C:/Documents and Settings/Profesor/Escritorio/Cubo/Cubo/Cubo/Cubo 2011/Cubo2011-13-01/Vol13 n\2721/Art N\2605 .dvi


CUBO A Mathematical Journal
Vol.13, No¯ 01, (61–71). March 2011

q− Fractional Inequalities

George A. Anastassiou

Department of Mathematical Sciences,

University of Memphis,

Memphis, TN 38152, U.S.A.,

email: ganastss@gmail.com

ABSTRACT

Here we give q−fractional Poincaré type, Sobolev type and Hilbert-Pachpatte type

integral inequalities, involving q−fractional derivatives of functions. We give also their

generalized versions.

RESUMEN

Estudiamos el tipo q−fraccional Poincaré, el tipo Sobolev y el tipo integral de in-

ecuaciones de Hilbert-Pachpatte, involucrando a q−fraccional derivados de funciones.

Damos también las versiones generalizadas.

Keywords: q−fractional derivative, q−fractional integral, q−fractional Poincaré inequality, q−

fractional Sobolev inequality, q−fractio- nal Hilbert-Pachpatte inequality.

AMS Subject Classification: 26A24, 26A33, 26A39, 26D10, 26D15, 33D05, 33D60, 81P99.

1 Introduction

Here we follow [4] in all of this section, see also [3].



62 George A. Anastassiou CUBO
13, 1 (2011)

Let q ∈ (0, 1), we define

[α]
q

:=
1 − qα

1 − q
, (α ∈ R) . (1)

The q−analog of the Pochhammer symbol (q−shifted factorial) is defined by:

(a; q)0 = 1, (a; q)k =

k−1
∏

i=0

(

1 − aqi
)

(k ∈ N ∪ {∞}) .

The expansion to reals is

(a; q)
α

=
(a; q)∞

(aqα; q)∞
(α ∈ R) ; (2)

also define the q−analog

(a − b)
(α)

= aα
(

b
a
; q
)

∞
(

qα b
a
; q
)

∞

, a, b ∈ R, a 6= 0.

Notice that

(a − b)
(α)

= aα
(

b

a
; q

)

α

.

The q−gamma function is defined by

Γq (x) =
(q; q)∞
(qx; q)∞

(1 − q)
1−x

, (x ∈ R − {0, −1, −2, ...}) . (3)

Clearly

Γq (x + 1) = [x]q Γq (x) . (4)

The q−derivative of a function f (x) is defined by

(Dqf ) (x) =
f (x) − f (qx)

x − qx
, (x 6= 0) , (5)

(Dqf ) (0) = lim
x→0

(Dqf ) (x) , (6)

and the q−derivatives of higher order:

D
0
q f = f, D

n
q f = Dq

(

D
n−1
q f

)

, n = 1, 2, 3, ... (7)

The q−integral is defined by

(Iq,0f ) (x) =

∫ x

0

f (t) dqt = x (1 − q)

∞
∑

k=0

f
(

xq
k
)

q
k, (0 < q < 1) , (8)

and

(Iq,af ) (x) =

∫ x

a

f (t) dqt =

∫ x

0

f (t) dqt −

∫ a

0

f (t) dqt. (9)



CUBO
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q− Fractional Inequalities 63

By [2], we see that: if f (x) ≥ 0, then it is not necessarily true that

∫ b

a

f (x) dqx ≥ 0.

In the case of a = xqn, then (9) becomes

∫ x

xqn
f (t) dqt = x (1 − q)

n−1
∑

k=0

f
(

xq
k
)

q
k
, (10)

see also [2].

Double q−integration is defined the usual iterative way.

Also we define

I
0
q,af = f, I

n
q,af = Iq,a

(

I
n−1
q,a f

)

, n = 1, 2, 3, ... (11)

The following are valid:

(DqIq,af ) (x) = f (x) , (12)

(Iq,aDqf ) (x) = f (x) − f (a) . (13)

Denote

[n]
q
! = [1]

q
[2]

q
... [n]

q
, n ∈ N;

[0]
q
! = 1,

[

n

k

]

q

=
[n]

q
!

[k]
q
! [n − k]

q
!
.

In the next we work on (0, b), b > 0, and let a ∈ (0, b). Also the required q−derivatives and

q−integrals do exist.

Definition 1. The fractional q−integral is

(

I
α
q,af

)

(x) =
xα−1

Γq (α)

∫ x

a

(

q
t

x
; q

)

α−1

f (t) dqt (14)

=
1

Γq (α)

∫ x

a

(x − qt)
(α−1)

f (t) dqt,
(

a < x, α ∈ R+
)

.

The usual fractional integral (see also [1]) is the limit case of (14) as q ↑ 1, since

lim
q↑1

x
α−1

(

q
t

x
; q

)

α−1

= (x − t)
α−1

. (15)

Clearly
(

I
α
q,af

)

(a) = 0. (16)

We mention



64 George A. Anastassiou CUBO
13, 1 (2011)

Theorem 2. Let α, β ∈ R+. The q−fractional integration has the semigroup property

(

I
β
q,aI

α
q,af

)

(x) =
(

I
α+β
q,a f

)

(x) , (a < x) . (17)

Corollary 3. For α ≥ n (n ∈ N) it holds

(

D
n
q I

α
q,af

)

(x) =
(

I
α−n
q,a f

)

(x) , (a < x) . (18)

We mention the fractional q−derivative of Caputo type:

Definition 4. The fractional q−derivative of Caputo type is

(

∗D
α
q,af

)

(x) =







(

I−αq,a f
)

(x) , α ≤ 0;
(

I
⌈α⌉−α
q,a D

⌈α⌉
q f (x)

)

, α > 0,
(19)

where ⌈.⌉ denotes the ceiling of the number.

Next we mention the highlight of this introductory section. Again all here come from [4]. So

the following is the fractional q−Taylor formula of Caputo type.

Theorem 5. Let α ∈ R+ − N, a < x. Then

(

I
α
q,a ∗D

α
q,af

)

(x) = f (x) −

⌈α⌉−1
∑

k=0

(

Dkq f
)

(a)

[k]
q
!

x
k
(

a

x
; q
)

k
. (20)

Also we give

Theorem 6. Let α ∈ R+ − N, β ∈ R+, α > β > 0, a < x. Then

(

I
β
q,a ∗D

α
q,af

)

(x) =
(

∗D
α−β
q,a f

)

(x) − (21)

⌈α⌉−1
∑

k=⌈α−β⌉

(

Dkq f
)

(a)

Γq (k − α + β + 1)
x

k−α+β
(

a

x
; q
)

k−α+β
.

2 Main Results

We need the following q−Hölder’s inequality.

Proposition 7. Let x > 0, 0 < q < 1; p1, q1 > 1 such that
1
p1

+ 1
q1

= 1; n ∈ N. Then

∫ x

xqn
|f (t)| |g (t)| dqt ≤

(
∫ x

xqn
|f (t)|

p1 dqt

)
1

p
1

(
∫ x

xqn
|g (t)|

q1 dqt

)
1

q
1

. (22)

Proof. By the discrete Hölder’s inequality we have

∫ x

xqn
|f (t)| |g (t)| dqt = x (1 − q)

n−1
∑

k=0

∣

∣f
(

xq
k
)∣

∣

∣

∣g
(

xq
k
)∣

∣q
k =



CUBO
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q− Fractional Inequalities 65

x (1 − q)
n−1
∑

k=0

(

∣

∣f
(

xq
k
)∣

∣

(

q
k
)

1

p
1

)(

∣

∣g
(

xq
k
)∣

∣

(

q
k
)

1

q
1

)

≤

(

x (1 − q)

n−1
∑

k=0

∣

∣f
(

xq
k
)∣

∣

p1
q

k

)
1

p
1

(

x (1 − q)

n−1
∑

k=0

∣

∣g
(

xq
k
)∣

∣

q1
q

k

)
1

q
1

=

(
∫ x

xqn
|f (t)|

p1 dqt

)
1

p
1

(
∫ x

xqn
|g (t)|

q1 dqt

)
1

q
1

.

We present a q−fractional Poincaré type inequality.

Theorem 8. Let x > 0, 0 < w ≤ x, 0 < q < 1; α > 0, p1, q1 > 1 such that
1
p1

+ 1
q1

= 1; n ∈ N. Set

∆ (w) := f (w) −

⌈α⌉−1
∑

k=0

(

Dkq f
)

(wqn)

[k]
q
!

w
k (qn; q)

k
.

Then

∫ x

0

|∆ (w)|
q1

wq1(α−1)
dqw ≤

1

(Γq (α))
q1

·

(

∫ x

0

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)q1

dqw

)

1

p
1

·

(
∫ x

0

(
∫ w

wqn

∣

∣

∗D
α
q,wqn f (t)

∣

∣

q1
dqt

)q1

dqw

)
1

q
1

. (23)

Proof. By q−fractional Taylor’s formula (20) we get

∆ (w) =
(

I
α
q,wqn ∗D

α
q,wqn f

)

(w) =
wα−1

Γq (α)

∫ w

wqn

(

q
t

w
; q

)

α−1

(

∗D
α
q,wqn f

)

(t) dqt. (24)

Here by (14) and (19), we see that

(

∗D
α
q,wqn f

)

(t) =
t⌈α⌉−α−1

Γq (⌈α⌉ − α)

∫ w

wqn

(

q
s

t
; q
)

⌈α⌉−α−1
D

⌈α⌉
q f (s) dq (s) , (25)

all wqn ≤ t ≤ w.

Here we observe trivially that

∣

∣

∣

∣

∫ x

xqn
f (t) dqt

∣

∣

∣

∣

≤

∫ x

xqn
|f (t)| dqt. (26)

Furthermore we see that

(

q
t

w
; q

)

α−1

=

(

q t
w

; q
)

∞
(

qα t
w

; q
)

∞

=

∏∞
i=0

(

1 − q t
w

qi
)

∏∞
i=0

(

1 − qα t
w

qi
) =

∏∞
i=0

(

1 − t
w

qi+1
)

∏∞
i=0

(

1 − t
w

qi+α
) > 0. (27)



66 George A. Anastassiou CUBO
13, 1 (2011)

Hence by (22) we obtain

|∆ (w)| ≤
wα−1

Γq (α)

∫ w

wqn

(

q
t

w
; q

)

α−1

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣dqt ≤

wα−1

Γq (α)

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)
1

p
1

·

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)
1

q
1

. (28)

Consequently we derive

|∆ (w)|

wα−1
≤

1

Γq (α)

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)
1

p
1

· (29)

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)
1

q
1

,

and

|∆ (w)|
q1

wq1(α−1)
≤

1

(Γq (α))
q1

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)

q
1

p
1

· (30)

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)

.

Applying q−Hölder’s inequality (which is also valid on [0, x]) on (30), we see that

∫ x

0

|∆ (w)|
q1

wq1(α−1)
dqw ≤

1

(Γq (α))
q1
·

∫ x

0





(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)

q
1

p
1

·

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)



dqw

≤
1

(Γq (α))
q1

·

(

∫ x

0

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)q1

dqw

)

1

p
1

·

(
∫ x

0

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)q1

dqw

)
1

q
1

, (31)

proving the claim.

Next we give a q−fractional Sobolev type inequality.

Theorem 9. Here all terms and assumptions as in Theorem 8. Additionaly let r1, r2 > 1 :
1
r1

+ 1
r2

=

1. Then
(
∫ x

0

(

|∆ (w)|

wα−1

)r1

dqw

)
1

r
1

≤
1

Γq (α)
·



CUBO
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q− Fractional Inequalities 67







∫ x

0

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)

r
2

1

p
1

dqw







1

r2
1

·

(

∫ x

0

(
∫ w

wqn

∣

∣

∗D
α
q,wqn f (t)

∣

∣

q1
dqt

)

r
1

r
2

q
1

dqw

)
1

r
1

r
2

. (32)

Proof. As in the proof of Theorem 8 we get (29), so that

(

|∆ (w)|

wα−1

)r1

≤
1

(Γq (α))
r1

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)

r
1

p
1

· (33)

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)

r
1

q
1

.

Hence
∫ x

0

(

|∆ (w)|

wα−1

)r1

dqw ≤
1

(Γq (α))
r1
· (34)

∫ x

0





(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)

r
1

p
1

·

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)

r
1

q
1



dqw

(by q−Hölder’s inequality on [0, x])

≤
1

(Γq (α))
r1







∫ x

0

(

∫ w

wqn

(

(

q
t

w
; q

)

α−1

)p1

dqt

)

r
2

1

p
1

dqw







1

r
1

· (35)

(

∫ x

0

(
∫ w

wqn

∣

∣

(

∗D
α
q,wqn f

)

(t)
∣

∣

q1
dqt

)

r
1

r
2

q
1

dqw

)
1

r
2

,

proving the claim.

It follows a q−fractional Hilbert-Pachpatte type inequality.

Theorem 10. Let for i = 1, 2 that xi > 0, 0 < wi ≤ xi, 0 < q < 1; α > 0, p1, q1 > 1 such that
1
p1

+ 1
q1

= 1; n ∈ N. Call

∆i (wi) = fi (wi) −

⌈α⌉−1
∑

k=0

(

Dkq fi
)

(wiq
n)

[k]
q
!

w
k
i (q

n; q)
k
,

F (w1) =

∫ w1

w1q
n

(

q
t1

w1
; q

)p1

α−1

dqt1, (36)



68 George A. Anastassiou CUBO
13, 1 (2011)

and

G (w2) =

∫ w2

w2q
n

(

q
t2

w2
; q

)q1

α−1

dqt2.

Then
∫ x1

0

∫ x2

0

|∆1 (w1)| |∆2 (w2)|

(w1w2)
α−1

(

F (w1)
p1

+
G(w2)

q1

)dqw1dqw2 ≤ (37)

x
1

p
1

1 x
1

q
1

2

(Γq (α))
2

(
∫ x1

0

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)

dqw1

)
1

q
1

·

(
∫ x2

0

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)

dqw2

)
1

p
1

.

Proof. We notice by (20) that

∆i (wi) =
w

α−1
i

Γq (α)

∫ wi

wiq
n

(

q
ti

wi
; q

)

α−1

(

∗D
α
q,wiq

n fi
)

(ti) dqti, (38)

for i = 1, 2.

Therefore we derive

|∆1 (w1)| ≤
w

α−1
1

Γq (α)

∫ w1

w1q
n

(

q
t1

w1
; q

)

α−1

∣

∣

(

∗D
α
q,w1q

n f1
)

(t1)
∣

∣dqt1 ≤

w
α−1
1

Γq (α)

(

∫ w1

w1q
n

(

q
t1

w1
; q

)p1

α−1

dqt1

)
1

p
1

·

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)
1

q
1

. (39)

Similarly we obtain

|∆2 (w2)| ≤
w

α−1
2

Γq (α)

∫ w2

w2q
n

(

q
t2

w2
; q

)

α−1

∣

∣

(

∗D
α
q,w2q

n f2
)

(t2)
∣

∣dqt2 ≤

w
α−1
2

Γq (α)

(

∫ w2

w2q
n

(

(

q
t2

w2
; q

)

α−1

)q1

dqt2

)
1

q
1

·

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)
1

p
1

. (40)

Consequently we get

|∆1 (w1)| |∆2 (w2)| ≤
(w1w2)

α−1

(Γq (α))
2

(F (w1))
1

p
1 (G (w2))

1

q
1 ·

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)
1

q
1

·

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)
1

p
1

(41)

(by Young’s inequality)

≤
(w1w2)

α−1

(Γq (α))
2

(

F (w1)

p1
+

G (w2)

q1

)

·



CUBO
13, 1 (2011)

q− Fractional Inequalities 69

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)
1

q
1

·

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)
1

p
1

. (42)

Therefore
∫ x1

0

∫ x2

0

|∆1 (w1)| |∆2 (w2)|

(w1w2)
α−1

(

F (w1)
p1

+
G(w2)

q1

)dqw1dqw2 ≤

1

(Γq (α))
2

(

∫ x1

0

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)
1

q
1

dqw1

)

·

(

∫ x2

0

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)
1

p
1

dqw2

)

≤ (43)

x
1

p
1

1 x
1

q
1

2

(Γq (α))
2

(
∫ x1

0

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)

dqw1

)
1

q
1

·

(
∫ x2

0

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)

dqw2

)
1

p
1

, (44)

proving the claim.

We continue with a generalized q−fractional Poincaré type inequality.

Theorem 11. Let x > 0, 0 < w ≤ x, 0 < q < 1; α > β > 0, p1, q1 > 1 :
1
p1

+ 1
q1

= 1; n ∈ N. Set

K (w) =
(

∗D
α−β
q,wqn f

)

(w) −

⌈α⌉−1
∑

k=⌈α−β⌉

(

Dkq f
)

(wqn)

Γq (k − α + β + 1)
w

k−α+β (qn; q)
k−α+β .

Then
∫ x

0

(

|K (w)|

wβ−1

)q1

dqw ≤
1

(Γq (β))
q1
· (45)

(

∫ x

0

(

∫ w

wqn

(

(

q
t

w
; q

)

β−1

)p1

dqt

)q1

dqw

)

1

p
1

·

(
∫ x

0

(
∫ w

wqn

∣

∣

∗D
α
q,wqn f (t)

∣

∣

q1
dqt

)q1

dqw

)
1

q
1

.

Proof. By (21) we get

K (w) = I
β
q,wqn

(

∗D
α
q,wqn f

)

(w) =
wβ−1

Γq (β)

∫ w

wqn

(

q
t

w
; q

)

β−1

(

∗D
α
q,wqn f

)

(t) dqt. (46)

Rest of proof goes as in the proof of Theorem 8.

Next comes a generalized q−fractional Sobolev’s type inequality.



70 George A. Anastassiou CUBO
13, 1 (2011)

Theorem 12. Here all terms and assumptions as in Theorem 11. Additionaly let r1, r2 > 1 :
1
r1

+ 1
r2

= 1. Then
(
∫ x

0

(

|K (w)|

wβ−1

)r1

dqw

)
1

r
1

≤
1

Γq (β)
· (47)







∫ x

0

(

∫ w

wqn

(

(

q
t

w
; q

)

β−1

)p1

dqt

)

r
2

1

p
1

dqw







1

r2
1

·

(

∫ x

0

(
∫ w

wqn

∣

∣

∗D
α
q,wqn f (t)

∣

∣

q1
dqt

)

r
1

r
2

q
1

dqw

)
1

r
1

r
2

.

Proof. As in the Theorem 9, using (46).

We finish with a generalized q−fractional Hilbert-Pachpatte type inequality.

Theorem 13. Let for i = 1, 2 that xi > 0, 0 < wi ≤ xi, 0 < q < 1; α > β > 0, p1, q1 > 1 :
1
p1

+ 1
q1

=

1; n ∈ N. Call

Ki (wi) =
(

∗D
α−β
q,wiq

n fi

)

(wi) −

⌈α⌉−1
∑

k=⌈α−β⌉

(

Dkq fi
)

(wiq
n)

Γq (k − α + β + 1)
w

k−α+β
i (q

n; q)
k−α+β ,

F
∗ (w1) =

∫ w1

w1q
n

(

q
t1

w1
; q

)p1

β−1

dqt1, (48)

G
∗ (w2) =

∫ w2

w2q
n

(

q
t2

w2
; q

)q1

β−1

dqt2.

Then
∫ x1

0

∫ x2

0

|K1 (w1)| |K2 (w2)|

(w1w2)
β−1

(

F ∗(w1)
p1

+
G∗(w2)

q1

)dqw1dqw2 ≤
x

1

p
1

1 x
1

q
1

2

(Γq (β))
2
· (49)

(
∫ x1

0

(
∫ w1

w1q
n

∣

∣

∗D
α
q,w1q

n f1
∣

∣

q1
(t1) dqt1

)

dqw1

)
1

q
1

·

(
∫ x2

0

(
∫ w2

w2q
n

∣

∣

∗D
α
q,w2q

n f2
∣

∣

p1
(t2) dqt2

)

dqw2

)
1

p
1

.

Proof. Similar to the proof of Theorem 10, using (21).

Received: October 2009. Revised: November 2009.



CUBO
13, 1 (2011)

q− Fractional Inequalities 71

References

[1] George Anastassiou, Fractional Differentiation Inequalities, Springer, N. York, Heidelberg,

2009.

[2] H. Gauchman, Integral inequalities in q−Calculus, Computers and Mathematics with Ap-

plications, 47 (2004), 281-300.

[3] P. Rajkovic, S. Marinkovic, M. Stankovic, Fractional integrals and derivatives in

q−Calculus, Applicable Analysis and Discrete Mathematics, 1 (2007), 311-323.

[4] M. Stankovic, P. Rajkovic, S. Marinkovic, On q−fractional derivatives of Riemann-

Liouville and Caputo type, arXiv: 0909.0387 v1[math.CA] 2 Sept. 2009.