On an Inequality of H. G. Hardy

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

Research Article

On an Inequality of H. G. Hardy

Sajid Iqbal,

¹

Kristina Kruli´c,

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

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 ≤ ∞. ByC^ma, 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 AC^ma, b, we denote the space of all functions g ∈ C^ma, bwithg^m−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, see 3. Let a, b, −∞ < a < b < ∞ be a finite interval on the real axis ^Ê. The Riemann-Liouville fractional integralsI_a^αfandI_b^α₋fof orderα >0 are defined by

I_a^αf

x 1 Γα

_x

a

ftx−t^α−1dt, x > a, 1.1

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I_b^α

−f

x 1 Γα

_b

x

ftt−x^α−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 operatorsI_a^αfandI_b^α₋f of orderα >0, see also4. The first result yields that the fractional integral operatorsI_a^αf andI_b^α

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

I_a^αf p≤K f p, I_b^α₋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

k x, y

dμ2

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

k x, y

f y

dμ2

y

, 2.2

wheref:Ω2 → ^Êis 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 functionx →uxkx, y/Kxis integrable onΩ1for each fixedy∈Ω2. DefinevonΩ2by

v y

:

Ω1

uxk x, y

Kx dμ1x<∞. 2.3

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

Ω1

uxφ

gx Kx

dμ1x≤

Ω2

v y

φf

ydμ2

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

uxφ

gx Kx

dμ₁x

Ω1

uxφ

1 Kx

Ω2

k x, y

f y

dμ₂ y

dμ₁x

≤

Ω1

ux Kx

Ω2

k x, y

φf

ydμ₂ y

dμ₁x

Ω2

φf

y

Ω1

uxk x, y Kx dμ₁x

dμ₂

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.I_a^αf denotes the Riemann-Liouville fractional integral off. Definevona, bby

v y

:α _b

y

ux

x−y_α−1

x−a^α dx <∞. 2.6

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

a

uxφ

Γα1

x−a^αI_a^αfx dx≤ _b

a

v y

φf

ydy 2.7

holds.

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Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ1x dx,dμ2y dy,

k x, y

⎧⎪

⎨

⎪⎩

x−y_α−1

Γα , a≤y≤x, 0, x < y≤b,

2.8

we get thatKx x−a^α/Γα1andgx I_a^αfx, so2.7follows.

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

_b

a

x−a^αφ

Γα1

x−a^αI_a^αfx dx≤ _b

a

b−y_α 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:^Ê → ^Êbe defined byφx x^q, then2.9reduces to

_b

a

x−a^α

Γα1

x−a^αI_a^αfx ^q dx≤ _b

a

b−yαfy^qdy. 2.10

Sincex∈a, bandα1−q<0, then we obtain that the left hand side of2.10is _b

a

x−a^α

Γα1

x−a^α|I_a^αfx| ^qdx≥b−a^α1−qΓα1^q _b

a

I_a^αfx^qdx 2.11

and the right-hand side of2.10is _b

a

b−y_αfy^qdy≤b−a^α _b

a

fy^qdy. 2.12

Combining2.11and2.12, we get _b

a

I_a^αfx^qdx≤

b−a^α Γα1

q_b

a

fy^qdy. 2.13

Taking power 1/qon both sides, we obtain1.3.

Corollary 2.4. Letube a weight function ona, bandα > 0.I_b^α

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

v y

:α _y

a

ux

y−x_α−1

b−x^α dx <∞. 2.14

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a

uxφ

Γα1 b−x^αI_b^α

−fx dx≤ _b

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 b−x^α, x∈a, binCorollary 2.4, we obtain the inequality

_b

a

b−x^αφ

Γα1

b−x^αI_b^α₋fx dx≤ _b

a

y−a_α φf

ydy. 2.16

Letq >1 and the functionφ:^Ê → ^Êbe defined byφx x^q, then2.16reduces to _b

a

b−x^α

Γα1

b−x^αI_b^α₋fx ^qdx≤ _b

a

y−a_αfy^qdy. 2.17

Sincex∈a, bandα1−q<0, then we obtain that the left hand side of2.17is _b

a

b−x^α

Γα1

b−x^αI_b^α₋fx ^qdx≥b−a^α1−qΓα1^q _b

a

I_b^α₋fx^qdx 2.18

and the right-hand side of2.17is _b

a

y−a_αfy^qdy≤b−a^α _b

a

fy^qdy. 2.19

Combining2.18and2.19, we get _b

a

I_b^α₋fx^qdx≤

b−a^α Γα1

q_b

a

fy^qdy. 2.20

Taking power 1/qon both sides, we obtain1.3.

Theorem 2.6. Letp, q >1, 1/p1/q1,α > 1/q,I_a^αf andI_b^α

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

_b

a

I_a^αfx^qdx≤C _b

a

f

y^qdy, 2.21

_b

a

I_b^α₋fx^qdx≤C _b

a

fy^qdy 2.22 hold, whereC b−a^qα/Γα^qqαpα−1 1^q−1.

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

x≤ 1 Γα

_x

a

ftx−t^α−1dt. 2.23

Then by the H ¨older inequality, the right-hand side of the above inequality is

≤ 1 Γα

_x

a

x−t^pα−1dt

1/p_x

a

|ft|^qdt

1/q

1 Γα

x−a^α−11/p pα−1 1_1/p

_x

a

|ft|^qdt

1/q

≤ 1 Γα

x−a^α−11/p pα−1 1_1/p

_b

a

|ft|^qdt _1/q

.

2.24

Thus, we have I_a^αf

x≤ 1 Γα

x−a^α−11/p pα−1 1_1/p

_b

a

|ft|^qdt _1/q

, for everyx∈a, b. 2.25

Consequently, we find

I_a^αfx^q≤ 1 Γα^q

x−a^qα−1q/p pα−1 1_q/p

_b

a

ft^qdt

, 2.26

and we obtain _b

a

I_a^αfx^qdx≤ b−a^qα−1q/p1

Γα^q

qα−1 q/p1

pα−1 1_q/p _b

a

ft^qdt. 2.27

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

qα

pα−1 1_q−1

≥qα^q> α^q, so C <

b−a^α 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

D_a^αfx 1 Γn−α

d dx

n_x

a

x−y_n−α−1 f

y

dy, 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

1D_a^α−kf ∈Ca, b,k1, . . . , n α 1, 2D_a^α−1f ∈ACa, b,

3D_a^α∈L_∞a, b.

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

Lemma 2.8. Letβ > α≥0 and letf ∈L1a, bhave anL_∞fractional derivativeD^β_afina, band let

D_a^β−kfa 0, k1, . . . , β

1, 2.30

then

D_a^αfx 1 Γ

β−α _x

a

x−y_β−α−1 D^β_af

y

dy, 2.31

for alla≤x≤b.

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

Definevona, bby

v y

: β−α

b y

ux

x−y_β−α−1

x−a^β−α dx <∞. 2.32

a

uxφ Γ

β−α1

x−a^β−α D_a^αfx dx≤

_b

a

v y

φDa^βf y

dy 2.33

holds.

k x, y

⎧⎪

⎨

⎪⎩

x−y_β−α−1 Γ

β−α , a≤y≤x, 0, x < y≤b,

2.34

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we get thatKx x−a^β−α/Γβ−α1. ReplacefbyD_a^βf. Then, byLemma 2.8, gx D^α_afxand we get2.33.

Remark 2.10. In particular for the weight functionux x−a^β−α,x∈a, binCorollary 2.9, we obtain the inequality

_b

a

x−a^β−αφ Γ

β−α1

x−a^β−α D^α_afx dx≤

_b

a

b−y_β−α

φDa^βf y

dy. 2.35

Letq > 1 and the functionφ:^Ê → ^Êbe defined byφx x^q, then after some calculation, we obtain

_b

a

D^α_afx^qdx≤

b−a^β−α Γβ−α1

_q_b

a

D^βafy^qdy. 2.36

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

C^νa, b

f∈Cⁿa, b:I_a^n−ν1fⁿ∈C¹a, b

, 2.37

ν >0, n ν. Letf ∈C^νa, b. We define the generalizedν-fractional derivative offover a, bas

D^ν_af

I_a^n−ν1fⁿ

, 2.38

the derivative with respect tox.

Lemma 2.11. Let ν ≥ γ 1, whereγ ≥ 0 and f ∈ C^νa, b. Assume that fⁱa 0, i 0,1, . . . ,ν−1, then

D^γ_af

x 1 Γ

ν−γ _x

a

x−t^ν−γ−1 D^ν_af

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

: ν−γ

b y

ux

x−y_ν−γ−1

x−x0^ν−γ dx <∞. 2.40

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a

uxφ Γ

ν−γ1

x−a^ν−γ D^γafx

dx≤ _b

a

v y

φD^ν_af

ydy 2.41

holds.

Proof. Similar to the proof ofCorollary 2.9.

Remark 2.13. In particular for the weight functionux x−a^ν−γ,x∈a, binCorollary 2.12, we obtain the inequality

_b

a

x−a^ν−γφ Γ

ν−γ1

x−a^ν−γ D_a^γfx

dx≤ _b

a

b−y_ν−γ

φD_a^νf

ydy. 2.42

Letq >1 and the functionφ:^Ê → ^Êbe defined byφx x^q, then2.42reduces to Γ

ν−γ1q_b

a

x−a^ν−γ1−qDa^γfx^qdx≤ _b

a

b−y_ν−γD_a^νfy^qdy. 2.43

Sincex∈a, bandν−γ1−q≤0, then we obtain _b

a

D^γ_afx^qdx≤

b−a^ν−γ Γν−γ1

_q_b

a

D_a^νfy^qdy. 2.44

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

D^γ_afx q≤ b−a^ν−γ Γ

ν−γ1 D_a^νf y

q. 2.45

Whenγ0, we find that

Γν1^q _b

a

x−a^ν1−qfx^qdx≤ _b

a

b−yνD^ν_af

y^qdy, 2.46

that is,

f q≤ b−a^ν Γν1 D_a^νf

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,g∈ACⁿa, b. The Caputo fractional derivative is given by

D_∗a^αgt 1 Γn−α

_x

a

gⁿ y

x−y_α−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.D^α_∗agdenotes the Caputo fractional derivative ofg. Definevona, bby

v y

: n−α _b

y

ux

x−y_n−α−1

x−a^n−α dx <∞. 2.49

a

uxφ

Γn−α1

x−a^n−α D_∗a^αgx dx≤ _b

a

v y

φgⁿ y

dy 2.50

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ₁x dx,dμ₂y dy,

k x, y

⎧⎪

⎨

⎪⎩

x−y_n−α−1

Γn−α , a≤y≤x, 0, x < y≤b,

2.51

we get thatKx x−a^n−α/Γn−α1. Replacefbygⁿ, sogbecomesD_∗a^αgand2.50 follows.

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

_b

a

x−a^n−αφ

Γn−α1

x−a^n−α D^α_∗agx dx≤ _b

a

b−y_n−α

φgⁿ y

dy. 2.52

_b

a

D^α_∗agx^q dx≤

b−a^n−α Γn−α1

q_b

a

gⁿy^qdy. 2.53

Taking power 1/qon both sides, we obtain

D^α_∗agx q≤ b−a^n−α Γn−α1 gⁿ

y

q. 2.54

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Theorem 2.16. Letp, q > 1, 1/p1/q 1,n−α >1/q,D_∗a^αfxdenotes the Caputo fractional derivative off, then the following inequality

_b

a

D^α_∗afx^qdx≤ b−a^qn−α Γn−α^q

pn−α−1 1_q/p

qn−α _b

a

fⁿy^qdy 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 thatf^ka 0, k0,1, . . . , n−1, andD_∗a^αf∈L_∞a, b, thenD^γ_∗af ∈Ca, b,and

D^γ_∗afx 1 Γ

α−γ _x

a

x−y_α−γ−1 D_∗a^αf

y

dy, 2.56

for alla≤x≤b.

Corollary 2.18. Letube a weight function ona, bandα >0.D^α_∗af denotes the Caputo fractional derivative off, and assumptions inLemma 2.17are satisfied. Definevona, bby

v y

: α−γ

b y

ux

x−y_α−γ−1

x−a^α−γ dx <∞. 2.57

a

uxφ Γ

α−γ1 x−a^α−γ

D^γ_∗afx

dx≤ _b

a

v y

φD^α_∗af

ydy 2.58

holds.

Proof. ApplyingTheorem 2.1withΩ1 Ω2 a, b,dμ₁x dx,dμ₂y dy,

k x, y

⎧⎪

⎨

⎪⎩

x−y_α−γ−1 Γ

α−γ , a≤y≤x, 0, x < y≤b,

2.59

we get thatKx x−a^α−γ/Γα−γ1. ReplacefbyD_∗a^αf, sogbecomesD^γ_∗afand2.58 follows.

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Remark 2.19. In particular for the weight functionux x−a^α−γ,x∈a, binCorollary 2.18, we obtain the inequality

_b

a

x−a^α−γφ Γ

α−γ1 x−a^α−γ

D^γ_∗afx

dx≤ _b

a

b−y_α−γ

φD^α_∗af

ydy. 2.60

_b

a

D^γ∗afx^qdx≤

b−a^α−γ Γα−γ1

_q_b

a

D^α_∗afy^qdy. 2.61

Forγ0, we obtain _b

a

fx^qdx≤

b−a^α Γα1

q_b

a

D^α_∗afy^qdy. 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 line^Êandα > 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

I_a;g^α f

x 1 Γα

_x

a

gtftdt

gx−gt_1−α, x > a, 2.63

I_b−;g^α f

x 1 Γα

_b

x

gtftdt

gt−gx_1−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. I_a^α_;gf denotes the left-sided fractional integral of a functionfwith respect to another functiongina, b. Definevona, bby

v y

:αg y

b y

ux

gx−g y_α−1

gx−gaα dx <∞. 2.65

a

uxφ

Γα1

gx−ga_αI_a^α_;gfx

dx≤ _b

a

v y

φf

ydy 2.66

holds.

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k x, y

⎧⎪

⎨

⎪⎩ 1 Γα1

g y gx−g

y_1−α, a≤y≤x,

0, x < y≤b,

2.67

we get thatKx 1/Γα1gx−ga^α, so2.66follows.

Remark 2.21. In particular for the weight functionux gxgx−ga^α,x ∈ a, bin Corollary 2.20, we obtain the inequality

_b

a

gx

gx−ga_α φ

Γα1

gx−ga_αI_a^α_;gfx

dx

≤ _b

a

g y

gb−g y_α

φf

ydy.

2.68

Letq >1 and the functionφ:^Ê → ^Êbe defined byφx x^q, then2.68reduces to

Γα1^q _b

a

gx

gx−ga_α1−qI_a^α_;gfx^qdx

≤ _b

a

g y

gb−g

yαfy^qdy.

2.69

Sincex∈a, bandα1−q<0,gis increasing, thengx−ga^α1−q >gb−ga^α1−q andgb−gy^α<gb−ga^αand we obtain

_b

a

gxI_a^α_;gfx^qdx≤

gb−ga_α Γα1

q_b

a

g

yfy^qdy. 2.70

Remark 2.22. If gx x, then I_a^α_;xfx reduces to I_a^αfx 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.I_b^α₋_;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

−gx_α−1

gb−gx_α dx <∞. 2.71

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a

uxφ

Γα1

gb−gx_αI_b^α₋_;gfx

dx≤ _b

a

v y

φf

ydy 2.72

holds.

Remark 2.24. In particular for the weight functionux gxgb−gx^α,x∈a, band for functionφx x^q,q >1, we obtain after some calculation

_b

a

gxI_b^α₋_;gfx^qdx≤

gb−gaα

Γα1

_q_b

a

g

yfy^qdy. 2.73

Remark 2.25. If gx x, then I_b^α

−;xfx reduces to I_b^α

−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,I_a^α_;gf and I_b^α

−;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

I_a^α_;gfx^qgxdx≤

gb−gaαq

αqΓα^q

pα−1 1_q/p _b

a

f y^qg

y dy,

_b

a

I_b^α₋_;gfx^qgxdx≤

gb−gaαq

αqΓα^q

pα−1 1_q/p _b

a

f y^qg

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-axis^Ê andα > 0.

The left- and right-sided Hadamard fractional integrals of orderαare given by

J_a^α f

x 1 Γα

_x

a

logx

y

α−1f y

dy

y , x > a, 2.75

J_b−^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

J_a^α fx^qdx x ≤

logb/a_α Γα1

q_b

a

f y^qdy

y , 2.77

and2.73becomes _b

a

J_b−^αf

x^qdx x ≤

logb/a_α Γα1

_q_b

a

f y^qdy

y . 2.78

Also, fromTheorem 2.26we obtain refinements of2.77and2.78, forα >1/q, _b

a

J_a^αfx^qdx x ≤

logb/aqα

qαΓα^q

pα−1 1_q/p _b

a

f y^qdy

y , _b

a

J_b−^αfx^qdx x ≤

logb/a_qα qαΓα^q

pα−1 1_q/p _b

a

f y^qdy

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 operatorsJ_a^αfandJ_b−^αfare bounded inLpa, bas follows:

J_a^αf _p≤K₁ f p, J_b−^αf p≤K₂ f p, 2.80

where

K1 1 Γα

_logb/a

0

t^α−1e^t/pdt, K2 1 Γα

_logb/a

0

t^α−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-axis^Ê. Also let α >0,σ >0, andη∈^Ê. We consider the left- and right-sided integrals of orderα∈^Êdefined by

I_a^α_;σ;ηf

x σx^−σαη Γα

_x

a

t^σησ−1ftdt

x^σ−t^σ^1−α , 2.82

I_b^α₋_;σ;ηf

x σx^ση Γα

_b

x

t^{σ1−η−α−1}ftdt

t^σ−x^σ^1−α , 2.83 respectively. Integrals2.82and2.83are called the Erd´elyi-Kober type fractional integrals.

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

v y

:ασy^σησ−1 _b

y

ux x^−ση

x^σ−y^σ_α−1 x^σ−a^σ^α 2F₁

α,−η;α1; 1−a/x^σdx <∞. 2.84

a

uxφ

Γα1 1−a/x^σα

2F1

α,−η;α1; 1−a/x^σI_a^α_;σ;ηfx

dx

≤ _b

a

v y

φf ydy

2.85

holds.

k x, y

⎧⎪

⎨

⎪⎩ 1 Γα

σx^−σαη

x^σ−y^σ_1−αy^σησ−1, a≤y≤x,

0, x < y≤b,

2.86

we get thatKx 1/Γα11−a/x^σ^α2F₁α,−η;α1; 1−a/x^σ, so2.85follows.

Remark 2.28. In particular for the weight function ux x^σ−1x^σ − a^σ^α2F1x where 2F1x 2F1α,−η;α1; 1−a/x^σinCorollary 2.27, we obtain the inequality

_b

a

x^σ−1x^σ−a^σ^α 2F₁xφ

Γα1 1−a/x^σ_α

2F1xI_a^α_;σ;ηfx

dx

≤ _b

a

y^σ−1

b^σ−y^σ_α

2F1

y φf

ydy,

2.87

where ₂F₁y 2F₁α, η;α1; 1−a/y^σ.

Corollary 2.29. Let u be a weight function on a, b, 2F1a, b;c;z denotes the hypergeometric function, andI_b^α₋_;σ;ηfdenotes the Erd´elyi-Kober type fractional right-sided integral. Definevby

v y

:ασy^{σ1−α−η−1} _y

a

ux x^σηα

y^σ−x^σ_α−1 b^σ−x^σ^α 2F1

α, αη;α1; 1−b/x^σdx <∞. 2.88

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a

uxφ

Γα1 b/x^σ−1_α

2F₁

α, αη;α1; 1−b/x^σI_b^α₋_;σ;ηfx

dx

≤ _b

a

v y

φf ydy

2.89

holds.

k x, y

⎧⎪

⎨

⎪⎩ 1 Γα

σx^ση

y^σ−x^σ_1−αy^{σ1−α−η−1}, x < y≤b,

0, a≤y≤x,

2.90

we get thatKx 1/Γα1b/x^σ−1^α 2F₁α, αη;α1; 1−b/x^σ, so2.89follows.

Remark 2.30. In particular for the weight function ux x^σ−1b^σ − x^σ^α2F1x where 2F1x 2F1α, αη;α1; 1−b/x^σinCorollary 2.29, we obtain the inequality

_b

a

x^σ−1b^σ−x^σ^α 2F1xφ

Γα1 b/x^σ−1_α

2F1x

I_b^α₋_;σ;ηfx

dx

≤ _b

a

y^σ−1

y^σ−a^σ_α

2F1

y φf

ydy,

2.91

where2F1y 2F1α,−α−η;α1; 1−b/y^σ.

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 ∈ ACⁿR1, R₂, for allω ∈ S^N−1, whereA R1, R2×S^N−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

∂r^ν : 1

Γn−ν _r

R1

r−t^n−ν−1∂ⁿftω

∂rⁿ dt, 2.92

wherex∈A, that is,xrω,r∈R1, R₂,ω∈S^N−1. Clearly,

∂⁰_∗R

1fx

∂r⁰ fx,

∂^ν_∗R

1fx

∂r^ν ∂^νfx

∂r^ν if ν∈^Æ, the usual radial derivative.

2.93