s+1
+ B (s + 1,α + 1) − B x−a

η(b,a)
(s + 1,α + 1)

)
|f′(b)|

)
(3.3)

holds for all x ∈ [a,a + η(b,a)].



150 B. MEFTAH

Proof. From Lemma 3.1, and properties of modulus, we have∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η (b,a)




x−a
η(b,a)∫

0

tα |f′(a + tη (b,a))|dt +
1∫
x−a
η(b,a)

(1 − t)α |f′(a + tη (b,a))|dt


 .

Using the s-preinvexity of |f′|, we obtain∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η (b,a)




x−a
η(b,a)∫

0

(
tα (1 − t)s |f′(a)| + tα+s |f′(b)|

)
dt

+

1∫
x−a
η(b,a)

(
(1 − t)α+s |f′(a)| + (1 − t)α ts |f′(b)|

)
dt




= η (b,a)






x−a
η(b,a)∫

0

tα (1 − t)s dt +
1∫
x−a
η(b,a)

(1 − t)α+s dt


 |f′(a)|

+




x−a
η(b,a)∫

0

tα+sdt +

1∫
x−a
η(b,a)

ts (1 − t)α dt


 |f′(b)|




= η (b,a)

((
B x−a
η(b,a)

(α + 1,s + 1) + 1
α+s+1

(
1 − x−a

η(b,a)

)α+s+1)
|f′(a)|

+

(
1

α+s+1

(
x−a
η(b,a)

)α+s+1
+ B (s + 1,α + 1) −B x−a

η(b,a)
(s + 1,α + 1)

)
|f′(b)|

)
,

(3.4)

where we use the facts that
x−a
η(b,a)∫

0

tα (1 − t)s dt = B x−a
η(b,a)

(α + 1,s + 1)

1∫
x−a
η(b,a)

(1 − t)α+s dt = 1
α+s+1

(
1 − x−a

η(b,a)

)α+s+1
x−a
η(b,a)∫

0

tα+sdt = 1
α+s+1

(
x−a
η(b,a)

)α+s+1
1∫
x−a
η(b,a)

ts (1 − t)α dt = B (s + 1,α + 1) −B x−a
η(b,a)

(s + 1,α + 1) . (3.5)

The proof is completed. �

Remark 3.1. In Theorem 3.1, if we put α = s = 1, we obtain Theorem 2.2 from [5].



FRACTIONAL OSTROWSKI TYPE INEQUALITIES 151

Corollary 3.1. In Theorem 3.1, if we choose x =
2a+η(b,a)

2
, then the following midpoint inequality

holds for fractional integrals

∣∣∣∣f( 2a+η(b,a)2 ) − 2α−1Γ(α+1)(η(b,a))α
(
Jα2a+η(b,a)

2

−f(a) + J
α
2a+η(b,a)

2

+f(a + η (b,a))

)∣∣∣∣
≤ η (b,a)

((
B1

2
(α + 1,s + 1) + 1

(α+s+1)2α+s+1

)
|f′(a)|

+
(

1
(α+s+1)2α+s+1

+ B (s + 1,α + 1) −B1
2

(s + 1,α + 1)
)
|f′(b)|

)
Remark 3.2. In Corollary 3.1, if we put α = s = 1, we obtain Corollary 3.2 from [18]. Moreover if
we take η (b,a) = b−a, we obtain Theorem 2.2 from [7].

Theorem 3.2. Let f : [a,a + η (b,a)] → R be a differentiable function such that η(b,a) > 0 and
f′ ∈ L ([a,a + η (b,a)]) and let q > 1 with 1

p
+ 1

q
= 1. If |f′|q is s-preinvex in the second sense for

some fixed s ∈ (0, 1], then the following inequality for fractional integrals

∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η(b,a)

(s+1)
1
q (αp+1)

1
p

((
x−a
η(b,a)

)α+ 1
p

((
x−a
η(b,a)

)s+1
|f′(b)|q

+

(
1 −

(
1 − x−a

η(b,a)

)s+1)
|f′(a)|q

)1
q

+
(

1 − x−a
η(b,a)

)α+ 1
p

×
((

1 − x−a
η(b,a)

)s+1
|f′(a)|q +

(
1 −

(
x−a
η(b,a)

)s+1)
|f′(b)|q

)1
q

)
(3.6)

holds for all x ∈ [a,a + η(b,a)].

Proof. From Lemma 3.1, properties of modulus, and Hölder’s inequality, we have

∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η (b,a)






x−a
η(b,a)∫

0

tαpdt




1
p



x−a
η(b,a)∫

0

|f′(a + tη (b,a))|q dt




1
q

+




1∫
x−a
η(b,a)

(1 − t)αp dt




1
p



1∫
x−a
η(b,a)

|f′(a + tη (b,a))|q dt




1
q




=
η(b,a)

(αp+1)
1
p



(
x−a
η(b,a)

)α+ 1
p




x−a
η(b,a)∫

0

|f′(a + tη (b,a))|q dt




1
q

+
(

1 − x−a
η(b,a)

)α+ 1
p




1∫
x−a
η(b,a)

|f′(a + tη (b,a))|q dt




1
q


 . (3.7)



152 B. MEFTAH

Since |f′|q is s-preinvex, we deduce∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η(b,a)

(αp+1)
1
p



(
x−a
η(b,a)

)α+ 1
p




x−a
η(b,a)∫

0

(1 − t)s |f′(a)|q + ts |f′(b)|q dt




1
q

+
(

1 − x−a
η(b,a)

)α+ 1
p




1∫
x−a
η(b,a)

(1 − t)s |f′(a)|q + ts |f′(b)|q dt




1
q




=
η(b,a)

(s+1)
1
q (αp+1)

1
p

((
x−a
η(b,a)

)α+ 1
p

((
x−a
η(b,a)

)s+1
|f′(b)|q

+

(
1 −

(
1 − x−a

η(b,a)

)s+1)
|f′(a)|q

)1
q

+
(

1 − x−a
η(b,a)

)α+ 1
p

×
((

1 − x−a
η(b,a)

)s+1
|f′(a)|q +

(
1 −

(
x−a
η(b,a)

)s+1)
|f′(b)|q

)1
q

)
,

which completes the proof. �

Corollary 3.2. In Theorem 3.2, if we choose x =
2a+η(b,a)

2
, then the following midpoint inequality

holds for fractional integrals∣∣∣∣f( 2a+η(b,a)2 ) − 2α−1Γ(α+1)(η(b,a))α
(
Jα2a+η(b,a)

2

−f(a) + J
α
2a+η(b,a)

2

+f(a + η (b,a))

)∣∣∣∣
≤ η(b,a)

2
α+ 1

p (s+1)
1
q (αp+1)

1
p

((
|f′(b)|q+(2s+1−1)|f′(a)|q

2s+1

)1
q

+

(
|f′(a)|q+(2s+1−1)|f′(b)|q

2s+1

)1
q

)
.

Remark 3.3. In Corollary 3.2, if we choose α = s = 1, and η (b,a) = b−a, we obtain Theorem 2.3
from [7].

Theorem 3.3. Let f : [a,a + η (b,a)] → R be a differentiable function such that η(b,a) > 0 and
f′ ∈ L ([a,a + η (b,a)]) and let q > 1. If |f′|q s-preinvex in the second sense for some fixed s ∈ (0, 1],
then the following inequality for fractional integrals∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣

≤ η(b,a)
(α+1)

1−
1
q

(((
x−a
η(b,a)

)(α+1)(1−1
q

))(
B x−a
η(b,a)

(α + 1,s + 1) |f′(a)|q

+ 1
α+s+1

(
x−a
η(b,a)

)α+s+1
|f′(b)|q

)1
q

+
(

1 − x−a
η(b,a)

)(α+1)(1−1
q

)

×
(

1
α+s+1

(
1 − x−a

η(b,a)

)α+s+1
|f′(a)|q ?

+
(
B (s + 1,α + 1) −B x−a

η(b,a)
(s + 1,α + 1)

)
|f′(b)|q

)1
q

)
(3.8)

holds for all x ∈ [a,a + η(b,a)].



FRACTIONAL OSTROWSKI TYPE INEQUALITIES 153

Proof. From Lemma 3.1, properties of modulus, and power mean inequality, we have∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η (b,a)






x−a
η(b,a)∫

0

tαdt




1−
1

q



x−a
η(b,a)∫

0

tα |f′(a + tη (b,a))|q dt




1
q

+




1∫
x−a
η(b,a)

(1 − t)α dt




1−
1

q



1∫
x−a
η(b,a)

(1 − t)α |f′(a + tη (b,a))|q dt




1
q




=
η(b,a)

(α+1)
1−

1
q



(
x−a
η(b,a)

)(α+1)(1−1
q

) 
x−a
η(b,a)∫

0

tα |f′(a + tη (b,a))|q dt




1
q

+
(

1 − x−a
η(b,a)

)(α+1)(1−1
q

) 
1∫
x−a
η(b,a)

(1 − t)α |f′(a + tη (b,a))|q dt




1
q


 .

(3.9)

Since |f′|q is s-preinvex, we deduce∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − Γ(α+1)(η(b,a))α (Jαx−f(a) + Jαx+f(a + η (b,a)))∣∣∣
≤ η(b,a)

(α+1)
1−

1
q

(((
x−a
η(b,a)

)(α+1)(1−1
q

))

×


|f′(a)|q

x−a
η(b,a)∫

0

tα(1 − t)sdt + |f′(b)|q
x−a
η(b,a)∫

0

tα+sdt




1
q

+
(

1 − x−a
η(b,a)

)(α+1)(1−1
q

)

×


|f′(a)|q

1∫
x−a
η(b,a)

(1 − t)α+sdt + |f′(b)|q
1∫
x−a
η(b,a)

ts (1 − t)α dt




1
q


 . (3.10)

Substituting (3.5) into (3.10), we obtain the desired result. �

Remark 3.4. In Theorem 3.3, If we take α = s = 1, we obtain Theorem 2.8 from [5].

Corollary 3.3. In Theorem 3.3, if we choose x =
2a+η(b,a)

2
, then the following midpoint inequality

holds for fractional integrals∣∣∣∣∣ 12α−1 f( 2a+η(b,a)2 ) − 2α−1Γ(α+1)(η(b,a))α
(
Jα

2a+η(b,a)
2

−f(a) + J
α
2a+η(b,a)

2

+f(a + η (b,a))

)∣∣∣∣∣
≤ η(b,a)

2
(α+1)(1−

1
q )(α+1)1−

1
q

((
B1

2
(α + 1,s + 1) |f′(a)|q + 1

(α+s+1)2α+s+1
|f′(b)|q

)1
q

+
(

1
(α+s+1)2α+s+1

|f′(a)|q +
(
B (s + 1,α + 1) −B1

2
(s + 1,α + 1)

)
|f′(b)|q

)1
q

)
.

Remark 3.5. In Corollary 3.3, if we put α = s = 1, we obtain Theorem 3.1 from [18].



154 B. MEFTAH

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Laboratoire des télécommunications, Faculté des Sciences et de la Technologie, University of 8 May

1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria.

∗Corresponding author: badrimeftah@yahoo.fr


	1. Introduction
	2. Preliminaries
	3. Main Results
	References