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# On an Inequality of H. G. Hardy

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Volume 2010, Article ID 264347,23pages doi:10.1155/2010/264347

## On an Inequality of H. G. Hardy

1

2

### and Josip Peˇcari´c

1, 2

1Abdus 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 uos2000@yahoo.com Received 18 June 2010; Accepted 16 October 2010

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 gCma, bwithgm−1ACa, b. For anyαÊ, we denote byαthe integral part ofαthe integerksatisfyingkα < k1, andαis the ceiling ofαmin{n∈Æ, nα}. ByL1a, b, we denote the space of all functions integrable on the intervala, b, and byLa, bthe set of all functions measurable and essentially bounded ona, b. Clearly,La, b⊂L1a, b.

We start with the definition of the Riemann-Liouville fractional integrals, see 3. Let a, b, −∞ < a < b < ∞ be a finite interval on the real axis Ê. The Riemann-Liouville fractional integralsIaαfandIbαfof orderα >0 are defined by

Iaαf

x 1 Γα

x

a

ftxtα−1dt, x > a, 1.1

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Ibα

f

x 1 Γα

b

x

fttxα−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 operatorsIaαfandIbαf of orderα >0, see also4. The first result yields that the fractional integral operatorsIaαf andIbα

fare bounded inLpa, b, 1≤p≤ ∞, that is

Iaαf pK f p, Ibαf pK f p, 1.3

where

K b−aα

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 k1×Ω2Êbe a nonnegative function, and

Kx

Ω2

k x, y

2

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 functionsg1Êwith the representation

gx

Ω2

k x, y

f y

2

y

, 2.2

wheref2Êis a measurable function.

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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 functionxuxkx, y/Kxis integrable onΩ1for each fixedy∈Ω2. DefinevonΩ2by

v y

:

Ω1

uxk x, y

Kx 1x<∞. 2.3

Ifφ:0,∞ → Êis convex and increasing function, then the inequality

Ω1

uxφ

gx Kx

1x≤

Ω2

v y

φf

ydμ2

y

2.4

holds for all measurable functionsf2Êand for all functionsgUk.

Proof. By using Jensen’s inequality and the Fubini theorem, sinceφis increasing function, we find that

Ω1

uxφ

gx Kx

1x

Ω1

uxφ

1 Kx

Ω2

k x, y

f y

2 y

1x

Ω1

ux Kx

Ω2

k x, y

φf

ydμ2 y

1x

Ω2

φf

y

Ω1

uxk x, y Kx 1x

2

y

Ω2

v y

φf

ydμ2

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.Iaαf denotes the Riemann-Liouville fractional integral off. Definevona, bby

v y

:α b

y

ux

xyα−1

x−aα dx <∞. 2.6

Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ

Γα1

x−aαIaαfx dxb

a

v y

φf

ydy 2.7

holds.

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Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩

xyα−1

Γα , ayx, 0, x < yb,

2.8

we get thatKx xaα/Γα1andgx Iaαfx, so2.7follows.

Remark 2.3. In particular for the weight functionux xaα, x∈a, binCorollary 2.2, we obtain the inequality

b

a

x−aαφ

Γα1

x−aαIaαfx dxb

a

byα φf

ydy. 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:ÊÊbe defined byφx xq, then2.9reduces to

b

a

x−aα

Γα1

x−aαIaαfx q dxb

a

byαfyqdy. 2.10

Sincex∈a, bandα1q<0, then we obtain that the left hand side of2.10is b

a

x−aα

Γα1

x−aα|Iaαfx| qdx≥b−aα1−qΓα1q b

a

Iaαfxqdx 2.11

and the right-hand side of2.10is b

a

byαfyqdy≤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.Ibα

f denotes the Riemann-Liouville fractional integral off. Definevona, bby

v y

:α y

a

ux

yxα−1

b−xα dx <∞. 2.14

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Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ

Γα1 b−xαIbα

fx dxb

a

v y

φf

ydy 2.15

holds.

Proof. Similar to the proof ofCorollary 2.2.

Remark 2.5. In particular for the weight functionux bxα, x∈a, binCorollary 2.4, we obtain the inequality

b

a

b−xαφ

Γα1

b−xαIbαfx dxb

a

yaα φf

ydy. 2.16

Letq >1 and the functionφ:ÊÊbe defined byφx xq, then2.16reduces to b

a

b−xα

Γα1

b−xαIbαfx qdxb

a

yaαfyqdy. 2.17

Sincex∈a, bandα1q<0, then we obtain that the left hand side of2.17is b

a

b−xα

Γα1

b−xαIbαfx qdx≥b−aα1−qΓα1q b

a

Ibαfxqdx 2.18

and the right-hand side of2.17is b

a

yaαfyqdy≤b−aα b

a

fyqdy. 2.19

Combining2.18and2.19, we get b

a

Ibαfxqdx

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,Iaαf andIbα

fdenote the Riemann-Liouville fractional integral off, then the following inequalities

b

a

IaαfxqdxC b

a

f

yqdy, 2.21

b

a

IbαfxqdxC b

a

fyqdy 2.22 hold, whereC b−a/Γαqqαpα−1 1q−1.

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Proof. We will prove only inequality2.21, since the proof of2.22is analogous. We have Iaαf

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 −1 11/p

x

a

|ft|qdt

1/q

≤ 1 Γα

x−aα−11/p −1 11/p

b

a

|ft|qdt 1/q

.

2.24

Thus, we have Iaαf

x≤ 1 Γα

x−aα−11/p −1 11/p

b

a

|ft|qdt 1/q

, for everyx∈a, b. 2.25

Consequently, we find

Iaαfxq≤ 1 Γαq

x−aqα−1q/p −1 1q/p

b

a

ftqdt

, 2.26

and we obtain b

a

Iaαfxqdx≤ b−aqα−1q/p1

Γαq

−1 q/p1

−1 1q/p b

a

ftqdt. 2.27

Remark 2.7. Forα≥1, inequalities2.21and2.22are refinements of1.3since

−1 1q−1

q> 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.

Next, we give results with respect to the generalized Riemann-Liouville fractional derivative. Let us recall the definition, for details see1, page 448.

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We define the generalized Riemann-Liouville fractional derivative off of orderα >0 by

Daαfx 1 Γn−α

d dx

nx

a

xyn−α−1 f

y

dy, 2.29

wheren α 1, x∈a, b.

Fora, bÊ, we say thatfL1a, bhas anLfractional derivativeDαaf α > 0in a, b, if and only if

1Daα−kfCa, b,k1, . . . , n α 1, 2Daα−1fACa, b,

3DaαLa, b.

Next, lemma is very useful in the upcoming corollarysee1, page 449and2.

Lemma 2.8. Letβ > α0 and letfL1a, bhave anLfractional derivativeDβafina, band let

Daβ−kfa 0, k1, . . . , β

1, 2.30

then

Daαfx 1 Γ

βα x

a

xyβ−α−1 Dβaf

y

dy, 2.31

for allaxb.

Corollary 2.9. Letube a weight function ona, b, and let assumptions inLemma 2.8be satisfied.

Definevona, bby

v y

: βα

b y

ux

xyβ−α−1

x−aβ−α dx <∞. 2.32

Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ Γ

βα1

x−aβ−α Daαfx dx

b

a

v y

φDaβf y

dy 2.33

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩

xyβ−α−1 Γ

βα , ayx, 0, x < yb,

2.34

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we get thatKx xaβ−α/Γβα1. ReplacefbyDaβf. Then, byLemma 2.8, gx Dαafxand 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 Γ

βα1

x−aβ−α Dαafx dx

b

a

byβ−α

φDaβf y

dy. 2.35

Letq > 1 and the functionφ:ÊÊbe defined byφx xq, then after some calculation, we obtain

b

a

Dαafxqdx

b−aβ−α Γβ−α1

qb

a

Dβafyqdy. 2.36

Next, we define Canavati-type fractional derivativeν-fractional derivative off, for details see1, page 446. We consider

Cνa, b

fCna, b:Ian−ν1fnC1a, b

, 2.37

ν >0, n ν. LetfCνa, b. We define the generalizedν-fractional derivative offover a, bas

Dνaf

Ian−ν1fn

, 2.38

the derivative with respect tox.

Lemma 2.11. Let νγ 1, whereγ0 and fCνa, b. Assume that fia 0, i 0,1, . . . ,ν−1, then

Dγaf

x 1 Γ

νγ x

a

x−tν−γ−1 Dνaf

tdt, 2.39

for allx∈a, b.

Corollary 2.12. Letube a weight function ona, b, and let assumptions inLemma 2.11be satisfied.

Definevona, bby

v y

: νγ

b y

ux

xyν−γ−1

x−x0ν−γ dx <∞. 2.40

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Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ Γ

νγ1

x−aν−γ Dγafx

dxb

a

v y

φDνaf

ydy 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 Γ

νγ1

x−aν−γ Daγfx

dxb

a

byν−γ

φDaνf

ydy. 2.42

Letq >1 and the functionφ:ÊÊbe defined byφx xq, then2.42reduces to Γ

νγ1qb

a

x−aν−γ1−qDaγfxqdxb

a

byν−γDaνfyqdy. 2.43

Sincex∈a, bandν−γ1q≤0, then we obtain b

a

Dγafxqdx

b−aν−γ Γν−γ1

qb

a

Daνfyqdy. 2.44

Taking power 1/qon both sides of2.44, we obtain

Dγafx q≤ b−aν−γ Γ

νγ1 Daνf y

q. 2.45

Whenγ0, we find that

Γν1q b

a

x−aν1−qfxqdxb

a

byνDνaf

yqdy, 2.46

that is,

f q≤ b−aν Γν1 Daνf

y

q. 2.47

In the next corollary, we give results with respect to the Caputo fractional derivative. Let us recall the definition, for details see1, page 449.

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Letα≥0,nα,gACna, b. The Caputo fractional derivative is given by

D∗aαgt 1 Γnα

x

a

gn y

xyα−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

v y

: n−α b

y

ux

xyn−α−1

x−an−α dx <∞. 2.49

Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ

Γnα1

x−an−α D∗aαgx dxb

a

v y

φgn y

dy 2.50

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩

xyn−α−1

Γnα , ayx, 0, x < yb,

2.51

we get thatKx xan−α/Γnα1. Replacefbygn, sogbecomesD∗aαgand2.50 follows.

Remark 2.15. In particular for the weight function ux xan−α, x ∈ a, b in Corollary 2.14, we obtain the inequality

b

a

x−an−αφ

Γn−α1

x−an−α Dα∗agx dxb

a

byn−α

φgn y

dy. 2.52

Letq > 1 and the functionφ:ÊÊbe defined byφx xq, 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−an−α Γnα1 gn

y

q. 2.54

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Theorem 2.16. Letp, q > 1, 1/p1/q 1,nα >1/q,D∗aαfxdenotes the Caputo fractional derivative off, then the following inequality

b

a

Dα∗afxqdx≤ b−aqn−α Γn−αq

pnα−1 1q/p

qnα b

a

fnyqdy 2.55

holds.

Proof. Similar to the proof ofTheorem 2.6.

The following result is given1, page 450.

Lemma 2.17. Letαγ1,γ >0, andnα. Assume thatfACna, bsuch thatfka 0, k0,1, . . . , n−1, andD∗aαfLa, b, thenDγ∗afCa, b,and

Dγ∗afx 1 Γ

αγ x

a

xyα−γ−1 D∗aαf

y

dy, 2.56

for allaxb.

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

v y

: αγ

b y

ux

xyα−γ−1

x−aα−γ dx <∞. 2.57

Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ Γ

αγ1 x−aα−γ

Dγ∗afx

dxb

a

v y

φDα∗af

ydy 2.58

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩

xyα−γ−1 Γ

αγ , ayx, 0, x < yb,

2.59

we get thatKx xaα−γ/Γαγ1. ReplacefbyD∗aαf, sogbecomesDγ∗afand2.58 follows.

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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 Γ

αγ1 x−aα−γ

Dγ∗afx

dxb

a

byα−γ

φDα∗af

ydy. 2.60

Letq > 1 and the functionφ:ÊÊbe defined byφx xq, then after some calculation, we obtain

b

a

Dγ∗afxqdx

b−aα−γ Γα−γ1

qb

a

Dα∗afyqdy. 2.61

Forγ0, we obtain b

a

fxqdx

b−aα Γα1

qb

a

Dα∗afyqdy. 2.62 We continue with definitions and some properties of the fractional 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

Ia;gα f

x 1 Γα

x

a

gtftdt

gxgt1−α, x > a, 2.63

Ib−;gα f

x 1 Γα

b

x

gtftdt

gtgx1−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. Iaα;gf denotes the left-sided fractional integral of a functionfwith respect to another functiongina, b. Definevona, bby

v y

:αg y

b y

ux

gxg yα−1

gxgaα dx <∞. 2.65

Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ

Γα1

gxgaαIaα;gfx

dxb

a

v y

φf

ydy 2.66

holds.

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Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩ 1 Γα1

g y gxg

y1−α, ayx,

0, x < yb,

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

gx

gx−gaα φ

Γα1

gxgaαIaα;gfx

dx

b

a

g y

gbg yα

φf

ydy.

2.68

Letq >1 and the functionφ:ÊÊbe defined byφx xq, then2.68reduces to

Γα1q b

a

gx

gxgaα1−qIaα;gfxqdx

b

a

g y

gbg

yαfyqdy.

2.69

Sincex∈a, bandα1q<0,gis increasing, thengx−gaα1−q >gb−gaα1−q andgb−gyα<gb−gaαand we obtain

b

a

gxIaα;gfxqdx

gbgaα Γα1

qb

a

g

yfyqdy. 2.70

Remark 2.22. If gx x, then Iaα;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

v y

:αg y

y a

ux g

y

gxα−1

gbgxα dx <∞. 2.71

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Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ

Γα1

gb−gxαIbα;gfx

dxb

a

v y

φf

ydy 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

gbgaα

Γα1

qb

a

g

yfyqdy. 2.73

Remark 2.25. If gx x, then Ibα

;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,Iaα;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≤

gbgaαq

αqΓαq

−1 1q/p b

a

f yqg

y dy,

b

a

Ibα;gfxqgxdx≤

gb−gaαq

αqΓαq

−1 1q/p b

a

f yqg

y dy

2.74

hold.

We continue by defining Hadamard 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

Jaα f

x 1 Γα

x

a

logx

y

α−1f y

dy

y , x > a, 2.75

Jb−αf

x 1 Γα

b

x

logy

x α−1f

y dy

y , x < b, 2.76

respectively.

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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

Jaα fxqdx x

logb/aα Γα1

qb

a

f yqdy

y , 2.77

and2.73becomes b

a

Jb−αf

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

Jaαfxqdx x

logb/a

qαΓαq

−1 1q/p b

a

f yqdy

y , b

a

Jb−αfxqdx x

logb/a qαΓαq

−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 operatorsJaαfandJb−αfare bounded inLpa, bas follows:

Jaαf pK1 f p, Jb−αf pK2 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 the Erd´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, andηÊ. We consider the left- and right-sided integrals of orderαÊdefined by

Iaα;σ;ηf

x σx−σαη Γα

x

a

tσησ−1ftdt

xσtσ1−α , 2.82

Ibα;σ;ηf

x σxση Γα

b

x

tσ1−η−α−1ftdt

tσxσ1−α , 2.83 respectively. Integrals2.82and2.83are called the Erd´elyi-Kober type fractional integrals.

(16)

Corollary 2.27. Let u be a weight function on a,b, 2F1a, b;c;z denotes the hypergeometric function, andIaα;σ;ηfdenotes the Erd´elyi-Kober type fractional left-sided integral. Definevby

v y

:ασyσησ−1 b

y

ux x−ση

xσyσα−1 xσaσα 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/xσIaα;σ;ηfx

dx

b

a

v y

φf ydy

2.85

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩ 1 Γα

σx−σαη

xσyσ1−αyσησ−1, ayx,

0, x < yb,

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−1xσaσα2F1x where 2F1x 2F1α,−η;α1; 1−a/xσinCorollary 2.27, we obtain the inequality

b

a

xσ−1xσaσα 2F1

Γα1 1−a/xσα

2F1xIaα;σ;ηfx

dx

b

a

yσ−1

bσyσα

2F1

y φf

ydy,

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

v y

:ασyσ1−α−η−1 y

a

ux xσηα

yσxσα−1 bσxσα 2F1

α, αη;α1; 1−b/xσdx <∞. 2.88

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Ifφ:0,∞ → Êis convex and increasing function, then the inequality b

a

uxφ

Γα1 b/xσ−1α

2F1

α, αη;α1; 1−b/xσIbα;σ;ηfx

dx

b

a

v y

φf ydy

2.89

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,1x dx,dμ2y dy,

k x, y

⎧⎪

⎪⎩ 1 Γα

σxση

yσxσ1−αyσ1−α−η−1, x < yb,

0, ayx,

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−1bσxσα2F1x where 2F1x 2F1α, αη;α1; 1−b/xσinCorollary 2.29, we obtain the inequality

b

a

xσ−1bσxσα 2F1

Γα1 b/xσ−1α

2F1x

Ibα;σ;ηfx

dx

b

a

yσ−1

yσaσα

2F1

y φf

ydy,

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 : AÊ, ν ≥ 0,n : ν, such thatf·ωACnR1, R2, for allωSN−1, whereA R1, R2×SN−1 forNÆ andSN−1 : {x ∈ÊN : |x| 1}. We call the Caputo radial fractional derivative as the following function:

ν∗R

1fx

∂rν : 1

Γnν r

R1

r−tn−ν−1nftω

∂rn dt, 2.92

wherexA, that is,xrω,r∈R1, R2,ωSN−1. Clearly,

0∗R

1fx

∂r0 fx,

ν∗R

1fx

∂rν νfx

∂rν if νÆ, the usual radial derivative.

2.93

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