CUBO A Mathematical Journal
Vol.14, No¯ 01, (01–07). March 2012

Univariate right fractional Ostrowski inequalities

George A. Anastassiou

University of Memphis,

Department of Mathematical Sciences,

Memphis, TN 38152, U.S.A.

email: ganastss@memphis.edu

ABSTRACT

Very general univariate right Caputo fractional Ostrowski inequalities are presented.

One of them is proved sharp and attained. Estimates are with respect to ‖·‖
p
, 1 ≤

p ≤ ∞.
RESUMEN

Se presenta de manera muy general desigualdades univariadas derechas de Caputo

fraccionarias de Ostrowski. Se prueba que una de ellas es aguda Las estimaciones con

respecto a ‖·‖
p
, 1 ≤ p ≤ ∞.

Keywords and Phrases: Ostrowski inequality, right Caputo fractional derivative.

2010 AMS Mathematics Subject Classification: 26A33, 26D10, 26D15.



2 George A. Anastassiou CUBO
14, 1 (2012)

1 Introduction

In 1938, A. Ostrowski [7] proved the following important inequality:

Theorem 1. Let f : [a,b] → R be continuous on [a,b] and differentiable on (a,b) whose derivative
f′ : (a,b) → R is bounded on (a,b), i.e., ‖f′‖∞ := sup

t∈(a,b)
|f′ (t)| < +∞. Then

∣∣∣∣∣ 1b − a
∫b
a

f(t)dt − f(x)

∣∣∣∣∣ ≤
[
1

4
+

(
x − a+b

2

)2
(b − a)

2

]
· (b − a)‖f′‖∞ , (1)

for any x ∈ [a,b]. The constant 1
4

is the best possible.

Since then there has been a lot of activity around these inequalities with important applications

to Numerical Analysis and Probability.

This paper is greatly motivated and inspired also by the following result.

Theorem 2. (see [1]) Let f ∈ Cn+1 ([a,b]), n ∈ N and x ∈ [a,b] be fixed, such that f(k) (x) = 0,
k = 1, ...,n. Then it holds∣∣∣∣∣ 1b − a

∫b
a

f(y)dy − f(x)

∣∣∣∣∣ ≤
∥∥f(n+1)∥∥∞
(n + 2) !

·

(
(x − a)

n+2
+ (b − x)

n+2

b − a

)
. (2)

Inequality (2) is sharp. In particular, when n is odd is attained by f∗ (y) := (y − x)
n+1 · (b − a),

while when n is even the optimal function is

f(y) := |y − x|
n+α · (b − a) , α > 1.

Clearly inequality (2) generalizes inequality (1) for higher order derivatives of f.

Also in [2], see Chapters 24-26, we presented a complete theory of left fractional Ostrowski

inequalities.

2 Main Results

We need

Definition 3. ([3], [4], [5], [6], [8]) Let f ∈ L1 ([a,b]), α > 0. The right Riemann-Liouville fractional
operator of order α by

Iαb−f(x) =
1

Γ (α)

∫b
x

(J − x)
α−1

f(J)dJ, (3)

∀ x ∈ [a,b], where Γ is the gamma function. We set I0b− := I (the identity operator).
Definition 4. ([3], [4], [5], [6], [8]) Let f ∈ ACm ([a,b]) (f(m−1) is in AC([a,b])), m ∈ N, m = dαe,
α > 0 (d·e the ceiling of the number). We define the right Caputo fractional derivative of order
α > 0, by

Dαb−f(x) =
(−1)

m

Γ (m − α)

∫b
x

(J − x)
m−α−1

f(m) (J)dJ, ∀ x ≤ b. (4)



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Univariate right fractional Ostrowski inequalities 3

If α = m ∈ N, then
Dmb−f(x) = (−1)

m
f(m) (x) , ∀ x ∈ [a,b] .

If x > b we define Dαb−f(x) = 0.

We also need

Theorem 5. ([3]) Let f ∈ ACm ([a,b]), x ∈ [a,b], α > 0, m = dαe. Then

f(x) =

m−1∑
k=0

f(k) (b)

k!
(x − b)

k
+

1

Γ (α)

∫b
x

(J − x)
α−1

Dαb−f(J)dJ, (5)

the right Caputo fractional Taylor formula with integral remainder.

We present

Theorem 6. Let α > 0, m = dαe, f ∈ ACm ([a,b]). Assume f(k) (b) = 0, k = 1, ...,m − 1, and
Dαb−f ∈ L∞ ([a,b]). Then∣∣∣∣∣ 1b − a

∫b
a

f(x)dx − f(b)

∣∣∣∣∣ ≤
∥∥Dαb−f∥∥∞,[a,b]
Γ (α + 2)

(b − a)
α
. (6)

Proof. Let x ∈ [a,b]. We have

f(x) − f(b) =
1

Γ (α)

∫b
x

(J − x)
α−1

Dαb−f(J)dJ.

Then

|f(x) − f(b)| ≤
1

Γ (α)

∫b
x

(J − x)
α−1

∣∣Dαb−f(J)∣∣dJ
≤

1

Γ (α)

(∫b
x

(J − x)
α−1

dJ

)∥∥Dαb−f∥∥∞,[a,b]
=

1

Γ (α)

(
(J − x)

α

α

∣∣∣∣b
x

)∥∥Dαb−f∥∥∞,[a,b]
=

1

Γ (α + 1)
(b − x)

α
∥∥Dαb−f∥∥∞,[a,b] .

Therefore

|f(x) − f(b)| ≤
(b − x)

α

Γ (α + 1)

∥∥Dαb−f∥∥∞,[a,b] , ∀ x ∈ [a,b] .
Hence it holds ∣∣∣∣∣ 1b − a

∫b
a

f(x)dx − f(b)

∣∣∣∣∣ =
∣∣∣∣∣ 1b − a

∫b
a

(f(x) − f(b))dx

∣∣∣∣∣
≤

1

b − a

∫b
a

|f(x) − f(b)|dx ≤
1

b − a

∫b
a

(b − x)
α

Γ (α + 1)

∥∥Dαb−f∥∥∞,[a,b] dx



4 George A. Anastassiou CUBO
14, 1 (2012)

=

∥∥Dαb−f∥∥∞,[a,b]
(b − a)Γ (α + 1)

∫b
a

(b − x)
α
dx

=

∥∥Dαb−f∥∥∞,[a,b]
(b − a)Γ (α + 1)


−


 (b − x)α+1

α + 1

∣∣∣∣∣
b

a






=

∥∥Dαb−f∥∥∞,[a,b]
(b − a)Γ (α + 1)

(−1)

(
0 −

(b − a)
α+1

α + 1

)

=

∥∥Dαb−f∥∥∞,[a,b]
(b − a)Γ (α + 2)

· (b − a)α+1 =

∥∥Dαb−f∥∥∞,[a,b] · (b − a)α
Γ (α + 2)

,

proving the claim.

We also give

Theorem 7. Let α ≥ 1, m = dαe, f ∈ ACm ([a,b]). Assume that f(k) (b) = 0, k = 1, ...,m−1, and
Dαb−f ∈ L1 ([a,b]). Then∣∣∣∣∣ 1b − a

∫b
a

f(x)dx − f(b)

∣∣∣∣∣ ≤
∥∥Dαb−f∥∥L1([a,b])

Γ (α + 1)
(b − a)

α−1
. (7)

Proof. We have again

|f(x) − f(b)| ≤
1

Γ (α)

∫b
x

(J − x)
α−1

∣∣Dαb−f(J)∣∣dJ
≤

1

Γ (α)
(b − x)

α−1
∫b
x

∣∣Dαb−f(J)∣∣dJ
≤

1

Γ (α)
(b − x)

α−1
∥∥Dαb−f∥∥L1([a,b]) .

Hence

|f(x) − f(b)| ≤

∥∥Dαb−f∥∥L1([a,b])
Γ (α)

(b − x)
α−1

, ∀ x ∈ [a,b] .

Therefore ∣∣∣∣∣ 1b − a
∫b
a

f(x)dx − f(b)

∣∣∣∣∣ ≤ 1b − a
∫b
a

|f(x) − f(b)|dx

≤
1

b − a

∫b
a

∥∥Dαb−f∥∥L1([a,b])
Γ (α)

(b − x)
α−1

dx

=

∥∥Dαb−f∥∥L1([a,b])
(b − a)Γ (α)

∫b
a

(b − x)
α−1

dx

=

∥∥Dαb−f∥∥L1([a,b])
(b − a)Γ (α)

(b − x)
α

α
=

∥∥Dαb−f∥∥L1([a,b])
Γ (α + 1)

(b − x)
α−1

,

proving the claim.



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Univariate right fractional Ostrowski inequalities 5

We continue with

Theorem 8. Let p,q > 1 : 1
p
+ 1
q

= 1, α > 1 − 1
p

, m = dαe, f ∈ ACm ([a,b]). Assume that
f(k) (b) = 0, k = 1, ...,m − 1, and Dαb−f ∈ Lq ([a,b]). Then∣∣∣∣∣ 1b − a

∫b
a

f(x)dx − f(b)

∣∣∣∣∣ ≤
∥∥Dαb−f∥∥Lq([a,b])

Γ (α) (p(α − 1) + 1)
1
p

(
α + 1

p

) (b − a)α−1+ 1p . (8)
Proof. We have again

|f(x) − f(b)| ≤
1

Γ (α)

∫b
x

(J − x)
α−1

∣∣Dαb−f(J)∣∣dJ
≤

1

Γ (α)

(∫b
x

(J − x)
p(α−1)

dJ

)1
p
(∫b
x

∣∣Dαb−f(J)∣∣q dJ
)1

q

≤
1

Γ (α)

(b − x)
(α−1)+ 1

p

(p(α − 1) + 1)
1
p

(∫b
x

∣∣Dαb−f(J)∣∣q dJ
)1

q

≤
1

Γ (α)

(b − x)
(α−1)+ 1

p

(p(α − 1) + 1)
1
p

∥∥Dαb−f∥∥Lq([a,b]) .
Therefore

|f(x) − f(b)| ≤

∥∥Dαb−f∥∥Lq([a,b])
Γ (α) (p(α − 1) + 1)

1
p

(b − x)
α−1+ 1

p , ∀ x ∈ [a,b] .

Hence ∣∣∣∣∣ 1b − a
∫b
a

f(x)dx − f(b)

∣∣∣∣∣ ≤ 1b − a
∫b
a

|f(x) − f(b)|dx

≤

∥∥Dαb−f∥∥Lq([a,b])
(b − a)Γ (α) (p(α − 1) + 1)

1
p

∫b
a

(b − x)
α−1+ 1

p dx

=

∥∥Dαb−f∥∥Lq([a,b])
Γ (α) (p(α − 1) + 1)

1
p

(b − a)
α−1+ 1

p(
α + 1

p

) .

Corollary 9. Let α > 1
2

, m = dαe, f ∈ ACm ([a,b]). Assume f(k) (b) = 0, k = 1, ...,m − 1,
Dαb−f ∈ L2 ([a,b]). Then∣∣∣∣∣ 1b − a

∫b
a

f(x)dx − f(b)

∣∣∣∣∣ ≤
∥∥Dαb−f∥∥L2([a,b])

Γ (α)
(√
2α − 1

)(
α + 1

2

) (b − a)α− 12 . (9)
We finish with



6 George A. Anastassiou CUBO
14, 1 (2012)

Proposition 10. Inequality (6) is sharp, namely it is attained by

f(x) = (b − x)
α

, α > 0, α /∈ N, x ∈ [a,b] .

Proof. Notice that (b − x)
α ∈ ACm ([a,b]). We see that

f′ (x) = −α(b − x)
α−1

,

f′′ (x) = (−1)
2
α(α − 1) (b − x)

α−2
,

...,

f(m−1) (x) = (−1)
m−1

α(α − 1) (α − 2) ...(α − m + 2) (b − x)
α−m+1

,

and

f(m) (x) = (−1)
m
α(α − 1) (α − 2) ...(α − m + 2) (α − m + 1) (b − x)

α−m
.

Thus

Dαb−f(x) =
(−1)

2m

Γ (m − α)
α(α − 1) ...(α − m + 1)

∫b
x

(J − x)
m−α−1

(b − J)
α−m

dJ

=
α(α − 1) ...(α − m + 1)

Γ (m − α)

∫b
x

(b − J)
(α−m+1)−1

(J − x)
(m−α)−1

dJ

=
α(α − 1) ...(α − m + 1)

Γ (m − α)

Γ (α − m + 1)Γ (m − α)

Γ (1)

= α(α − 1) ...(α − m + 1)Γ (α − m + 1) = Γ (α + 1) .

That is

Dαb−f(x) = Γ (α + 1) , ∀ x ∈ [a,b] .

Also we see that f(k) (b) = 0, k = 0,1, ...,m−1, and Dαb−f ∈ L∞ ([a,b]). So f fulfills all assumptions.
Next we see

R.H.S.(6) =
Γ (α + 1)

Γ (α + 2)
(b − a)

α
=

(b − a)
α

(α + 1)
.

L.H.S.(6) =
1

b − a

∫b
a

(b − x)
α
dx

=
1

b − a

(b − a)
α+1

(α + 1)
=

(b − a)
α

α + 1
,

proving attainability and sharpness of (6).

Received: November 2010. Revised: March 2011.



CUBO
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Univariate right fractional Ostrowski inequalities 7

References

[1] G.A. Anastassiou, Ostrowski type inequalities, Proc. AMS 123 (1995), 3775-3781.

[2] G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New

York, 2009.

[3] G.A. Anastassiou, On Right Fractional Calculus, Chaos, Solitons and Fractals, 42 (2009), 365-

376.

[4] A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic

Journal of Theoretical Physics, Vol. 3, No. 12 (2006), 81-95.

[5] G.S. Frederico, D.F.M. Torres, Fractional Optimal Control in the sense of Caputo and the

fractional Noether’s theorem, International Mathematical Forum, Vol. 3, No. 10 (2008), 479-

493.

[6] R. Gorenflo, F. Mainardi, Essentials of Fractional Calculus, 2000, Maphysto Center,

http://www.maphysto.dk/oldpages/events/LevyCAC2000/ MainardiNotes/fm2k0a.ps.

[7] A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funcktion von ihrem Inte-

gralmittelwert, Comment. Math. Helv., 10 (1938), 226-227.

[8] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Ap-

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Minsk, 1987)].


	Introduction
	Main Results