EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 419-439
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Weighted Opial–type inequalities for fractional integral
and differential operators involving generalized

Mittag–Leffler functions

Zivord Tomovski1,2 Josip Pečarić3, Ghulam Farid4,∗

1 Department of Mathematics, University of Rijeka, Croatia
2 Institute of Mathematics, Faculty of Natural Sciences and Mathematics,St. Cyril and
Methodius University, Skopje, Republic of Macedonia
3 Faculty of Textile Technology, University of Zagreb, Croatia
4 Department of Mathematics, COMSATS Institute of Information Technology,
Attock Campus, Pakistan

Abstract. In this paper, by using Hölder integral inequality we give generalization of wighted
Opial–type inequalities by using generalized fractional integral and differential operators involving
generalized Mittag–Leffler functions.

2010 Mathematics Subject Classifications: 26A33, 26D15, 33E12

Key Words and Phrases: Opial–type inequality, fractional integral, fractional derivative, Mittag–
Leffler function

1. Introduction and preliminaries

In 1960 Opial established the following integral inequality [17].

Let x(t) ∈ C(1)[0, h] be such that x(0) = x(h) = 0, and x(t) > 0 in (0, h).
Then ∫ h

0
|x(t)x′(t)|dt ≤ h

4

∫ h

0

(
x′(t)

)2
dt, (1)

where constant h
4 is the best possible.

Opial’s inequality [3, 4, 5, 6, 7, 14] is studied extensively by many researchers. It
recognizes as a fundamental result in the theory of differential and difference equations
and other areas of mathematics, and has attracted a great deal of attention in the literature

∗Corresponding author.

Email addresses: zivorad.tomovski@math.uniri.hr, zivoradt@yahoo.com (Z. Tomovski),
pecaric@element.hr (J. Pečarić), faridphdsms@hotmail.com, ghlmfarid@ciit-attock.edu.pk (G.
Farid)

http://www.ejpam.com 419 c© 2017 EJPAM All rights reserved.



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 420

(see, for instance, [1, 2]). In [3, 4, 5, 6, 7] Opial–type integral inequalities were considered
for different kinds of fractional derivative and fractional integral operators for example
Riemann-Liouville, Caputo, Canvati etc were established. Our paper is motivated by the
work of Koliha and Pecaric [14] on Opial inequalities for fractional differential operators
and presents a class of very general weighted Opial type inequalities using integral and
differential operators in fractional calculus involving generalized Mittag-Leffler functions.

The following hypotheses are assumed throughout this section: Let I be a closed
interval in R, a is a fixed point in I, let Φ be a continuous function nonnegative on I × I,
and let y, h ∈ C (I) . We assume that the following condition involving Φ, h and y is
satisfied:

|y (x)| ≤

∣∣∣∣∣∣
x∫
a

Φ (x, t) |h (t)| dt

∣∣∣∣∣∣ , x ∈ I. (2)

Koliha and Pecaric in [14] proved the following weighted Opial type inequalities by
application of Hölder integral inequality.

Theorem 1. Assume that (2) holds. Let x ∈ I, let α, β > 0, r > max (1, α) , and let
U, V ∈ C (I) be such that U (s) ≥ 0 and V (s) > 0 for all s ∈ I. Then∣∣∣∣∣∣

x∫
a

U (s) |y (s)|β |h (s)|α ds

∣∣∣∣∣∣ ≤ C (x)

∣∣∣∣∣∣
x∫
a

V (s) |h (s)|r ds

∣∣∣∣∣∣
(α+β)/r

(3)

where

C (x) =

(
α

α+ β

)α/r x∫
a

(
U r (s)V −α (s)

)1/(r−α)
P (s)β(r−1)/(r−α) ds

(r−α)/r

, (4)

P (s) =

∣∣∣∣∣∣
s∫
a

V (t)−1/(r−1) Φ (s, t)r/(r−1) dt

∣∣∣∣∣∣ . (5)

Theorem 2. Assume that (2) holds. Let x ∈ I, α, β > 0, r > max (1, α) , and let U, V ∈
C (I) be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I.Then

∣∣∣∣∣∣
x∫
a

U (s) |y (s)|β |h (s)|α ds

∣∣∣∣∣∣ ≤
x∫
a

U (ω)

∣∣∣∣∣∣
ω∫
a

V (t) Φ (ω, t) dt

∣∣∣∣∣∣
r−α
r

dω ‖V ‖β∞ ‖h‖
α+β
∞ . (6)

If the exponents α, β and r in Theorem 2 are not necessarily positive, in this case the
inequality (2) must be strengthened to equality



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 421

|y (s)| =

∣∣∣∣∣∣
s∫
a

Φ (s, t) |h (t)| dt

∣∣∣∣∣∣ , s ∈ I, (7)

where Φ is a nonnegative continuous function on I × I, and y, h ∈ C (I) .

Theorem 3. Assume that (7) holds. Let x ∈ I, U, V ∈ C (I) be such that U (s) ≥ 0 and
V (s) > 0 for all s ∈ I. Consider real numbers α, β, r and the following relations:

(i) r > 1, β > 0, 0 < α < r;
(ii) r < α < 0, β < 0;
(iii) −α < β < 0, 0 < α < r < 1;
(iv) β > 0, 0 < r < min (α, 1) ;
(v) α < 0 < r < 1, 0 < β < −α;
(vi) β < 0, α < 0, r > 1;
(vii) 1 < r < α, −α < β < 0;
(viii) β > 0, r < 0 < α;
(ix) α < r < 0, 0 < β < −α.
If one of the conditions (i)-(iii) is satisfied, then∣∣∣∣∣∣

x∫
a

U (s) |y (s)|β |h (s)|α ds

∣∣∣∣∣∣ ≤ C (x)

∣∣∣∣∣∣
x∫
a

V (s) |h (s)|r ds

∣∣∣∣∣∣
(α+β)/r

. (8)

If one of the conditions (iv)-(ix) is satisfied, then

∣∣∣∣∣∣
x∫
a

U (s) |y (s)|β |h (s)|α ds

∣∣∣∣∣∣ ≥ C (x)

∣∣∣∣∣∣
x∫
a

V (s) |h (s)|r ds

∣∣∣∣∣∣
(α+β)/r

, (9)

where C(x) is defined by (4) and (5).

2. Fractional differential and integral operators involving Mittag-Leffler
functions

Fractional calculus refers to integration and differentiation of fractional order. Several
mathematicians contributed to this subject over the years. People like Liouville, Riemann,
and Weyl made major contributions to the theory of fractional calculus. The story on
the fractional calculus continued with contributions from Fourier, Abel, Lacroix, Leibniz,
Grunwald and Letnikov. For a historical survey the reader may see [12, 15, 16].

Fractional integral inequalities are useful in establishing the uniqueness of solutions for
certain fractional partial differential equations. They also provide upper and lower bounds
for the solutions of fractional boundary value problems. These considerations have led
various researchers in the field of integral inequalities to explore certain extensions and
generalizations by involving fractional calculus operators (see, [9, 11, 18, 20, 21] ).



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 422

Let x > 0. By L1 (0, x) we denote the space of all Lebesgue integrable functions on the
interval (0, x) . For any f ∈ L1 (0, x) the Riemann-Liouvill fractional integral of f of order
ν is defined by(

Iνa+f
)

(s) =
1

Γ(ν)

∫ s

a
(x− t)ν−1f(t)dt = (f ∗Kν) (s) , s ∈ [0, x] , ν > 0, (10)

where Kν (s) = sν−1

Γ(ν) . The integral on the right side of (10) exists for almost s ∈ [0, x] and

Iνa+f ∈ L1 (0, x) .The Riemann-Liouville fractional derivative of f ∈ L1 (0, x) of order ν
is defined by (

Dν
a+f

)
(x) =

(
d

dx

)n (
In−νa+ f

)
(x) , (ν > 0, n = [ν] + 1) (11)

By Cm [0, x] we denote the space of all functions which have continuous derivatives up
to order m, and AC [0, x] is the space of all absolutely continuous functions on [0, x] . By
ACm [0, x] we denote the space of all functions f ∈ Cm [0, x] with f (m−1) ∈ AC [0, x] . By
L∞ (0, x) we denote the space of all measurable functions essentially bounden on [0, x] .
Let µ > 0, m = [µ] + 1, f ∈ ACm [a, b] . The Caputo derivative of order µ > 0 is defined as(

CDµ
a+f

)
(x) =

(
Im−µa+

dm

dxm
f

)
(x) (12)

=
1

Γ (m− µ)

x∫
a

(x− s)m−µ−1 dm

dsm
f (s) ds.

Definition 1. [21] Let f ∈ L1 [a, b] , f ∗K(1−ν)(1−µ) ∈ AC1 [a, b] . The fractional derivative
operator Dµ,ν

a+ of order 0 < µ < 1 and type 0 ≤ ν ≤ 1 with respect to x ∈ [a, b] is defined
by (

Dµ,ν
a+ f

)
(x) =

(
I
ν(1−µ)
a+

d

dx

(
I

(1−ν)(1−µ)
a+ f

))
(x) (13)

whenever the right hand side exists.

This generalization gives the classical Riemann-Liouville fractional differentiation op-
erator if ν = 0. For ν = 1 it gives the fractional differential operator introduced by
Caputo. We denote it by Dµ,1

a+f =C Dµ
a+f.

Several authors (see, [9, 20]) called (13) the Hilfer fractional derivative. Applications
of Dµ,ν

a+ are given in [9, 20, 21, 23].
The purpose of this paper is to give weighted Opial type integral inequalities involving

different kinds of fractional differential operators. For 0 < µ < 1 and 0 < ν ≤ 1, the Hilfer
fractional differentiation operator Dµ,ν

a+ can be rewritten in the form(
Dµ,ν
a+ f

)
(x) =

(
I
ν(1−µ)
a+

(
Dµ+ν−µν
a+ f

))
(x) (14)

=
1

Γ (ν (1− µ))

x∫
a+

(x− τ)ν(1−µ)−1
(
Dµ+ν−µν
a+ f

)
(τ) dτ.



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 423

Definition of this generalized fractional integral operator containing Mittag–Leffler func-
tion is as follows.

Definition 2. (Prabhakar [18]) Let µ, ν, γ be positive real numbers and ω ∈ R. Then the
generalized fractional integral operator εγµ,ν,ω,a+ for a real-valued continuous function f is
defined by:

(εγµ,ν,ω,a+f)(x) =

x∫
a+

(x− t)ν−1Eγµ,ν(ω(x− t)µ)f(t)dt, (15)

where the function Eγµ,ν is generalized Mittag–Leffler function defined as

Eγµ,ν(t) =

∞∑
n=0

(γ)n
n!Γ(µn+ ν)

tn, (16)

and (γ)n is the Pochhammer symbol: (γ)n = γ(γ + 1)...(γ + n− 1), (γ)0 = 1.

The integral operaor εγ
µ,ν,ω,a+

is bounded in the space C(I) with a finite norm ‖f‖C =

max
x∈I
|f (x)| , and there exists a positive constant M > 0, such that (see [11])

∥∥∥εγµ,ν,ω,a+f∥∥∥C ≤M ‖f‖C .

For ω = 0 in (15), integral operator εγ
µ,ν,ω,a+

would correspond essentially to the
Riemann-Liouville fractional integral operator Iνa+f .

Let eγµ,ν (t, ω) = tν−1Eγµ,ν (−ωtµ). In [22] Tomovski et al. proved the following uniform
estimate for the function eγµ,ν (ω, t) :

Lemma 1. If µ ∈ (0, 1) , γ, ω > 0, µγ > ν − 1 > 0, then the following uniform bound
holds true ∣∣eγµ,ν (t, ω)

∣∣ ≤ Γ
(
γ − ν−1

µ

)
Γ
(
ν−1
µ

)
πµω

ν−1
µ Γ (γ)

[
cos
(πµ

2

)]γ− ν−1
µ

, t > 0. (17)

Lemma 2. [22] If µ ∈ (0, 1) , γ, ω > 0, ν ≥ µγ, then eγµ,ν (t, ω) > 0, for all t > 0.

We define a variant of Sobolev space:

Wm,1 [a, b] =

{
f ∈ L1 [a, b] :

dm

dtm
f ∈ L1 [a, b]

}
. (18)

Definition 3. (Prabhakar derivative [9]) Let f ∈ L1 [0, b] , 0 < t < b ≤ ∞, µ, ν, γ > 0, and

f ∗ e−γµ,m−ν,ω ∈Wm,1 [0, b] , m = [ν]. Then the Prabhakar derivative is defined by following
relation (

Dγ
µ,ν,ω,0+f

)
(t) =

dm

dtm
ε−γµ,m−ν,ω,0+f (t) . (19)



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 424

Definition 4. (Caputo-Prabhakar derivative [9]) Let f ∈ L1 [0, b] , 0 < t < b ≤ ∞,

µ, ν, γ > 0 , m = [ν]. Then the Caputo-Prabhakar derivative for f ∈ ACm [0, b] is defined
by following relation (

CDγ
µ,ν,ω,0+f

)
(t) = ε−γµ,m−ν,ω,0+

dm

dtm
f (t) (20)

=
(
Dγ
µ,ν,ω,0+f

)
(t)−

m−1∑
k=0

tk−µE−γµ,k−ν+1 (ωtµ) f (k) (0+) .

Remark 1. Let µ, ν, γ > 0 and f ∈ ACm [0, b] , 0 < t < b ≤ ∞, then

(
CDγ

µ,ν,ω,0+f
)

(t) = Dγ
µ,ν,ω,0+

(
f (t)−

m−1∑
k=0

tk

k!
f (k) (0+)

)
. (21)

Moreover, if f (k) (0+) = 0, k = 0, 1, 2, ...m− 1, then(
CDγ

µ,ν,ω,0+f
)

(t) =
(
Dγ
µ,ν,ω,0+f

)
(t) .

Definition 5. (Hilfer-Prabhakar derivative [9]). Let µ ∈ (0, 1) , ν ∈ [0, 1] , and let f ∈
L1 [a, b] , 0 < t < b ≤ ∞, f ∗ e−γ(1−ν)

ρ, (1−ν)(1−µ), ω ∈ AC
1 [0, b] . The Hilfer-Prabhakar derivative

is defined by(
Dγ, µ, ν
ρ, ω, 0+f

)
(t) =

(
ε−γνρ, ν(1−µ), ω, 0+

d

dt

(
ε
−γ(1−ν)
ρ, (1−ν)(1−µ), ω, 0+f

))
(t) , (22)

where γ, ω ∈ R, ρ > 0, and ε0ρ, 0, ω, 0+f = f. Moreover,(
Dγ, µ
ρ, ω, 0+f

)
(t) =

(
Dγ, µ, 1
ρ, ω, 0+f

)
(t) =

(
ε−γρ, 1−µ, ω, 0+

d

dt
f

)
(t) . (23)

3. Main Results

Our first main result is given in the following theorem. Namely, we present Opial type
inequalities for Hilfer fractional operator (13).

Theorem 4. Let x > 0, α, β > 0, µ ∈ (0, 1) , ν ∈ (0, 1] and U, V ∈ C (I) be such that
U (s) ≥ 0, V (s) > 0 for all s ∈ I. Then let f ∈ L (0, x) have an integrable fractional
derivative Dµ+ν−µν

0+ f ∈ L∞ (0, x) .

(i) If r > max
{

1, α, (ν (1− µ))−1
}

, then

x∫
0

U (s)
∣∣(Dµ,ν

0+ f
)

(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≤ Ω (x)×



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 425 x∫
0

V (s)
∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣r ds


α+β
r

, (24)

where

Ω (x) =

(
α

α+ β

)α
r

 x∫
0

(
U r (s)V −α (s)

) 1
r−α (∆ (s))

β(r−1)
r−α ds

 r−α
r

, (25)

∆ (s) =

s∫
0

(V (t))−
1
r−1

[
1

Γ (ν (1− µ))
(s− t)ν(1−µ)−1

] r
r−1

dt. (26)

(ii) If 0 < r < min
{
α, 1, (ν (1− µ))−1

}
, then

x∫
0

U (s)
∣∣(Dµ,ν

0+ f
)

(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≥ Ω (x)×

 x∫
0

V (s)
∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣r ds


α+β
r

, (27)

where Ω (x) and ∆ (s) are given by (25) and (26).
Proof. According to (14),

(
Dµ,ν

0+ f
)

(s) =
1

Γ (ν (1− µ))

s∫
0

(s− τ)ν(1−µ)−1
(
Dµ+ν−µν

0+ f
)

(τ) dτ, s ∈ [0, x] . (28)

Setting

y (s) =
(
Dµ,ν

0+ f
)

(s) , h (s) =
(
Dµ+ν−µν

0+ f
)

(s) , Φ (s, t) =
(s− t)ν(1−µ)−1

Γ (ν (1− µ))
,

we observe that condition (2) is satisfied with a = 0 and I = [0, x] :

|y (s)| ≤
s∫

0

Φ (s, t) |h (t)| dt, 0 ≤ s ≤ x.

The rest of the proof of (i) is the same as Theorem 4.2 of [14].

For ν = 1, we obtain the following Opial type inequalities for Caputo fractional deriva-
tive, defined by (12).

Corollary 1. Let x > 0, α, β > 0, µ ∈ (0, 1) and U, V ∈ C (I) be such that U (s) ≥ 0,
V (s) > 0 for all s ∈ I and let f ∈ AC1 (0, x) .



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 426

(i) If r > max
{

1, α, (1− µ)−1
}
, then

x∫
0

U (s)
∣∣(CDµ

0+f
)

(s)
∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds ≤ Ω (x)

 x∫
0

V (s)

∣∣∣∣ ddsf (s)

∣∣∣∣r ds


α+β
r

, (29)

where

Ω (x) =

(
α

α+ β

)α
r

 x∫
0

(
U r (s)V −α (s)

) 1
r−α (∆ (s))

β(r−1)
r−α ds

 r−α
r

, (30)

∆ (s) =

s∫
0

(V (t))−
1
r−1

[
1

Γ (1− µ) (s− t)µ
] r
r−1

dt. (31)

(ii) If 0 < r < min
{
α, 1, (1− µ)−1

}
, then

x∫
0

U (s)
∣∣(CDµ

0+f
)

(s)
∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds ≥ Ω (x)

 x∫
0

V (s)

∣∣∣∣ ddsf (s)

∣∣∣∣r ds


α+β
r

, (32)

where Ω (x) and ∆ (s) are given by (30) and (31).

Corollary 2. Let x > 0, α, β > 0, µ ∈ (0, 1) , ν ∈ (0, 1] and let f ∈ L (0, x) have an
integrable fractional derivative Dµ+ν−µν

0+ f ∈ L∞ (0, x) .

(i) If r > max
{

1, α, (ν (1− µ))−1
}

, then

x∫
0

∣∣(Dµ,ν
0+ f

)
(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≤ Ω1x

βν(1−µ)−α+1
r

+1×

 x∫
0

∣∣∣(Dµ+ν−µν
0+ f

)
(s)
∣∣∣r ds


α+β
r

, (33)

where

Ω1 =

(
α

α+β

)α
r

(Γ (ν (1− µ)))−β
(

r−1
ν(1−µ)r−1

)β(r−1)
r

[
β[ν(1−µ)r−1]

r−α + 1
] r−α

r

. (34)

(ii) If 0 < r < min
{
α, 1, (ν (1− µ))−1

}
, then



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 427

x∫
0

∣∣(Dµ,ν
0+ f

)
(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≥ Ω1x

βν(1−µ)−α+1
r

+1×

 x∫
0

∣∣∣(Dµ+ν−µν
0+ f

)
(s)
∣∣∣r ds


α+β
r

, (35)

where Γ is the Euler Gamma function and Ω1 is given by (34).
Proof. By Theorem 4,

x∫
0

∣∣(Dµ,ν
0+ f

)
(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≤ Ω (x)

 x∫
0

∣∣∣(Dµ+ν−µν
0+ f

)
(s)
∣∣∣r ds


α+β
r

, (36)

where

Ω (x) =

(
α

α+ β

)α
r

 x∫
0

 s∫
0

[
(s− t)ν(1−µ)−1

Γ (ν (1− µ))

] r
r−1

dt


β(r−1)
r−α

ds


r−α
r

=

(
α

α+ β

)α
r

(Γ (ν (1− µ)))−β
(

r − 1

ν (1− µ) r − 1

)β(r−1)
r

 x∫
0

s
β[ν(1−µ)r−1]

r−α
ds

 r−α
r

=

(
α

α+β

)α
r

(Γ (ν (1− µ)))−β
(

r−1
ν(1−µ)r−1

)β(r−1)
r

[
β[ν(1−µ)r−1]

r−α + 1
] r−α

r

xβν(1−µ)−α+1
r

+1.

Corollary 3. Let x > 0, α, β > 0, p > q > 0, µ ∈ (0, 1) , ν ∈ (0, 1]. Then let f ∈ L (0, x)
have an integrable fractional derivative Dµ+ν−µν

0+ f ∈ L∞ (0, x) .

(i) If r > max
{

1, α, 1 + q, (ν (1− µ))−1
}

, then

x∫
0

sp
∣∣(Dµ,ν

0+ f
)

(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≤ Ω2x

βν(1−µ)+p− (β+α)(q+1)
r

+1 (37)

×

 x∫
0

sq
∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣r ds


α+β
r

,

where

Ω2 =

(
α

α+β

)α
r

(Γ (ν (1− µ)))−β
(
B
(
r−1−q
r−1 , ν(1−µ)r−1

r−1

))β(r−1)
r

[
β[ν(1−µ)r−1−q]+pr−qα

r−α + 1
] r−α

r

. (38)



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 428

(ii) If 0 < r < min
{
α, 1, 1 + q, (ν (1− µ))−1

}
, then

x∫
0

sp
∣∣(Dµ,ν

0+ f
)

(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≥ Ω2x

βν(1−µ)+p− (α+β)(q+1)
r

+1 (39)

×

 x∫
0

sq
∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣r ds


α+β
r

,

where Γ and B are the Euler Gamma and Beta functions and Ω2 is given by (38).
Proof. By Theorem 4,

x∫
0

sp
∣∣(Dµ,ν

0+ f
)

(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds ≤ Ω (x)

 x∫
0

sq
∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣r ds


α+β
r

,

(40)
where

Ω (x) =

(
α

α+ β

)α
r

×

 x∫
0

s
pr−qα
r−α

 s∫
0

(tq)−
1
r−1

[
(s− t)ν(1−µ)−1

Γ (ν (1− µ))

] r
r−1

dt


β(r−1)
r−α

ds


r−α
r

=

(
α

α+ β

)α
r

(Γ (ν (1− µ)))−β

×

 x∫
0

s
pr−qα
r−α

 s∫
0

t
r−1−q
r−1

−1 (s− t)
ν(1−µ)r−1

r−1
−1 dt


β(r−1)
r−α

ds


r−α
r

=

(
α

α+ β

)α
r

(Γ (ν (1− µ)))−β
(
B

(
r − 1− q
r − 1

,
ν (1− µ) r − 1

r − 1

))β(r−1)
r

×

 x∫
0

s
β[ν(1−µ)r−1−q]+pr−qα

r−α
ds

 r−α
r

=

(
α

α+β

)α
r

(Γ (ν (1− µ)))−β
(
B
(
r−1−q
r−1 , ν(1−µ)r−1

r−1

))β(r−1)
r

[
β[ν(1−µ)r−1−q]+pr−qα

r−α + 1
] r−α

r

×xβν(1−µ)+p− (β+α)(q+1)
r

+1.



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 429

Corollary 4. Let x > 0, α, β > 0, µ ∈ (0, 1) and let f ∈ AC1 (0, x) .

(i) If r > max
{

1, α, (1− µ)−1
}
, then

x∫
0

∣∣(CDµ
0+f

)
(s)
∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds ≤ Ω3x
β(1−µ)−α+1

r
+1

 x∫
0

∣∣∣∣ ddsf (s)

∣∣∣∣r ds


α+β
r

, (41)

where

Ω3 =

(
α

α+β

)α
r

(Γ (1− µ))−β
(

r−1
(1−µ)r−1

)β(r−1)
r

[
β[(1−µ)r−1]

r−α + 1
] r−α

r

. (42)

(ii) If 0 < r < min
{
α, 1, (1− µ)−1

}
, then

x∫
0

∣∣(CDµ
0+f

)
(s)
∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds ≥ Ω3x
β(1−µ)−α+1

r
+1

 x∫
0

∣∣∣∣ ddsf (s)

∣∣∣∣r ds


α+β
r

, (43)

where Γ is the Euler Gamma function and Ω3 is given by (42).

Example 1. If we put α = 1, β = 1, r = 2, µ, ν ∈ (0, 1) , 2ν (1− µ) ≥ 1, γ, ω > 0, ν ≥ µγ,
U (s) = 1, V (s) = 1

eγµ,ν(s,ω)
in Theorem 4, by using the integral formula (see [18])

x∫
0

(x− t)ν−1Eγµ,ν (ω (x− t)µ) tδ−1dt = Γ (δ)xν+δ−1Eγµ,ν+δ (ωxµ)

we obtain

∆ (s) =

s∫
0

eγµ,ν (t, ω)

[
1

Γ (ν (1− µ))
(s− t)ν(1−µ)−1

]2

dt

=
1

Γ2 (ν (1− µ))

s∫
0

tν−1Eγµ,ν (−ωtµ) (s− t)2[ν(1−µ)−1] dt

=
1

Γ2 (ν (1− µ))

s∫
0

(s− t)ν−1Eγµ,ν (−ω (s− t)µ) t2[ν(1−µ)−1]dt

=
Γ (2ν (1− µ)− 1)

Γ2 (ν (1− µ))
sν+2[ν(1−µ)−1]Eγµ,ν+2ν(1−µ)−1 (−ωsµ)

=
1

Γ (2ν (1− µ) + 1)
eγµ,ν+2ν(1−µ)−1 (s, ω) .



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 430

Hence,

Ω (x) =

√
2

2Γ (2ν (1− µ) + 1)

 x∫
0

eγµ,ν (s, ω) eγµ,ν+2ν(1−µ)−1 (s, ω) ds

1/2

. (44)

By Lemma 2.2, we obtain that eγµ,ν (s, ω) eγµ,ν+2ν(1−µ)−1 (s, ω) > 0, for all s ∈ (0, x] , i.e.

Ω (x) > 0, for all x > 0. By Theorem 4, we obtain,

x∫
0

∣∣(Dµ,ν
0+ f

)
(s)
∣∣ ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣ ds ≤ Ω (x)

 x∫
0

∣∣∣(Dµ+ν−µν
0+ f

)
(s)
∣∣∣2

eγµ,ν (s, ω)
ds

 , (45)

where Ω (x) is given by (44).

Theorem 5. Let x > 0, x ∈ I, α, β > 0, r > max (1, α) , µ ∈ (0, 1) , ν ∈ (0, 1] and let
U, V ∈ C (I) be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I. If f ∈ L (0, x) have an
integrable fractional derivative Dµ+ν−µν

0+ f ∈ L∞ (0, x) , then∣∣∣∣∣∣
x∫

0

U (s)
∣∣(Dµ,ν

0+ f
)

(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds

∣∣∣∣∣∣
≤
(

1

Γ (ν (1− µ))

) r−α
r

x∫
0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) (λ− t)µ+ν−µν dt

∣∣∣∣∣∣
r−α
r

dλ×

‖V ‖β∞
∥∥∥(Dµ+ν−µν

0+ f
)∥∥∥α+β

∞
.

If we take U = V = 1 in Theorem 5, since

x∫
0

∣∣∣∣∣∣
λ∫

0

(λ− t)µ+ν−µν dt

∣∣∣∣∣∣
r−α
r

dλ

=
1

(µ+ ν − µν + 1)
r−α
r

x∫
0

λ(µ+ν−µν+1) r−α
r dλ

=
x(µ+ν−µν+1) r−α

r
+1

(µ+ ν − µν + 1)
r−α
r

[
(µ+ ν − µν + 1)

r−α
r + 1

] ,
we get the following special inequality of Theorem 5.

Corollary 5. Let x > 0, α, β > 0, r > max (1, α) , µ ∈ (0, 1) , ν ∈ (0, 1]. If f ∈ L (0, x)
have an integrable fractional derivative Dµ+ν−µν

0+ f ∈ L∞ (0, x) , then



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 431

∣∣∣∣∣∣
x∫

0

∣∣(Dµ,ν
0+ f

)
(s)
∣∣β ∣∣∣(Dµ+ν−µν

0+ f
)

(s)
∣∣∣α ds

∣∣∣∣∣∣ ≤
(

1

Γ (ν (1− µ)) (µ+ ν − µν + 1)

) r−α
r

(46)

× x(µ+ν−µν+1) r−α
r

+1[
(µ+ ν − µν + 1)

r−α
r + 1

] ∥∥∥(Dµ+ν−µν
0+ f

)∥∥∥α+β

∞
.

Corollary 6. Let x > 0, x ∈ I, α, β > 0, r > max (1, α) , µ ∈ (0, 1) and let U, V ∈ C (I)
be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I. If f ∈ L (0, x) have an integrable fractional
derivative f ∈ AC1 (0, x) , then∣∣∣∣∣∣

x∫
0

U (s)
∣∣(CDµ

0+f
)

(s)
∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds
∣∣∣∣∣∣ (47)

≤
(

1

Γ (1− µ)

) r−α
r

x∫
0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) (λ− t) dt

∣∣∣∣∣∣
r−α
r

dλ ‖V ‖β∞

∥∥∥∥( d

ds
f (s)

)∥∥∥∥α+β

∞
.

Moreover for U (s) = V (s) = 1, there holds∣∣∣∣∣∣
x∫

0

∣∣(CDµ
0+f

)
(s)
∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds
∣∣∣∣∣∣ ≤

(
rx

3r − 2α

)(
x2

2Γ (1− µ)

) r−α
r
∥∥∥∥( d

ds
f (s)

)∥∥∥∥α+β

∞
.

(48)

We present some interesting Opial type inequalities regarding Prabhakar integral op-
erator (15) and Riemamn-Liouville integral operator.

Theorem 6. Let x > 0, x ∈ I, α, β > 0, r > max (1, α) , µ ∈ (0, 1) , γ, ω > 0, µγ >
ν − 1 > 0 and let U, V ∈ C (I) be such that U (s) ≥ 0 and V (s) > 0 for all s ∈ I. If
h ∈ L (0, x) , then∣∣∣∣∣∣

x∫
0

U (s)
∣∣∣(εγµ,ν,ω,0+h)(s)

∣∣∣β |h (s)|α ds

∣∣∣∣∣∣ ≤ C (x)

∣∣∣∣∣∣
x∫

0

V (s) |h (s)|r ds

∣∣∣∣∣∣
(α+β)/r

, (49)

where

C (x) =

(
α

α+ β

)α/r Γ
(
γ − ν−1

µ

)
Γ
(
ν−1
µ

)
πµω

ν−1
µ Γ (γ)

[
cos
(πµ

2

)]γ− ν−1
µ

β

(50)

×

 x∫
0

(
U r (s)V −α (s)

)1/(r−α)

 s∫
0

(V (t))−1/(r−1) dt

β(r−1)/(r−α)

ds


(r−α)/r

.



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 432

Corollary 7. Let x > 0 and h ∈ L (0, x) . If α, β > 0, r > max (1, α) , µ ∈ (0, 1) , γ, ω > 0,
µγ > ν − 1 > 0, then

∣∣∣∣∣∣
x∫

0

∣∣∣(εγµ,ν,ω,0+h)(s)
∣∣∣β |h (s)|α ds

∣∣∣∣∣∣ ≤ C (x)

∣∣∣∣∣∣
x∫

0

|h (s)|r ds

∣∣∣∣∣∣
(α+β)/r

, (51)

where

C (x) =

(
α

α+β

)α/r( Γ
(
γ− ν−1

µ

)
Γ
(
ν−1
µ

)
πµω

ν−1
µ Γ(γ)[cos(πµ2 )]

γ− ν−1
µ

)β
(
β(r−1)
r−α + 1

) r−α
r

x
β(r−1)+r−α

r . (52)

Theorem 7. Let x > 0, x ∈ I, α, β > 0, r > max (1, α) , µ, ν, γ > 0, ω ∈ R and let
U, V, h ∈ C (I) be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I. Then the following inequality
holds true: ∣∣∣∣∣∣

x∫
0

U (s)
∣∣∣(εγµ,ν,ω,0+h)(s)

∣∣∣β |h (s)|α ds

∣∣∣∣∣∣ (53)

≤
x∫

0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) eγµ,ν (λ− t, ω) dt

∣∣∣∣∣∣
r−α
r

dλ ‖V ‖β∞ ‖h‖
α+β
∞ .

If we take U = V = 1 and ω = 0 in Theorem 7, since

x∫
0

∣∣∣∣∣∣
λ∫

0

[
1

Γ (ν)
(λ− t)ν−1

]
dt

∣∣∣∣∣∣
r−α
r

dλ =
rx

r(ν + 1)− να

(
xν

Γ (ν + 1)

) r−α
r

we get the following Opial type inequality regarding Riemann-Liouville integral operator
Iν0+h :

Corollary 8. Let x > 0, x ∈ I, α, β > 0, r > max (1, α) , ν > 0. If h ∈ L (0, x) and
h ∈ C(I), then

x∫
0

∣∣(Iν0+h)(s)
∣∣β |h (s)|α ds ≤ rx

r(ν + 1)− να

(
xν

Γ (ν + 1)

) r−α
r

‖h‖α+β
∞ . (54)

We present some Opial type inequalities for Prabhakar operator (15) and Caputo-
Prabhakar derivative (12).



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 433

Theorem 8. Let x > 0, x ∈ I, f ∈ L (0, x), and let U, V ∈ C (I) be such that U (s) ≥ 0,
V (s) > 0 for all s ∈ I, α, β > 0, µ, ν, γ > 0,m = [ν], and f ∈ ACm [0, x] .
(i) If r > max (1, α) , then

x∫
0

U (s)
∣∣∣(CDγ

µ,ν,ω,0+f
)

(s)
∣∣∣β ∣∣∣f (m) (s)

∣∣∣α ds ≤ Ω4 (x)

 x∫
0

V (s)
∣∣∣f (m) (s)

∣∣∣r ds


α+β
r

, (55)

where

Ω4 (x) =

(
α

α+ β

)α
r

 x∫
0

(
U r (s)V −α (s)

) 1
r−α (∆ (s))

β(r−1)
r−α ds

 r−α
r

, (56)

∆ (s) =

s∫
0

(V (t))−
1
r−1
[
eγµ,m−ν (s− t, ω)

] r
r−1 dt. (57)

(ii) If r < max (1, α) , then

x∫
0

U (s)
∣∣∣(CDγ

µ,ν,ω,0+f
)

(s)
∣∣∣β ∣∣∣f (m) (s)

∣∣∣α ds ≥ Ω4 (x)

 x∫
0

V (s)
∣∣∣f (m) (s)

∣∣∣r ds


α+β
r

, (58)

where Ω4 (x) is given by (56) and (57).

Corollary 9. Let x > 0 and f ∈ L (0, x) , and let U, V ∈ C (I) be such that U (s) ≥ 0,
V (s) > 0 for all s ∈ I, α, β > 0, µ, ν, γ > 0, ω ∈ R and f ∗ e−γµ,m−ν,ω ∈Wm,1(0, x), m = [ν]

, f ∈ ACm [0, x] , f (k) (0+) = 0, k = 0, 1, 2, ...m− 1.

(i) If r > max (1, α) , then

x∫
0

U (s)
∣∣∣(Dγ

µ,ν,ω,0+f
)

(s)
∣∣∣β ∣∣∣f (m) (s)

∣∣∣α ds ≤ Ω4 (x)

 x∫
0

V (s)
∣∣∣f (m) (s)

∣∣∣r ds


α+β
r

. (59)

(ii) If r < max (1, α) , then

x∫
0

U (s)
∣∣∣(Dγ

µ,ν,ω,0+f
)

(s)
∣∣∣β ∣∣∣f (m) (s)

∣∣∣α ds ≥ Ω4 (x)

 x∫
0

V (s)
∣∣∣f (m) (s)

∣∣∣r ds


α+β
r

, (60)

where Ω4 (x) is given by (56) and (57).

Theorem 9. Let x > 0, h ∈ L (0, x) , x ∈ I, α, β > 0, r > max (1, α) , µ, ν, γ > 0 and let
U, V ∈ C (I) be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I. If m = [ν] , h ∈ ACm [0, x] ,
then ∣∣∣∣∣∣

x∫
0

U (s)
∣∣∣(CDγ

µ,ν,ω,0+h
)

(s)
∣∣∣β ∣∣∣h(m) (s)

∣∣∣α ds
∣∣∣∣∣∣ (61)



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 434

≤
x∫

0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) e−γµ,m−ν (λ− t, ω) dt

∣∣∣∣∣∣
r−α
r

dλ ‖V ‖β∞
∥∥∥h(m)

∥∥∥α+β

∞
.

Theorem 10. Let x > 0, h ∈ L (0, x) , x ∈ I, and let U, V ∈ C (I) be such that U (s) ≥ 0,
V (s) > 0 for all s ∈ I, α, β > 0, r > max (1, α) , µ, ν, γ > 0. If h ∗ e−γµ,m−ν,ω ∈ Wm,1(0, x),

m = [ν] , h ∈ ACm [0, x] , h(k) (0+) = 0, k = 0, 1, 2, ...,m− 1, then∣∣∣∣∣∣
x∫

0

U (s)
∣∣∣(Dγ

µ,ν,ω,0+h
)

(s)
∣∣∣β ∣∣∣h(m) (s)

∣∣∣α ds
∣∣∣∣∣∣ (62)

≤
x∫

0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) e−γµ,m−ν (λ− t, ω) dt

∣∣∣∣∣∣
r−α
r

dλ ‖V ‖β∞
∥∥∥h(m)

∥∥∥α+β

∞
.

Theorem 11. Let x > 0, α, β, ρ > 0, µ ∈ (0, 1) , ν ∈ [0, 1] , γ, ω ∈ R and U, V ∈ C (I) be

such that U (s) ≥ 0, V (s) > 0 for all s ∈ I. Also let f ∈ L (0, x) and f ∗ e−γ(1−ν)
ρ, (1−ν)(1−µ), ω ∈

AC1(0, x).
(i) If r > max {1, α} , then

x∫
0

U (s)
∣∣∣(Dγ, µ, ν

ρ, ω, 0+f
)

(s)
∣∣∣β ∣∣∣∣ dds (ε−γ(1−ν)

ρ, (1−ν)(1−µ), ω, 0+f
)

(s)

∣∣∣∣α ds (63)

≤ Ω5 (x)

 x∫
0

V (s)

∣∣∣∣ dds (ε−γ(1−ν)
ρ, (1−ν)(1−µ), ω, 0+f

)
(s)

∣∣∣∣r ds


α+β
r

,

where

Ω5 (x) =

(
α

α+ β

)α
r

 x∫
0

(
U r (s)V −α (s)

) 1
r−α (∆ (s))

β(r−1)
r−α ds

 r−α
r

, (64)

∆ (s) =

s∫
0

(V (t))−
1
r−1

[
e−γνρ, ν(1−µ) (s− t, ω)

] r
r−1

dt. (65)

(ii) If 0 < r < min {α, 1}, then

x∫
0

U (s)
∣∣∣(Dγ, µ, ν

ρ, ω, 0+f
)

(s)
∣∣∣β ∣∣∣∣ dds (ε−γ(1−ν)

ρ, (1−ν)(1−µ), ω, 0+f
)

(s)

∣∣∣∣α ds (66)

≥ Ω5 (x)

 x∫
0

V (s)

∣∣∣∣ dds (ε−γ(1−ν)
ρ, (1−ν)(1−µ), ω, 0+f

)
(s)

∣∣∣∣r ds


α+β
r

,

where Ω5 (x) and ∆ (s) are given by (64) and (65).



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 435

Corollary 10. Let x > 0, f ∈ W 1,1 (0, x) , α, β, ρ > 0, µ ∈ (0, 1) , γ, ω ∈ R and U, V ∈
C (I) be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I.

(i) If r > max {1, α}, then

x∫
0

U (s)
∣∣∣(Dγ, µ

ρ, ω, 0+f
)

(s)
∣∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds ≤ Ω6 (x)

 x∫
0

V (s)

∣∣∣∣ ddsf (s)

∣∣∣∣r ds


α+β
r

, (67)

where

Ω6 (x) =

(
α

α+ β

)α
r

 x∫
0

(
U r (s)V −α (s)

) 1
r−α (∆ (s))

β(r−1)
r−α ds

 r−α
r

, (68)

∆ (s) =

s∫
0

(V (t))−
1
r−1

[
e−γρ, 1−µ (s− t, ω)

] r
r−1

dt. (69)

(ii) If 0 < r < min {α, 1}, then

x∫
0

U (s)
∣∣∣(Dγ, µ

ρ, ω, 0+f
)

(s)
∣∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds ≥ Ω6 (x)

 x∫
0

V (s)

∣∣∣∣ ddsf (s)

∣∣∣∣r ds


α+β
r

, (70)

where Ω6 (x) and ∆ (s) are given by (68) and (69).

Finally, we present Opial type inequalities regarding Hilfer-Prabhakar operator.

Theorem 12. Let x > 0, α, β, ρ > 0, µ ∈ (0, 1) , ν ∈ [0, 1] , γ, ω ∈ R and U, V ∈ C (I)

be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I and f ∈ L (0, x) , f ∗ e−γ(1−ν)
ρ, (1−ν)(1−µ), ω ∈

AC1(0, x). If r > max {1, α} , then∣∣∣∣∣∣
x∫

0

U (s)
∣∣∣(Dγ, µ, ν

ρ, ω, 0+f
)

(s)
∣∣∣β ∣∣∣∣ dds (ε−γ(1−ν)

ρ, (1−ν)(1−µ), ω, 0+f
)

(s)

∣∣∣∣α ds
∣∣∣∣∣∣

≤
x∫

0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) e−γνρ, ν(1−µ) (λ− t, ω) dt

∣∣∣∣∣∣
r−α
r

dλ×

‖V ‖β∞

∥∥∥∥ dds (ε−γ(1−ν)
ρ, (1−ν)(1−µ), ω, 0+f

)
(s)

∥∥∥∥α+β

∞
.

Corollary 11. Let x > 0, α, β, ρ > 0, µ ∈ (0, 1) , γ, ω ∈ R and U, V ∈ C (I) be such that
U (s) ≥ 0, V (s) > 0 for all s ∈ I and f ∈ L (0, x) , f ∈ AC1(0, x). If r > max {1, α},
then ∣∣∣∣∣∣

x∫
0

U (s)
∣∣∣(Dγ, µ

ρ, ω, 0+f
)

(s)
∣∣∣β ∣∣∣∣ ddsf (s)

∣∣∣∣α ds
∣∣∣∣∣∣ (71)



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 436

≤
x∫

0

U (λ)

∣∣∣∣∣∣
λ∫

0

V (t) e−γρ, (1−µ) (λ− t, ω) dt

∣∣∣∣∣∣
r−α
r

dλ ‖V ‖β∞

∥∥∥∥ ddsf (s)

∥∥∥∥α+β

∞
.

4. Further Generalizations

In this section we give Opial–type integral inequalities for fractional integral operator
containing more generalized Mittag–Leffler function in the kernel [19].

Definition 6. Let µ, ν, k, l, γ be positive real numbers and ω ∈ R. Then the generalized
fractional integral operator containing Mittag–Leffler function εγ,δ,k

µ,ν,l,ω,a+
for a real valued

continuous function f is defined by:

(εγ,δ,k
µ,ν,l,ω,a+

f)(x) =

∫ x

a
(x− t)ν−1Eγ,δ,kµ,ν,l (ω(x− t)µ)f(t)dt, (72)

where the function Eγ,δ,kµ,ν,l is generalized Mittag–Leffler function defined as

Eγ,δ,kµ,ν,l (t) =
∞∑
n=0

(γ)kn
Γ(µn+ ν)

tn

(δ)ln
. (73)

If δ = l = 1 in (72), then integral operator εγ,δ,k
µ,ν,l,ω,a+

reduces to an integral operator

containing generalized Mittag–Leffler function Eγ,1,kµ,ν,1 introduced by Srivastava, and To-
movski in [20]. Along δ = l = 1 in addition if k = 1 (72) reduces to an integral operator
defined by Prabhakar in [18] containing Mittag-Leffler function Eγµ,ν . For ω = 0 in (72),

integral operator εγ,δ,k
µ,ν,l,ω,a+

would correspond essentially to the right-handed Riemann–
Liouville fractional integral operators.

Here we present some general results involving generalized fractional integral operator,
εγ,δ,k
α,β,l,ω,a+

containing more general form of Mittag–Leffler function Eγ,δ,kµ,ν,l .

Theorem 13. Let x > 0, α, β, µ, ν, k, l, γ > 0 with k < l+µ, ν > 1 and r > max
{

1, α, 1
ν

}
,

also let U, V ∈ C (I) be such that U (s) ≥ 0, V (s) > 0 for all s ∈ I. Then for ω ∈ R and
f ∈ L (0, x) we have

x∫
0

U (s)
∣∣∣(εγ,δ,kµ,ν,l,ω,a+

f
)

(s)
∣∣∣β |f (s)|α ds ≤ Ω (x)

 x∫
0

V (s) |f (s)|r ds


α+β
r

, (74)

where

Ω (x) =

(
α

α+ β

)α
r

 x∫
0

(
U r (s)V −α (s)

) 1
r−α (∆ (s))

β(r−1)
r−α ds

 r−α
r

, (75)

∆ (s) =

s∫
0

(V (t))−
1
r−1

(
Eγ,δ,kµ,ν,l (ω(s− t)µ)(s− t)ν−1

) r
r−1

dt. (76)



Z. Tomovski, J. Pečarić and G. Farid / Eur. J. Pure Appl. Math, 10 (3) (2017), 419-439 437

Proof. According to (72),

(εγ,δ,k
µ,ν,l,ω,a+

f)(s) =

∫ s

a
(s− t)ν−1Eγ,δ,kµ,ν,l (ω(s− t)µ)f(t)dt,

by setting

y (s) = (εγ,δ,k
µ,ν,l,ω,a+

f)(s), h (s) = f (s) , Φ (s, t) = Eγ,δ,kµ,ν,l (ω(s− t)µ) (s− t)ν−1 ,

we observe that condition (2) is satisfied with a = 0 and I = [0, x] :

|y (s)| ≤
s∫

0

Φ (s, t) |h (t)| dt, 0 ≤ s ≤ x.

The rest of the proof of is the same as Theorem 4.2 of [14].

Corollary 12. Let x > 0, α, β, µ, ν, k, l, γ > 0 with k < l+µ, ν > 1 and r > max
{

1, α, 1
ν

}
.

Then for ω ∈ R and f ∈ L (0, x) we have

x∫
0

∣∣∣(εγ,δ,kµ,ν,l,ω,a+
f
)

(s)
∣∣∣β |f (s)|α ds ≤

(
α

α+ β

)α
r

×

∫ x

0

(∫ s

0

(
Eγ,δ,kµ,ν,l (ω(s− t)µ)(s− t)ν−1

) r
r−1

dt

)β(r−1)
r−α

ds


(r−α)
r
 x∫

0

|f (s)|r ds


α+β
r

.

(77)

Remark 2. If δ = l = 1 in above results, then we obtain results involving integral operator
εγ,1,k
µ,ν,1,ω,a+

containing generalized Mittag–Leffler function Eγ,1,kµ,ν,1 introduced by Srivastava,

and Tomovski in [20]. Along δ = l = 1 in addition if k = 1, then we obtain results
involving integral operator defined by Prabhakar in [18] containing Mittag-Leffler function
Eγµ,ν . If ω = 0, then we obtain results right-handed Riemann–Liouville fractional integral
operator (see, [19]). Similar inequalities of Opial type can also be obtained for integral
operators which contain multinomial Mittag–Leffler function [10] and Multiindex Mittag-
Leffler function [13] in the kernel.

Acknowledgements

The author Zivorad Tomovski is supported under the European Commission and the
Croatian Ministry of Science, Education and Sports Co-Financing Agreement No. 291823.
In particular, ZT acknowledges project financing from the Maria Curie FP7-PEOPLE-
2011-COFUND program NEWFELPRO Grant Agreement No. 37 – Anomalous diffusion.
Author Ghulam Farid is supported by COMSATS Institute of Information Technology,
Islamabad Pakistan.



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