pISSN 1225-6951 eISSN 0454-8124 c
° Kyungpook Mathematical Journal
New Generalizations of Ostrowski-Like Type Inequalities for
Fractional Integrals
C¸ etin Yildiz∗
Atat¨urk University, K. K. Education Faculty, Department of Mathematics, 25240, Campus, Erzurum, Turkey
e-mail : [email protected] Muhamet Emin ¨Ozdemir
Uluda˘g University, Education Faculty, Department of Mathematics, 16059, Bursa, Turkey
e-mail : [email protected] Mehmet Zeki Sarikaya
D¨uzce University, Department of Mathematics, Faculty of Science and Arts, D¨uzce, Turkey
e-mail : [email protected]
Abstract. In this paper, we use the Riemann-Liouville fractional integrals to establish several new inequalities for some differantiable mappings that are connected with the celebrated Ostrowski type integral inequality.
1. Introduction
In 1938, the classical integral inequality established by Ostrowski as follows: Theorem 1.1. Let f : [a, b] → R be a differentiable mapping on (a, b) whose derivative f0 : (a, b) → R is bounded on (a, b), i.e., kf0k
∞ = sup
t∈(a,b)
|f0(t)| < ∞.
Then, we have the inequality: ¯ ¯ ¯ ¯ ¯f (x) − 1 b − a Z b a f (t)dt ¯ ¯ ¯ ¯ ¯≤ " 1 4 + ¡ x − a+b 2 ¢2 (b − a)2 # (b − a) kf0k ∞ * Corresponding Author.
Received February 24, 2014; accepted February 5, 2016.
2010 Mathematics Subject Classification: 26A15, 26A51, 26D10.
Key words and phrases: Ostrowski’s Inequality, Convex(Concave) Functions, Riemann-Liouville Fractional Integration, H¨older Inequality, Power-mean Inequality.
for all x ∈ [a, b]. The constant 1
4 is the best possible.
Recently, several generalizations of the Ostrowski integral inequality for map-pings of bounded variation and for Lipschitzian, monotonic, absolutely continuous and n-times differentiable mappings with error estimates for some special means and for some numerical quadrature rules are considered by many authors.
In [6], M. Alomari and M. Darus proved some Ostrowski’s type inequality for the class of convex(concave) functions:
Theorem 1.2. Let f : I ⊂ [0, ∞) → R, be a differentiable mapping on I◦(the
interior of I) such that f0 ∈ L[a, b], where a, b ∈ I with a < b. If |f0|p−1p is convex
on [a, b], then the following inequality holds: (1.1) ¯ ¯ ¯ ¯ ¯f (x) − 1 b − a Z b a f (x)dx ¯ ¯ ¯ ¯ ¯ ≤ 1 (b − a)(p + 1)1/p " (b − x)2 µ |f0(x)|q + |f0(b)|q 2 ¶1 q + (x − a)2 µ |f0(x)|q + |f0(a)|q 2 ¶1 q #
for each x ∈ [a, b], where 1
p+1q = 1.
Theorem 1.3. Let f : I ⊂ [0, ∞) → R, be a differentiable mapping on I◦ such
that f0∈ L[a, b], where a, b ∈ I with a < b. If |f0|p−1p is concave on [a, b], then, the
following inequality holds: ¯ ¯ ¯ ¯ ¯f (x) − 1 b − a Z b a f (x)dx ¯ ¯ ¯ ¯ ¯ ≤ b − a (p + 1)p1 "µ b − x b − a ¶2¯¯ ¯ ¯f0 µ b + x 2 ¶¯¯ ¯ ¯ (1.2) + µ x − a b − a ¶2¯¯ ¯ ¯f0 µ a + x 2 ¶¯¯ ¯ ¯ # , for each x ∈ [a, b], where p > 1.
Theorem 1.4. Let f : I ⊂ [0, ∞) → R, be a differentiable mapping on I◦ such
that f0 ∈ L[a, b], where a, b ∈ I with a < b. If |f0|q
is concave on [a, b], q ≥ 1 and |f0(x)| ≤ M, then the following inequality holds:
¯ ¯ ¯ ¯ ¯f (x) − 1 b − a Z b a f (x)dx ¯ ¯ ¯ ¯ ¯ ≤ (b − a) 2 "µ b − x b − a ¶2¯¯ ¯ ¯f0 µ b + 2x 3 ¶¯ ¯ ¯ ¯ (1.3) + µ x − a b − a ¶2¯¯ ¯ ¯f0 µ a + 2x 3 ¶¯¯ ¯ ¯ # .
In recent years, such inequalities were studied extensively by many researchers and numerious generalizations, extensions and variants of them appeared in number of papers see ([1]-[8])
Now, we give some necessary definitions and mathematical preliminaries of frac-tional calculus theory which are used throughout this paper, see([18]).
Definition 1.1. Let f ∈ L1[a, b]. The Riemann-Liouville integrals Jaα+f and Jbα−f
of order α > 0 with a ≥ 0 are defined by Jaα+f (x) = 1 Γ(α) Z x a (x − t)α−1f (t)dt, x > a and Jbα−f (x) = 1 Γ(α) Z b x (t − x)α−1f (t)dt, x < b respectively where Γ(α) =∞R 0
e−uuα−1du. Here is J0
a+f (x) = Jb0−f (x) = f (x).
In the case of α = 1, the fractional integral reduces to the classical integral. For some recent results connected with fractional integral inequalities see ([9]-[17]).
2. Main Results
In order to prove our main theorems, we need the following lemma:
Lemma 2.1. Let f : I ⊂ R → R be a differentiable mapping on Io where a, b ∈ I
with a < b. If f0∈ L[a, b], then, for all x ∈ [a, b] and α > 0 we have: · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] = Z 1 0 m(t)f0(ta + (1 − t)b)dt
for each t ∈ [a, b], where
m(t) = −tα, t ∈h0,b−x b−a i (1 − t)α, t ∈³b−x b−a, 1 i , for all x ∈ [a, b].
Proof. By integration by parts, we can obtain I = Z 1 0 m(t)f0(ta + (1 − t)b)dt = Z b−x b−a 0 (−tα) f0(ta + (1 − t)b)dt + Z 1 b−x b−a (1 − t)αf0(ta + (1 − t)b)dt = µ b − x b − a ¶α f (x) b − a− α b − a Z b−x b−a 0 tα−1f (ta + (1 − t)b)dt + µ x − a b − a ¶α f (x) b − a− α b − a Z 1 b−x b−a (1 − t)α−1f (ta + (1 − t)b)dt.
Using the change of the variable u = ta + (1 − t)b for t ∈ [0, 1] , which gives
I = (b − x) α (b − a)α+1f (x) − α (b − a)α+1 Z b x (b − u)α−1f (u)du + (x − a) α (b − a)α+1f (x) − α (b − a)α+1 Z x a (u − a)α−1f (u)du = · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] .
This is completed the proof. 2
The main results may be stated as follows:
Theorem 2.1. Let f : [a, b] → R, be a differentiable mapping on (a, b) with a < b such that f0 ∈ L[a, b]. If |f0| is convex on [a, b] and x ∈ [a, b], then the following
inequality for fractional integrals with α > 0 holds: ¯ ¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ¯ ≤ 1 α + 2 ½µ (b − x)α+2 (b − a)α+2 + (x − a)α+1 (b − a)α+1 · 1 α + 1+ b − x b − a ¸¶ |f0(a)| (2.1) + µ (x − a)α+2 (b − a)α+2 + (b − x)α+1 (b − a)α+1 · 1 α + 1 + x − a b − a ¸¶ |f0(b)| ¾
Proof. From Lemma and since |f0| is convex on [a, b], we have ¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ≤ Z b−x b−a 0 |−t|α|f0(ta + (1 − t)b)| dt + Z 1 b−x b−a |(1 − t)α| |f0(ta + (1 − t)b)| dt ≤ Z b−x b−a 0 tα[t |f0(a)| + (1 − t) |f0(b)|] dt + Z 1 b−x b−a (1 − t)α[t |f0(a)| + (1 − t) |f0(b)|] dt = · 1 α + 2 (b − x)α+2− (x − a)α+2 (b − a)α+2 + 1 α + 1 (x − a)α+1 (b − a)α+1 ¸ |f0(a)| + · 1 α + 2 (x − a)α+2− (b − x)α+2 (b − a)α+2 + 1 α + 1 (b − x)α+1 (b − a)α+1 ¸ |f0(b)| = 1 α + 2 µ (b − x)α+2 (b − a)α+2 + (x − a)α+1 (b − a)α+1 · 1 α + 1 + b − x b − a ¸¶ |f0(a)| + 1 α + 2 µ (x − a)α+2 (b − a)α+2 + (b − x)α+1 (b − a)α+1 · 1 α + 1 + x − a b − a ¸¶ |f0(b)|
which completes the proof. 2
Corollary 2.1. If we take x = a+b
2 in (2.1), we get ¯ ¯ ¯ ¯f µ a + b 2 ¶ −2α−1Γ(α + 1) (b − a)α · Jα (a+b 2 ) +f (b) + Jα (a+b 2 ) −f (a) ¸¯ ¯ ¯ ¯ (2.2) ≤ b − a 2 (α + 1) µ |f0(a)| + |f0(b)| 2 ¶ .
Theorem 2.2. Let f : [a, b] → R, be a differentiable mapping on (a, b) with a < b such that f0 ∈ L[a, b]. If |f0|q
is convex on [a, b], q > 1 and x ∈ [a, b], then the following inequality for fractional integrals holds:
¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ (2.3) ≤ 1 (b − a)α+1(αp + 1)p1 " (b − x)α+1 µ |f0(x)|q+ |f0(b)|q 2 ¶1 q +(x − a)α+1 µ |f0(x)|q+ |f0(a)|q 2 ¶1 q # where 1 p +1q = 1, α > 0.
Proof. From Lemma and using the well known H¨older inequality, we have ¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ≤ Z b−x b−a 0 |−t|α|f0(ta + (1 − t)b)| dt + Z 1 b−x b−a |(1 − t)α| |f0(ta + (1 − t)b)| dt ≤ ÃZ b−x b−a 0 tαpdt !1 pÃZ b−x b−a 0 |f0(ta + (1 − t)b)|q dt !1 q + ÃZ 1 b−x b−a (1 − t)αpdt !1 pÃZ 1 b−x b−a |f0(ta + (1 − t)b)|q dt !1 q . Since |f0| is convex, by Hermite-Hadamard inequality we have,
Z b−x b−a 0 |f0(ta + (1 − t)b)|q dt ≤ b − x b − a µ |f0(x)|q+ |f0(b)|q 2 ¶ , Z 1 b−x b−a |f0(ta + (1 − t)b)|qdt ≤ x − a b − a µ |f0(a)|q + |f0(x)|q 2 ¶
and by simple computation Z b−x b−a 0 tαpdt = 1 αp + 1 µ b − x b − a ¶αp+1 , Z 1 b−x b−a (1 − t)αpdt = 1 αp + 1 µ x − a b − a ¶αp+1 . Therefore ¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ≤ 1 (b − a)α+1(αp + 1)1p " (b − x)α+1 µ |f0(x)|q+ |f0(b)|q 2 ¶1 q +(x − a)α+1 µ |f0(x)|q+ |f0(a)|q 2 ¶1 q # where 1
p +1q = 1. Hence, using the formula Γ(α + 1) = αΓ(α) (α > 0) for Euler
Gamma function, the proof is complete. 2
Corollary 2.2. If we take x = a+b 2 in (2.3), we have ¯ ¯ ¯ ¯f µ a + b 2 ¶ −2 α−1Γ(α + 1) (b − a)α · Jα (a+b 2 ) +f (b) + Jα (a+b 2 ) −f (a) ¸¯¯ ¯ ¯ ≤ b − a 4 µ 1 αp + 1 ¶1 p à ¯ ¯f0(a+b 2 ) ¯ ¯q + |f0(b)|q 2 !1 q + à ¯ ¯f0(a+b 2 ) ¯ ¯q + |f0(a)|q 2 !1 q .
Theorem 2.3. Let f : [a, b] → R, be a differentiable mapping on (a, b) with a < b such that f0 ∈ L[a, b]. If |f0|q
is convex on [a, b], q ≥ 1 and x ∈ [a, b], then the following inequality for fractional integrals holds:
¯ ¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ¯ (2.4) ≤ µ 1 α + 1 ¶1−1 q µ 1 α + 2 ¶1 q × (µ b − x b − a ¶α+1·µ b − x b − a ¶ |f0(a)|q+ µ 1 α + 1+ x − a b − a ¶ |f0(b)|q ¸1 q + µ x − a b − a ¶α+1·µ 1 α + 1 + b − x b − a ¶ |f0(a)|q + µ x − a b − a ¶ |f0(b)|q ¸1 q )
where α > 0 and Γ is Euler Gamma function.
Proof. From Lemma and using the well known power mean inequality, we have ¯ ¯ ¯ ¯ · (x − a)α+ (b − x)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ (2.5) ≤ Z b−x b−a 0 |−t|α|f0(ta + (1 − t)b)| dt + Z 1 b−x b−a |(1 − t)α| |f0(ta + (1 − t)b)| dt ≤ ÃZ b−x b−a 0 tαdt !1−1 q ÃZ b−x b−a 0 tα|f0(ta + (1 − t)b)|q dt !1 q + ÃZ 1 b−x b−a (1 − t)αdt !1−1 qÃZ 1 b−x b−a (1 − t)α|f0(ta + (1 − t)b)|q dt !1 q .
Since |f0|q is convex, we have Z b−x b−a 0 tα|f0(ta + (1 − t)b)|q dt (2.6) ≤ Z b−x b−a 0 tα£t |f0(a)|q + (1 − t) |f0(b)|q¤ dt = 1 α + 2 µ b − x b − a ¶α+2 |f0(a)|q + Ã 1 α + 1 µ b − x b − a ¶α+1 − 1 α + 2 µ b − x b − a ¶α+2! |f0(b)|q and (2.7) Z 1 b−x b−a (1 − t)α|f0(ta + (1 − t)b)|qdt ≤ Z 1 b−x b−a (1 − t)α£t |f0(a)|q+ (1 − t) |f0(b)|q¤dt = Ã 1 α + 1 µ x − a b − a ¶α+1 − 1 α + 2 µ x − a b − a ¶α+2! |f0(a)|q + 1 α + 2 µ x − a b − a ¶α+2 |f0(b)|q.
Therefore, if we write (2.6) and (2.7) in (2.5), then we get (2.4) which is required.2 Corollary 2.3. If we take x = a+b
2 in (2.4), we have ¯ ¯ ¯ ¯f µ a + b 2 ¶ −2 α−1Γ(α + 1) (b − a)α · J(αa+b 2 ) +f (b) + Jα (a+b 2 ) −f (a) ¸¯ ¯ ¯ ¯ ≤ µ 1 α + 1 ¶1−1 qµ 1 α + 2 ¶1 qµ1 2 ¶α+1 q +1 × (µ |f0(a)|q + α + 3 (α + 1)|f 0(b)|q ¶1 q + µ α + 3 (α + 1)|f 0(a)|q + |f0(b)|q ¶1 q ) .
Theorem 2.4. Let f : [a, b] ⊂ [0, ∞) → R, be a differentiable mapping on (a, b) with a < b such that f0 ∈ L[a, b]. If |f0|q is concave on [a, b] and x ∈ [a, b], then the
following inequality for fractional integrals holds: ¯ ¯ ¯ ¯ ¯ · (x − a)α+ (x − b)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ¯ (2.8) ≤ 1 (αp + 1)1p(b − a)α+1 · (b − x)α+1 ¯ ¯ ¯ ¯f0 µ b + x 2 ¶¯¯ ¯ ¯ + (x − a)α+1 ¯ ¯ ¯ ¯f0 µ a + x 2 ¶¯¯ ¯ ¯ ¸
for each x ∈ [a, b], where p > 1.
Proof. From Lemma and using the H¨older inequality, we have ¯ ¯ ¯ ¯ · (x − a)α+ (x − b)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ (2.9) ≤ Z b−x b−a 0 |−t|α|f0(ta + (1 − t)b)| dt + Z 1 b−x b−a |(1 − t)α| |f0(ta + (1 − t)b)| dt ≤ ÃZ b−x b−a 0 tαpdt !1 pÃZ b−x b−a 0 |f0(ta + (1 − t)b)|qdt !1 q + ÃZ 1 b−x b−a (1 − t)αpdt !1 pÃZ 1 b−x b−a |f0(ta + (1 − t)b)|qdt !1 q . Since |f0|q
is concave on [a, b], by Hermite-Hadamard’s inequality we get (2.10) Z b−x b−a 0 |f0(ta + (1 − t)b)|q dt ≤ b − x b − a ¯ ¯ ¯ ¯f0 µ b + x 2 ¶¯¯ ¯ ¯ q and (2.11) Z 1 b−x b−a |f0(ta + (1 − t)b)|qdt ≤ x − a b − a ¯ ¯ ¯ ¯f0 µ a + x 2 ¶¯¯ ¯ ¯ q .
Therefore, if we write (2.10) and (2.11) in (2.9), we get (2.8). This completes the
proof. 2
Remark 2.2. In Theorem , if we choose α = 1, then the inequality (2.8) reduces the inequality (1.2) of Theorem .
Corollary 2.4. In Theorem if we take x = a+b
2 , we have ¯ ¯ ¯ ¯f µ a + b 2 ¶ −2α−1Γ(α + 1) (b − a)α · Jα (a+b 2 ) +f (b) + Jα (a+b 2 ) −f (a) ¸¯ ¯ ¯ ¯ ≤ b − a 4 1 (αp + 1)1p ·¯ ¯ ¯ ¯f0 µ a + 3b 4 ¶¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯f0 µ 3a + b 4 ¶¯ ¯ ¯ ¯ ¸ .
Theorem 2.5. Let f : [a, b] ⊂ [0, ∞) → R, be a differentiable mapping on (a, b) with a < b such that f0 ∈ L[a, b]. If |f0|q is concave on [a, b], q ≥ 1 and x ∈ [a, b],
then the following inequality for fractional integrals holds: (2.12) ¯ ¯ ¯ ¯ ¯ · (x − a)α+ (x − b)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ¯ ≤ 1 α + 1 "µ b − x b − a ¶α+1¯¯ ¯ ¯f0 µ b + (α + 1)x α + 2 ¶¯ ¯ ¯ ¯ + µ x − a b − a ¶α+1¯¯ ¯ ¯f0 µ a + (α + 1)x α + 2 ¶¯ ¯ ¯ ¯ # .
Proof. We note that by concavity of |f0|q and power-mean inequality, we have ¯ ¯ ¯ ¯ · (x − a)α+ (x − b)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ≤ Z b−x b−a 0 |−t|α|f0(ta + (1 − t)b)| dt + Z 1 b−x b−a |(1 − t)α| |f0(ta + (1 − t)b)| dt ≤ ÃZ b−x b−a 0 tαdt !1−1 qÃZ b−x b−a 0 tα|f0(ta + (1 − t)b)|q dt !1 q + ÃZ 1 b−x b−a (1 − t)αdt !1−1 q ÃZ 1 b−x b−a (1 − t)α|f0(ta + (1 − t)b)|q dt !1 q . Accordingly, by Lemma and the Jensen integral inequality, we obtain Z b−x b−a 0 tα|f0(ta + (1 − t)b)|q dt ≤ ÃZ b−x b−a 0 tαdt ! ¯¯ ¯ ¯ ¯ ¯f 0 Rb−x b−a 0 tα(ta + (1 − t)b)dt Rb−x b−a 0 tαdt ¯ ¯ ¯ ¯ ¯ ¯ q = 1 α + 1 µ b − x b − a ¶α+1¯¯ ¯ ¯f0 µ b + (α + 1)x α + 2 ¶¯¯ ¯ ¯ q and Z 1 b−x b−a (1 − t)α|f0(ta + (1 − t)b)|qdt ≤ ÃZ 1 b−x b−a (1 − t)αdt ! ¯¯¯ ¯ ¯ ¯f 0 R1 b−x b−a(1 − t) α(ta + (1 − t)b)dt R1 b−x b−a(1 − t) αdt ¯ ¯ ¯ ¯ ¯ ¯ q = 1 α + 1 µ x − a b − a ¶α+1¯¯ ¯ ¯f0 µ a + (α + 1)x α + 2 ¶¯ ¯ ¯ ¯ q .
Therefore ¯ ¯ ¯ ¯ ¯ · (x − a)α+ (x − b)α (b − a)α+1 ¸ f (x) − Γ(α + 1) (b − a)α+1[J α x+f (b) + Jxα−f (a)] ¯ ¯ ¯ ¯ ¯ ≤ 1 α + 1 "µ b − x b − a ¶α+1¯¯ ¯ ¯f0 µ b + (α + 1)x α + 2 ¶¯ ¯ ¯ ¯ + µ x − a b − a ¶α+1¯¯ ¯ ¯f0 µ a + (α + 1)x α + 2 ¶¯ ¯ ¯ ¯ #
which completes the proof. 2
Remark 2.3. In Theorem , if we choose α = 1, then the inequality (2.12) reduces the inequality (1.3) of Theorem .
Corollary 2.5. If we take x = a+b
2 in (2.12), we have ¯ ¯ ¯ ¯f µ a + b 2 ¶ −2 α−1Γ(α + 1) (b − a)α · J(αa+b 2 ) +f (b) + Jα (a+b 2 ) −f (a) ¸¯¯ ¯ ¯ ≤ b − a 4 (α + 1) ½¯ ¯ ¯ ¯f0 µ a (α + 1) + b (α + 3) 2 (α + 2) ¶¯ ¯ ¯ ¯ + ¯ ¯ ¯ ¯f0 µ a (α + 3) + b (α + 1) 2 (α + 2) ¶¯ ¯ ¯ ¯ ¾ .
References
[1] A. M. Ostrowski, ¨Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv., 10(1938), 226-227.
[2] S. S. Dragomir, The Ostrowski integral inequality for Lipschitzian mappings and
ap-plications, Comput. Math. Appl., 38 (1999), 33-37.
[3] Z. Liu, Some companions of an Ostrowski type inequality and application, J. Inequal. in Pure and Appl. Math., 10(2), 2009, Art. 52, 12 pp.
[4] B. G. Pachpatte, On an inequality of Ostrowski type in three independent variables, J. Math.Anal. Appl., 249(2000), 583-591.
[5] B. G. Pachpatte, On a new Ostrowski type inequality in two independent variables, Tamkang J. Math., 32(1)(2001), 45-49
[6] M. Alomari, M. Darus, Some Ostrowski type inequalities for convex functions with
applications, RGMIA 13(1)(2010) article No. 3. Preprint..
[7] M. Alomari, M. Darus, S. S. Dragomir, P. Cerone, Ostrowski type inequalities for
func-tions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23(2010),
1071-1076.
[8] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comeni-anae, Vol. LXXIX, 1(2010), pp. 129-134
[9] M. Z. Sarikaya, E. Set, H. Yaldız and N. Ba¸sak, Hermite-Hadamard’s inequalities
for fractional integrals and related fractional inequalities, Math. and Comput. Mod.,
[10] M. Z. Sarikaya, H. Ogunmez, On new inequalities via Riemann-Liouville fractional
integration, Abstract and Applied Analysis, 2012.
[11] C¸ . Yıldız, M. E. ¨Ozdemir and H. K. ¨Onalan, Fractional integral inequalities for
dif-ferent functions, New Trends in Mathematical Sciences, 3(2)2015, 110-117.
[12] E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex
in the second sense via fractional integrals, Comput. Math. Appl., 63(7)(2012),
1147-1154.
[13] B. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3)(2009), Art. 86.
[14] Z. Dahmani, New inequalities in fractional integrals, International Journal of Nonlin-ear Science, 9(4)(2010), 493-497.
[15] Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional
integration, Ann. Funct. Anal., 1(1)(2010), 51-58.
[16] Z. Dahmani, L. Tabharit and S. Taf, Some fractional integral inequalities, Nonl. Sci. Lett. A., 1(2)(2010), 155-160.
[17] Z. Dahmani, L. Tabharit and S. Taf, New generalizations of Gr¨uss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3)(2010), 93-99.
[18] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives
