Volume 2010, Article ID 264347,23pages doi:10.1155/2010/264347
Research Article
On an Inequality of H. G. Hardy
Sajid Iqbal,
1Kristina Kruli´c,
2and Josip Peˇcari´c
1, 21Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 28a, 10000 Zagreb, Croatia
Correspondence should be addressed to Sajid Iqbal,sajid [email protected]
Received 18 June 2010; Accepted 16 October 2010
Academic Editor: Q. Lan
Copyrightq2010 Sajid Iqbal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.
1. Introduction
First, let us recall some facts about fractional derivatives needed in the sequel, for more details see, for example,1,2.
Let 0< a < b ≤ ∞. ByCma, b, we denote the space of all functions ona, bwhich have continuous derivatives up to orderm, and ACa, b is the space of all absolutely continuous functions on a, b. By ACma, b, we denote the space of all functions g ∈
Cma, bwithgm−1 ∈ACa, b. For anyα∈
Ê, we denote byαthe integral part ofαthe
integerksatisfyingk≤α < k1, andαis the ceiling ofαmin{n∈Æ, n≥α}. ByL1a, b,
we denote the space of all functions integrable on the intervala, b, and byL∞a, bthe set
of all functions measurable and essentially bounded ona, b. Clearly,L∞a, b⊂L1a, b.
We start with the definition of the Riemann-Liouville fractional integrals, see3. Let
a, b, −∞ < a < b < ∞ be a finite interval on the real axis Ê. The Riemann-Liouville
fractional integralsIaαfandIbα−fof orderα >0 are defined by
Iaαfx 1
Γα
x
a
Ibα
−f
x 1
Γα
b
x
ftt−xα−1dt, x < b, 1.2
respectively. HereΓαis the Gamma function. These integrals are called the left-sided and the right-sided fractional integrals. We denote some properties of the operatorsIα
afandI
α b−f of orderα >0, see also4. The first result yields that the fractional integral operatorsIα
af
andIα
b−fare bounded inLpa, b, 1≤p≤ ∞, that is
Iα
af p≤K f p, I
α
b−f p≤K f p, 1.3
where
K b−a
α
αΓα . 1.4
Inequality1.3, that is the result involving the left-sided fractional integral, was proved by H. G. Hardy in one of his first papers, see5. He did not write down the constant, but the calculation of the constant was hidden inside his proof.
Throughout this paper, all measures are assumed to be positive, all functions are assumed to be positive and measurable, and expressions of the form 0· ∞,∞/∞, and 0/0 are taken to be equal to zero. Moreover, by a weightuux, we mean a nonnegative measurable function on the actual interval or more general set.
The paper is organized in the following way. After this Introduction, inSection 2we state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy since 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. We conclude this paper with new results involving mixed Riemann-Liouville fractional integrals.
2. The Main Results
LetΩ1,Σ1, μ1and Ω2,Σ2, μ2be measure spaces with positive σ-finite measures, and let
k:Ω1×Ω2 → Êbe a nonnegative function, and
Kx
Ω2
kx, ydμ2
y, x∈Ω1. 2.1
Throughout this paper, we suppose that Kx > 0 a.e. on Ω1, and by a weight function
shortly: a weight, we mean a nonnegative measurable function on the actual set. LetUk
denote the class of functionsg:Ω1 → Êwith the representation
gx
Ω2
kx, yfydμ2
y, 2.2
Our first result is given in the following theorem.
Theorem 2.1. Letube a weight function onΩ1,ka nonnegative measurable function onΩ1×Ω2,
andKbe defined onΩ1by2.1. Assume that the functionx →uxkx, y/Kxis integrable
onΩ1for each fixedy∈Ω2. DefinevonΩ2by
vy:
Ω1
uxk
x, y
Kx dμ1x<∞. 2.3
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
Ω1
uxφgx
Kx
dμ1x≤
Ω2
vyφfydμ2
y 2.4
holds for all measurable functionsf:Ω2 → Êand for all functionsg∈Uk.
Proof. By using Jensen’s inequality and the Fubini theorem, sinceφis increasing function, we find that
Ω1
uxφgx
Kx
dμ1x
Ω1
uxφ 1
Kx
Ω2
kx, yfydμ2
y
dμ1x
≤
Ω1
ux
Kx
Ω2
kx, yφfydμ2
y
dμ1x
Ω2
φfy
Ω1
uxk
x, y
Kx dμ1x
dμ2
y
Ω2
vyφfydμ2
y,
2.5
and the proof is complete.
As a special case ofTheorem 2.1, we get the following result.
Corollary 2.2. Letube a weight function ona, bandα >0.Iα
af denotes the Riemann-Liouville
fractional integral off. Definevona, bby
vy:α
b
y
ux
x−yα−1
x−aα dx <∞. 2.6
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα1 x−aαI
α
afx dx≤
b
a
vyφfydy 2.7
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
x−yα−1
Γα , a≤y≤x,
0, x < y≤b,
2.8
we get thatKx x−aα/Γα1andgx Iα
afx, so2.7follows.
Remark 2.3. In particular for the weight functionux x−aα, x∈a, binCorollary 2.2, we obtain the inequality
b
a
x−aαφ
Γα1 x−aαI
α
afx dx≤
b
a
b−yαφfydy. 2.9
Although 2.4 holds for all convex and increasing functions, some choices of φ are of particular interest. Namely, we will consider power function. Let q > 1 and the function
φ:Ê → Êbe defined byφx x
q, then2.9reduces to
b
a
x−aα
Γ
α1
x−aαI
α afx
q
dx≤
b
a
b−yαfyqdy. 2.10
Sincex∈a, bandα1−q<0, then we obtain that the left hand side of2.10is
b
a
x−aα
Γ
α1
x−aα|I
α afx|
q
dx≥b−aα1−qΓα1q b
a
Iα afx
q
dx 2.11
and the right-hand side of2.10is
b
a
b−yαfyqdy≤b−aα
b
a
fyqdy. 2.12
Combining2.11and2.12, we get
b
a
Iaαfxqdx≤
b−aα
Γα1
qb
a
fyqdy. 2.13
Taking power 1/qon both sides, we obtain1.3.
Corollary 2.4. Letube a weight function ona, bandα > 0.Iα
b−f denotes the Riemann-Liouville
fractional integral off. Definevona, bby
vy:α
y
a
ux
y−xα−1
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα1 b−xα
Iα
b−fx
dx≤
b
a
vyφfydy 2.15
holds.
Proof. Similar to the proof ofCorollary 2.2.
Remark 2.5. In particular for the weight functionux b−xα, x∈a, binCorollary 2.4, we obtain the inequality
b
a
b−xαφ
Γα1 b−xα
Ibα−fx dx≤
b
a
y−aαφfydy. 2.16
Letq >1 and the functionφ:Ê → Êbe defined byφx x
q, then2.16reduces to
b
a
b−xα
Γ
α1
b−xα
Ibα−fx
q
dx≤
b
a
y−aαfyqdy. 2.17
Sincex∈a, bandα1−q<0, then we obtain that the left hand side of2.17is b
a
b−xα
Γ
α1
b−xα
Ibα−fx
q
dx≥b−aα1−qΓα1q b
a
Ibα−fxqdx 2.18
and the right-hand side of2.17is
b
a
y−aαfyqdy≤b−aα
b
a
fyqdy. 2.19
Combining2.18and2.19, we get
b
a
Ibα−fxqdx≤
b−aα
Γα1
qb
a
fyqdy. 2.20
Taking power 1/qon both sides, we obtain1.3.
Theorem 2.6. Letp, q >1,1/p1/q1,α > 1/q,Iα
af andIbα−fdenote the Riemann-Liouville
fractional integral off, then the following inequalities
b
a
Iaαfxqdx≤C
b
a
fyqdy, 2.21
b
a
Ibα−fxqdx≤C
b
a
fyqdy 2.22
Proof. We will prove only inequality2.21, since the proof of2.22is analogous. We have
Iα af
x≤ 1
Γα
x
a
ftx−tα−1dt. 2.23
Then by the H ¨older inequality, the right-hand side of the above inequality is
≤ 1
Γα
x
a
x−tpα−1dt
1/px
a
|ft|qdt
1/q
1
Γα
x−aα−11/p
pα−1 11/p
x
a
|ft|qdt
1/q
≤ 1
Γα
x−aα−11/p
pα−1 11/p
b
a
|ft|qdt
1/q
.
2.24
Thus, we have
Iα af
x≤ 1
Γα
x−aα−11/p
pα−1 11/p
b
a
|ft|qdt
1/q
, for everyx∈a, b. 2.25
Consequently, we find
Iα afx
q≤ 1
Γαq
x−aqα−1q/p
pα−1 1q/p
b
a
ftq
dt
, 2.26
and we obtain
b
a
Iaαfxqdx≤ b−a
qα−1q/p1
Γαqqα−1 q/p1pα−1 1q/p b
a
ftqdt. 2.27
Remark 2.7. Forα≥1, inequalities2.21and2.22are refinements of1.3since
qαpα−1 1q−1≥qαq> αq, so C <
b−aα αΓα
q
. 2.28
We proved that Theorem 2.6 is a refinement of 1.3, and Corollaries 2.2 and 2.4 are generalizations of1.3.
We define the generalized Riemann-Liouville fractional derivative off of orderα >0 by
Daαfx 1
Γn−α
d dx
nx
a
x−yn−α−1fydy, 2.29
wheren α 1, x∈a, b.
Fora, b ∈Ê, we say thatf ∈L1a, bhas anL∞fractional derivativeD
α
af α > 0in
a, b, if and only if
1Daα−kf ∈Ca, b,k1, . . . , n α 1, 2Dα−1
a f ∈ACa, b,
3Dα
a ∈L∞a, b.
Next, lemma is very useful in the upcoming corollarysee1, page 449and2.
Lemma 2.8. Letβ > α≥0and letf ∈L1a, bhave anL∞fractional derivativeDβafina, band
let
Daβ−kfa 0, k1, . . . ,
β1, 2.30
then
Daαfx 1
Γβ−α
x
a
x−yβ−α−1Dβaf
ydy, 2.31
for alla≤x≤b.
Corollary 2.9. Letube a weight function ona, b, and let assumptions inLemma 2.8be satisfied. Definevona, bby
vy:β−α
b
y
ux
x−yβ−α−1
x−aβ−α dx <∞.
2.32
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γβ−α1 x−aβ−α
Daαfx
dx≤
b
a
vyφDaβf
ydy 2.33
holds.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
x−yβ−α−1
Γβ−α , a≤y≤x,
0, x < y≤b,
we get thatKx x−aβ−α/Γβ−α1. ReplacefbyDaβf. Then, byLemma 2.8, gx
Dα
afxand we get2.33.
Remark 2.10. In particular for the weight functionux x−aβ−α,x∈a, binCorollary 2.9, we obtain the inequality
b
a
x−aβ−αφ
Γβ−α1 x−aβ−α
Dα afx
dx≤
b
a
b−yβ−αφDaβf
ydy. 2.35
Letq > 1 and the functionφ:Ê → Êbe defined byφx x
q, then after some calculation,
we obtain
b
a
Dα afx
q
dx≤
b−aβ−α
Γβ−α1 qb
a
Dβafy q
dy. 2.36
Next, we defineCanavati-type fractional derivativeν-fractional derivative off, for details see1, page 446. We consider
Cνa, b f∈Cna, b:Ian−ν1fn∈C1a, b, 2.37
ν >0, n ν. Letf ∈Cνa, b. We define the generalizedν-fractional derivative offover
a, bas
DνafIan−ν1fn, 2.38
the derivative with respect tox.
Lemma 2.11. Let ν ≥ γ 1, where γ ≥ 0 and f ∈ Cνa, b. Assume that fia 0, i
0,1, . . . ,ν−1, then
Dγaf
x 1
Γν−γ
x
a
x−tν−γ−1Dνaftdt, 2.39
for allx∈a, b.
Corollary 2.12. Letube a weight function ona, b, and let assumptions inLemma 2.11be satisfied. Definevona, bby
vy:ν−γ
b
y
ux
x−yν−γ−1
x−x0ν−γ
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γν−γ1 x−aν−γ
Dγafx
dx≤
b
a
vyφDaνfydy 2.41
holds.
Proof. Similar to the proof ofCorollary 2.9.
Remark 2.13. In particular for the weight functionux x−aν−γ,x∈a, binCorollary 2.12, we obtain the inequality
b
a
x−aν−γφ
Γν−γ1 x−aν−γ
Daγfx
dx≤
b
a
b−yν−γφDν af
ydy. 2.42
Letq >1 and the functionφ:Ê → Êbe defined byφx x
q, then2.42reduces to
Γν−γ1q b
a
x−aν−γ1−qDaγfx q
dx≤
b
a
b−yν−γDaνfyqdy. 2.43
Sincex∈a, bandν−γ1−q≤0, then we obtain
b
a
Dγafx q
dx≤
b−aν−γ
Γν−γ1 qb
a
Dν
afyqdy. 2.44
Taking power 1/qon both sides of2.44, we obtain
Dγafx q≤ b−a
ν−γ
Γν−γ1 D
ν af
y q. 2.45
Whenγ0, we find that
Γν1q b
a
x−aν1−qfxqdx≤
b
a
b−yνDνafyqdy, 2.46
that is,
f q≤
b−aν
Γν1 D
ν af
y q. 2.47
Letα≥0,nα,g∈ACna, b. The Caputo fractional derivative is given by
D∗αagt 1
Γn−α
x
a
gny
x−yα−n1dy, 2.48
for allx∈a, b. The above function exists almost everywhere forx∈a, b.
Corollary 2.14. Letube a weight function ona, bandα >0.Dα
∗agdenotes the Caputo fractional
derivative ofg. Definevona, bby
vy: n−α
b
y
ux
x−yn−α−1
x−an−α dx <∞. 2.49
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γn−α1 x−an−α D
α
∗agx dx≤
b
a
vyφgnydy 2.50
holds.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
x−yn−α−1
Γn−α , a≤y≤x,
0, x < y≤b,
2.51
we get thatKx x−an−α/Γn−α1. Replacefbygn, sogbecomesDα
∗agand2.50
follows.
Remark 2.15. In particular for the weight function ux x − an−α, x ∈ a, b in
Corollary 2.14, we obtain the inequality
b
a
x−an−αφ
Γn−α1 x−an−α D
α
∗agx dx≤
b
a
b−yn−αφgnydy. 2.52
Letq > 1 and the functionφ:Ê → Êbe defined byφx x
q, then after some calculation,
we obtain
b
a
Dα∗agxq dx≤
b−an−α
Γn−α1 qb
a
gnyqdy. 2.53
Taking power 1/qon both sides, we obtain
Dα∗agx q≤ b−a
n−α
Γn−α1 g
ny
Theorem 2.16. Letp, q > 1,1/p1/q 1,n−α >1/q,Dα
∗afxdenotes the Caputo fractional
derivative off, then the following inequality
b
a
Dα
∗afx q
dx≤ b−a
qn−α
Γn−αqpn−α−1 1q/pqn−α
b
a
fnyqdy 2.55
holds.
Proof. Similar to the proof ofTheorem 2.6.
The following result is given1, page 450.
Lemma 2.17. Letα≥γ1,γ >0, andnα. Assume thatf∈ACna, bsuch thatfka 0,
k0,1, . . . , n−1, andDα
∗af∈L∞a, b, thenD γ
∗af ∈Ca, b,and
Dγ∗afx
1 Γα−γ
x
a
x−yα−γ−1D∗αafydy, 2.56
for alla≤x≤b.
Corollary 2.18. Letube a weight function ona, bandα >0.Dα
∗af denotes the Caputo fractional
derivative off, and assumptions inLemma 2.17are satisfied. Definevona, bby
vy:α−γ
b
y
ux
x−yα−γ−1
x−aα−γ dx <∞. 2.57
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα−γ1 x−aα−γ
Dγ∗afx
dx≤
b
a
vyφDα∗afydy 2.58
holds.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
x−yα−γ−1
Γα−γ , a≤y≤x,
0, x < y≤b,
2.59
we get thatKx x−aα−γ/Γα−γ1. ReplacefbyDα
∗af, sogbecomesD γ
∗afand2.58
Remark 2.19. In particular for the weight functionux x−aα−γ,x∈a, binCorollary 2.18, we obtain the inequality
b
a
x−aα−γφ
Γα−γ1 x−aα−γ
Dγ∗afx
dx≤
b
a
b−yα−γφDα∗afydy. 2.60
Letq > 1 and the functionφ:Ê → Êbe defined byφx x
q, then after some calculation,
we obtain
b
a
Dγ∗afx q
dx≤
b−aα−γ
Γα−γ1 qb
a
Dα
∗afy q
dy. 2.61
Forγ0, we obtain
b
a
fxq
dx≤
b−aα
Γα1
qb
a
Dα
∗afy q
dy. 2.62
We continue with definitions and some properties of thefractional integrals of a function
fwith respect to given functiong. For details see, for example,3, page 99.
Leta, b,−∞ ≤ a < b≤ ∞be a finite or infinite interval of the real lineÊandα > 0.
Also letgbe an increasing function ona, bandga continuous function ona, b. The left-and right-sided fractional integrals of a functionfwith respect to another functiongina, b
are given by
Iaα;gfx 1
Γα
x
a
gtftdt
gx−gt1−α, x > a, 2.63
Ibα−;gfx 1
Γα
b
x
gtftdt
gt−gx1−α, x < b, 2.64
respectively.
Corollary 2.20. Letube a weight function ona, b, and letgbe an increasing function ona, b, such thatg is a continuous function ona, b andα > 0.Iα
a;gf denotes the left-sided fractional
integral of a functionfwith respect to another functiongina, b. Definevona, bby
vy:αgy
b
y
ux
gx−gyα−1
gx−gaα dx <∞. 2.65
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα1
gx−gaα
Iaα;gfx
dx≤
b
a
vyφfydy 2.66
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
1 Γα1
gy
gx−gy1−α, a≤y≤x,
0, x < y≤b,
2.67
we get thatKx 1/Γα1gx−gaα, so2.66follows.
Remark 2.21. In particular for the weight functionux gxgx−gaα,x ∈ a, bin
Corollary 2.20, we obtain the inequality
b
a
gxgx−gaαφ
Γα1
gx−gaα
Iα
a;gfx
dx
≤
b
a
gygb−gyαφfydy.
2.68
Letq >1 and the functionφ:Ê → Êbe defined byφx x
q, then2.68reduces to
Γα1q b
a
gxgx−gaα1−qIaα;gfxqdx
≤
b
a
gygb−gyαfyqdy.
2.69
Sincex∈a, bandα1−q<0,gis increasing, thengx−gaα1−q >gb−gaα1−q
andgb−gyα<gb−gaαand we obtain
b
a
gxIaα;gfxqdx≤
gb−gaα
Γα1
qb
a
gyfyqdy. 2.70
Remark 2.22. If gx x, then Iα
a;xfx reduces to Iaαfx Riemann-Liouville fractional
integral and2.70becomes2.13.
Analogous toCorollary 2.20, we obtain the following result.
Corollary 2.23. Letube a weight function ona, b, and letgbe an increasing function ona, b, such thatgis a continuous function ona, bandα > 0.Ibα−;gf denotes the right-sided fractional integral of a functionfwith respect to another functiongina, b. Definevona, bby
vy:αgy
y
a
ux
gy−gxα−1
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα1
gb−gxα
Ibα−;gfx
dx≤
b
a
vyφfydy 2.72
holds.
Remark 2.24. In particular for the weight functionux gxgb−gxα,x∈a, band for functionφx xq,q >1, we obtain after some calculation
b
a
gxIbα−;gfxqdx≤
gb−gaα
Γα1
qb
a
gyfyqdy. 2.73
Remark 2.25. If gx x, then Iα
b−;xfx reduces to Ibα−fx Riemann-Liouville fractional integral and2.73becomes2.20.
The refinements of2.70and2.73forα >1/qare given in the following theorem.
Theorem 2.26. Letp, q > 1,1/p1/q 1,α >1/q,Iα
a;gf and Ibα−;gf denote the left-sided and
right-sided fractional integral of a functionf with respect to another functiong in a, b, then the following inequalities:
b
a
Iaα;gfxqgxdx≤
gb−gaαq
αqΓαqpα−1 1q/p
b
a
f
yqgydy,
b
a
Ibα−;gfxqgxdx≤
gb−gaαq
αqΓαqpα−1 1q/p
b
a
f
yqgydy
2.74
hold.
We continue by definingHadamard type fractional integrals.
Leta, b, 0 ≤ a < b≤ ∞be a finite or infinite interval of the half-axisÊ andα > 0.
The left- and right-sided Hadamard fractional integrals of orderαare given by
Jaαfx 1
Γα
x
a
logx
y
α−1fydy
y , x > a, 2.75
Jbα−fx 1
Γα
b
x
logy
x
α−1fydy
y , x < b, 2.76
Notice that Hadamard fractional integrals of orderαare special case of the left- and right-sided fractional integrals of a functionfwith respect to another functiongx logx
ina, b, where 0≤a < b≤ ∞, so2.70reduces to
b
a
Jaαfxqdx
x ≤
logb/aα
Γα1
qb
a
fyqdy
y , 2.77
and2.73becomes
b
a
Jα b−f
xqdx
x ≤
logb/aα
Γα1
qb
a
f
yqdy
y . 2.78
Also, fromTheorem 2.26we obtain refinements of2.77and2.78, forα >1/q,
b
a
Jα afx
qdx
x ≤
logb/aqα
qαΓαqpα−1 1q/p
b
a
f
yqdy
y ,
b
a
Jα b−fx
qdx
x ≤
logb/aqα
qαΓαqpα−1 1q/p
b
a
f
yqdy
y .
2.79
Some results involving Hadamard type fractional integrals are given in3, page 110. Here, we mention the following result that can not be compared with our result.
Letα >0, 1≤p≤ ∞, and 0≤a < b≤ ∞, then the operatorsJα
afandJbα−fare bounded
inLpa, bas follows:
Jα
af p≤K1 f p, Jbα−f p≤K2 f p, 2.80
where
K1 1
Γα
logb/a
0
tα−1et/pdt, K2 1
Γα
logb/a
0
tα−1e−t/pdt. 2.81
Now we present the definitions and some properties of theErd´elyi-Kober type fractional integrals. Some of these definitions and results were presented by Samko et al. in4.
Leta, b,0 ≤ a < b ≤ ∞be a finite or infinite interval of the half-axisÊ
. Also let
α >0,σ >0, andη∈Ê. We consider the left- and right-sided integrals of orderα∈Êdefined
by
Iaα;σ;ηf
x σx
−σαη
Γα
x
a
tσησ−1ftdt
xσ−tσ1−α , 2.82
Ibα−;σ;ηfx σx
ση
Γα
b
x
tσ1−η−α−1ftdt
tσ−xσ1−α , 2.83
Corollary 2.27. Let u be a weight function on a,b, 2F1a, b;c;z denotes the hypergeometric
function, andIα
a;σ;ηfdenotes the Erd´elyi-Kober type fractional left-sided integral. Definevby
vy:ασyσησ−1
b
y
ux x
−σηxσ−yσα−1
xσ−aσα
2F1
α,−η;α1; 1−a/xσdx <∞. 2.84
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα1
1−a/xσα 2F1
α,−η;α1; 1−a/xσ
Iaα;σ;ηfx
dx
≤
b
a
vyφfydy
2.85
holds.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
1 Γα
σx−σαη
xσ−yσ1−αy
σησ−1, a≤y≤x,
0, x < y≤b,
2.86
we get thatKx 1/Γα11−a/xσα2F1α,−η;α1; 1−a/xσ, so2.85follows.
Remark 2.28. In particular for the weight function ux xσ−1xσ − aσα
2F1x where
2F1x 2F1α,−η;α1; 1−a/xσinCorollary 2.27, we obtain the inequality
b
a
xσ−1xσ−aσα 2F1xφ
Γα1
1−a/xσα 2F1x
Iaα;σ;ηfx
dx
≤
b
a
yσ−1bσ−yσα 2F1
yφfydy,
2.87
where 2F1y 2F1α, η;α1; 1−a/yσ.
Corollary 2.29. Let u be a weight function on a, b, 2F1a, b;c;z denotes the hypergeometric
function, andIbα−;σ;ηfdenotes the Erd´elyi-Kober type fractional right-sided integral. Definevby
vy:ασyσ1−α−η−1
y
a
ux x
σηαyσ−xσα−1
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b
a
uxφ
Γα1
b/xσ−1α 2F1
α, αη;α1; 1−b/xσ
Ibα−;σ;ηfx
dx
≤
b
a
vyφfydy
2.89
holds.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,
kx, y
⎧ ⎪ ⎨ ⎪ ⎩
1 Γα
σxση
yσ−xσ1−αy
σ1−α−η−1, x < y≤b,
0, a≤y≤x,
2.90
we get thatKx 1/Γα1b/xσ−1α 2F1α, αη;α1; 1−b/xσ, so2.89follows.
Remark 2.30. In particular for the weight function ux xσ−1bσ − xσα
2F1x where
2F1x 2F1α, αη;α1; 1−b/xσinCorollary 2.29, we obtain the inequality
b
a
xσ−1bσ−xσα 2F1xφ
Γα1
b/xσ−1α 2F1x
Ibα−;σ;ηfx
dx
≤
b
a
yσ−1yσ−aσα 2F1
yφfydy,
2.91
where2F1y 2F1α,−α−η;α1; 1−b/yσ.
In the next corollary, we give some results related to the Caputo radial fractional derivative. Let us recall the following definition, see1, page 463.
Letf : A → Ê, ν ≥ 0,n : ν, such thatf·ω ∈ AC
nR
1, R2, for allω ∈ SN−1,
whereA R1, R2×SN−1 forN ∈ Æ andS
N−1 : {x ∈
Ê
N : |x| 1}. We call the Caputo
radial fractional derivative as the following function:
∂ν∗R
1fx
∂rν :
1 Γn−ν
r
R1
r−tn−ν−1∂
nftω
∂rn dt, 2.92
wherex∈A, that is,xrω,r∈R1, R2,ω∈SN−1.
Clearly,
∂0∗R
1fx
∂r0 fx,
∂ν∗R
1fx
∂rν
∂νfx
∂rν if ν∈Æ, the usual radial derivative.
Corollary 2.31. Letube a weight function onR1, R2, and∂ν∗R1fx/∂rνdenotes the Caputo radial
fractional derivative off. DefinevonR1, R2by
vt: n−ν
R2
t
urr−t
n−ν−1
r−R1n−ν
dr <∞. 2.94
Ifφ:0,∞ → Êis convex and increasing, then the inequality
R2
R1
urφ
Γn−ν1 r−R1n−ν
∂ν
∗R1fx
∂rν dr≤ R2 R1
vtφ∂
nftω
∂rn
dt 2.95
holds.
Proof. ApplyTheorem 2.1withΩ1 Ω2 R1, R2,dμ1x dr,dμ2y dt, and
kr, t
⎧ ⎪ ⎨ ⎪ ⎩
r−tn−ν−1
Γn−ν , R1 ≤t≤r,
0, r < t≤R2.
2.96
Then replacefxby∂nftω/∂rn, so2.95follows.
Remark 2.32. In particular for the weight functionur r−R1n−ν,r ∈R1, R2, we obtain
the following inequality:
R2
R1
r−R1n−νφ
Γn−ν1 r−R1n−ν
∂ν
∗R1fx
∂rν dr≤ R2 R1
R2−tn−νφ
∂nf∂rtωn
dt. 2.97
Letq >1 andφ:Ê → Êbe defined byφx x
q, then2.97becomes
Γn−ν1q R2
R1
r−R1n−ν1−q
∂ν
∗R1fx
∂rν q dr≤ R2 R1
R2−tn−ν
∂nf∂rtωn
qdt. 2.98
Sincer ∈R1, R2and1−qn−ν≤0, we obtain
R2 R1 ∂ν
∗R1fx
∂rν q dr ≤
R2−R1n−ν
Γn−ν1
qR2
R1
∂nf∂rtωn
qdt. 2.99
Taking power 1/qon both sides, we get
∂ν
∗R1fx
∂rν
q
≤ R2−R1n−ν
Γn−ν1
∂nf∂rtωn
q
Ifν0, then
fx q≤ R2−R1 n
Γn1
∂nf∂rtωn
q
. 2.101
Ifν∈Æ, then
∂ν∂rfνx
q
≤ R2−R1n−ν
Γn−ν1
∂nf∂rtωn
q
. 2.102
Now, we continue with theRiemann-Liouville radial fractional derivative of fof order β, but first we need to define the following: letBX stand for the Borel class on space X and
define the measureRNon0,∞,B0,∞by
RNΓ
Γr
N−1dr, anyΓ∈ B
0,∞. 2.103
Now, letf ∈L1A L1R1, R2×SN−1.
For a fixedω∈SN−1, we define
gωr:frω fx, 2.104
where
x∈A:B0, R2−B0, R1,
0< R1≤r≤R2, r|x|, ω x
r ∈S
N−1. 2.105
The above led to the following definition of Riemann-Liouville radial fractional derivative. For details see1, page 466. Letβ >0,m: β 1,f ∈L1A, andAis the spherical shell.
We define
∂βR
1fx
∂rβ D
β R1frω
⎧ ⎪ ⎨ ⎪ ⎩
1 Γm−β
∂ ∂r
mr
R1
r−tm−β−1ftωdt, ω∈SN−1−Kf,
0, ω∈Kf,
2.106
where
xrω∈A, r∈R1, R2, ω∈SN−1,
Kf:ω∈SN−1:f·ω/∈L 1
R1, R2,BR1,R2, RN
Ifβ0, define
∂βR
1fx
∂rβ :fx.
2.108
We call∂βR
1fx/∂r
βthe Riemann-Liouville radial fractional derivative offof orderβ.
The following result is given in1, page 466.
Lemma 2.33. Letν ≥ γ1,γ ≥ 0,n:ν,f : A → Êwithf ∈L1A. Assume thatf·ω ∈ ACnR1, R2, for everyω ∈ SN−1, and that ∂νR1f·ω/∂rν is measurable onR1, R2for every
ω ∈ SN−1. Also assume that there exists∂ν
R1frω/∂r
ν ∈
Êfor everyr ∈ R1, R2and for every ω∈SN−1, and∂ν
R1fω/∂r
νis measurable onA. Suppose that there existsM
1>0,
∂νR
1fω
∂rν
≤M1, for everyr, ω∈R1, R2×SN−1. 2.109
We suppose that∂jfR
1ω/∂rj0,j 0,1, . . . , n−1, for everyω∈SN−1, then
∂γR
1fx
∂rγ D
γ R1frω
1 Γν−γ
r
R1
r−tν−γ−1DRν
1f
tωdt 2.110
is valid for everyx∈A, that is, true for everyr∈R1, R2and for everyω∈SN−1,γ >0.
Corollary 2.34. Let ube a weight function onR1, R2. Let the assumption of theLemma 2.33be
satisfied, andDRγ
1frωdenotes the Riemann-Liouville radial fractional derivative off. Definevon R1, R2by
vt:ν−γ
R2
t
urr−t
ν−γ−1
r−R1ν−γ
dr <∞. 2.111
Ifφ:0,∞ → Êis convex and increasing, then the inequality
R2
R1
urφ
Γν−γ1 r−R1ν−γ
DRγ
1frω
dr≤
R2
R1
vtφDνR
1f
tωdt 2.112
holds.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 R1, R2,
kr, t
⎧ ⎪ ⎨ ⎪ ⎩
r−tν−γ−1
Γν−γ , R1≤t≤r,
0, r < t≤R2,
we get thatKr r−R1ν−γ/Γν−γ1. Replacef·byDRν1f·ω, and then from the above
Lemma 2.33, we getgr DγR
1frω. This will give us2.112.
Remark 2.35. In particular for the weight functionur r −R1ν−γ,r ∈ R1, R2in above
Corollary 2.34 and for φx xq, q > 1 we obtain, after some calculation, the following
inequality:
DRγ
1frω
q≤
R2−R1ν−γ
Γν−γ1 DRν
1ftω
q. 2.114
Ifγ0, then
frω q≤
R2−R1ν
Γν1 DνR
1ftω
q. 2.115
In the previous corollaries, we derived only inequalities over some subsets of Ê.
However, Theorem 2.1covers much more general situations. We conclude this paper with multidimensional fractional integrals. Such operations of fractional integration in the n -dimensional Euclidean spaceÊ
n,n ∈
Æ are natural generalizations of the corresponding
one-dimensional fractional integrals and fractional derivatives, being taken with respect to one or several variables.
Forx x1, . . . , xn∈Ê
n andα α
1, . . . , αn, we use the following notations:
Γα Γα1· · ·Γαn, a,b a1, b1× · · · ×an, bn, 2.116
and byx>a, we meanx1 > a1, . . . , xn> an.
Thepartial Riemann-Liouville fractional integralsof orderαk > 0 with respect to thekth
variablexkare defined by
Iαk akf
x 1
Γαk
xk
ak
fx1, . . . , xk−1, tk, xk1, . . . , xnxk−tkαk−1dtk, xk> ak, 2.117
Iαk bk−f
x 1
Γαk
bk
xk
fx1, . . . , xk−1, tk, xk1, . . . , xntk−xkαk−1dtk, xk< bk, 2.118
respectively. These definitions are valid for functionsfx fx1, . . . , xndefined forxk> ak
andxk< bk, respectively.
Next, we define themixed Riemann-Liouville fractional integralsof orderα >0 as
Iaαfx 1
Γα
x1
a1
· · ·
xn
an
ftx−tα−1dt, x>a,
Ibα−fx 1
Γα
b1
x1
· · ·
bn
xn
ftt−xα−1dt, x<b.
Corollary 2.36. Let u be a weight function ona,b and α > 0. Iα
af denotes the mixed partial
Riemann-Liouville fractional integral off. Definevona,bby
vy:α
b1
y1
· · ·
bn
yn
uxx−y
α−1
x−aα dx<∞. 2.120
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b1
a1
· · ·
bn
an uxφ
Γα1
x−aαIaαfx dx≤
b1
a1
· · ·
bn
an
vyφfydy 2.121
holds for all measurable functionsf:a,b → Ê.
Proof. ApplyingTheorem 2.1withΩ1 Ω2 a,b,
kx,y
⎧ ⎪ ⎨ ⎪ ⎩
x−yα−1
Γα , a≤y≤x,
0, x<y≤b,
2.122
we get thatKx x−aα/Γα1andgx Iα
afx, so2.121follows.
Corollary 2.37. Let ube a weight function on a,band α > 0. Ibα
−f denotes the mixed partial
Riemann-Liouville fractional integral off. Definevona,bby
vy:α
y1
a1
· · ·
yn
an
uxy−x
α−1
b−xα dx<∞. 2.123
Ifφ:0,∞ → Êis convex and increasing function, then the inequality
b1
a1
· · ·
bn
an uxφ
Γα1
b−xαIbα−fx
dx≤
b1
a1
· · ·
bn
an
vyφfydy 2.124
holds for all measurable functionsf:a,b → Ê.
Remark 2.38. Analogous to Remarks 2.3 and 2.5, we obtain multidimensional version of inequality1.3forq >1 as follows:
b1
a1
· · ·
bn
an Iα
afx
gdx≤b−aα
Γα1
qb1
a1
· · ·
bn
an
fyqdy,
b1
a1
· · ·
bn
an
Ibα−fxgdx≤
b−aα
Γα1
qb1
a1
· · ·
bn
an
fyqdy.
Acknowledgments
The authors would like to express their gratitude to professor S. G. Samko for his help and suggestions and also to the careful referee whose valuable advice improved the final version of this paper. The research of the second and third authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grant no. 117-1170889-0888.
References
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