Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 787939, 7 pages
doi:10.1155/2011/787939
Research Article
Fractional Quantum Integral Inequalities
Hasan Öğünmez and Umut Mutlu Özkan
Department of Mathematics, Faculty of Science and Arts, Kocatepe University,
03200 Afyonkarahisar, Turkey
Correspondence should be addressed to Umut Mutlu Özkan, umut ozkan@aku.edu.tr
Received 10 November 2010; Revised 19 January 2011; Accepted 16 February 2011
Academic Editor: J. Szabados
Copyright q 2011 H. Öğünmez and U. M. Özkan. 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.
The aim of the present paper is to establish some fractional q-integral inequalities on the specific
time scale, Ìt0 {t : t t0 qn , n a nonnegative integer} ∪ {0}, where t0 ∈ Ê, and 0 < q < 1.
1. Introduction
The study of fractional q-calculus in 1 serves as a bridge between the fractional qcalculus in the literature and the fractional q-calculus on a time scale Ìt0 {t : t t0 qn ,
n a nonnegative integer} ∪ {0}, where t0 ∈ Ê, and 0 < q < 1.
Belarbi and Dahmani 2 gave the following integral inequality, using the RiemannLiouville fractional integral: if f and g are two synchronous functions on 0, ∞, then
Γα 1 α
J α fg t ≥
J ftJ α gt,
tα
1.1
for all t > 0, α > 0.
Moreover, the authors 2 proved a generalized form of 1.1, namely that if f and g
are two synchronous functions on 0, ∞, then
tα
tβ
J β fg t J α fg t ≥ J α ftJ β gt J β ftJ α gt,
Γα 1
Γ β1
for all t > 0, α > 0, and β > 0.
1.2
2
Journal of Inequalities and Applications
Furthermore, the authors 2 pointed out that if fi i1,2,...,n are n positive increasing
functions on 0, ∞, then
n n
1−n fi t ≥ J α f1
J α fi t,
J
α
i1
1.3
i1
for any t > 0, α > 0.
In this paper, we have obtained fractional q-integral inequalities, which are quantum
versions of inequalities 1.1, 1.2, and 1.3, on the specific time scale Ìt0 {t : t t0 qn ,
n a nonnegative integer} ∪ {0}, where t0 ∈ Ê, and 0 < q < 1. In general, a time scale is an
arbitrary nonempty closed subset of the real numbers 3.
Many authors have studied the fractional integral inequalities and applications. For
example, we refer the reader to 4–6.
To the best of our knowledge, this paper is the first one that focuses on fractional qintegral inequalities.
2. Description of Fractional q-Calculus
Let t0 ∈ Ê and define
Ìt
0
t : t t0 qn , n a nonnegative integer ∪ {0},
0 < q < 1.
If there is no confusion concerning t0 , we will denote Ìt0 by Ì. For a function f :
nabla q-derivative of f is
f qt − ft
∇q ft q−1 t
2.1
Ì
→
Ê, the
2.2
for all t ∈ Ì \ {0}. The q-integral of f is
t
∞
fs∇s 1 − q t qi f tqi .
2.3
i0
0
The fundamental theorem of calculus applies to the q-derivative and q-integral; in particular,
∇q
t
fs∇s ft,
2.4
0
and if f is continuous at 0, then
t
0
∇q fs∇s ft − f0.
2.5
Journal of Inequalities and Applications
3
Let Ìt1 , Ìt2 denote two time scales. Let f : Ìt1 → Ê be continuous let g :
q-differentiable, strictly increasing, and g0 0. Then for b ∈ Ìt1 ,
b
ft∇q gt∇t gb 0
Ìt
1
→
f ◦ g −1 s∇s.
Ìt
2
be
2.6
0
The q-factorial function is defined in the following way: if n is a positive integer, then
t − sn t − s t − qs t − q2 s · · · t − qn−1 s .
2.7
If n is not a positive integer, then
t − sn tn
∞
1 − s/tqk
.
nk
k0 1 − s/tq
2.8
The q-derivative of the q-factorial function with respect to t is
∇q t − sn 1 − qn
t − s n−1 ,
1−q
2.9
and the q-derivative of the q-factorial function with respect to s is
∇q t − sn −
n−1
1 − qn .
t − qs
1−q
2.10
The q-exponential function is defined as
eq t ∞ 1 − qk t ,
eq 0 1.
2.11
k0
Define the q-Gamma function by
Γq ν 1
1−q
1 0
t
1−q
ν−1
eq qt ∇t,
ν ∈ Ê .
2.12
Note that
Γq ν 1 νq Γq ν,
ν ∈ Ê , where νq :
1 − qν
.
1−q
2.13
The fractional q-integral is defined as
∇−ν
q ft 1
Γq ν
t
0
ν−1
t − qs
fs∇s.
2.14
4
Journal of Inequalities and Applications
Note that
∇−ν
q 1 1
1 q − 1 ν
t tν .
ν
Γq ν q − 1
Γq ν 1
2.15
More results concerning fractional q-calculus can be found in 1, 7–9.
3. Main Results
In this section, we will state our main results and give their proofs.
Theorem 3.1. Let f and g be two synchronous functions on Ìt0 . Then for all t > 0, ν > 0, we have
Γq ν 1 −ν
∇−ν
∇q ft∇−ν
q fg t ≥
q gt.
tν
3.1
Proof. Since f and g are synchronous functions on Ìt0 , we get
fs − f ρ gs − g ρ ≥ 0
3.2
for all s > 0, ρ > 0. By 3.2, we write
fsgs f ρ g ρ ≥ fsg ρ f ρ gs.
3.3
Multiplying both side of 3.3 by t − qsν−1 /Γq ν, we have
ν−1
ν−1
t − qs
t − qs
fsgs f ρ g ρ
Γq ν
Γq ν
ν−1
ν−1
t − qs
t − qs
fsg ρ f ρ gs.
≥
Γq ν
Γq ν
3.4
Integrating both sides of 3.4 with respect to s on 0, t, we obtain
1
Γq ν
t
ν−1
fsgs∇s t − qs
0
1
≥
Γq ν
t
0
t − qs
ν−1
1
Γq ν
t
fsg ρ ∇s ν−1 f ρ g ρ ∇s
t − qs
0
1
Γq ν
t
0
ν−1 t − qs
f ρ gs∇s.
3.5
Journal of Inequalities and Applications
5
So,
∇−ν
q fg t f ρ g ρ
1
Γq ν
t
ν−1
∇s
t − qs
0
t
t
g ρ
f ρ
ν−1
ν−1
≥
t − qs
t − qs
fs∇s gs∇s.
Γq ν 0
Γq ν 0
3.6
Hence, we have
−ν −ν
−ν ∇−ν
q fg t f ρ g ρ ∇q 1 ≥ g ρ ∇q f t f ρ ∇q g t.
3.7
Multiplying both side of 3.7 by t − qρν−1 /Γq ν, we obtain
ν−1
ν−1
t − qρ
t − qρ
−ν ∇q fg t f ρ g ρ ∇−ν
q 1
Γq ν
Γq ν
ν−1
ν−1
t − qρ
t − qρ
−ν
≥
g ρ ∇q ft f ρ ∇−ν
q gt.
Γq ν
Γq ν
3.8
Integrating both side of 3.8 with respect to ρ on 0, t, we get
∇−ν
q fg t
≥
t 0
ν−1
t
∇−ν
t − qρ
ν−1
q 1
f ρ g ρ t − qρ
∇ρ
∇ρ Γq ν
Γq ν 0
t
∇−ν
q ft
Γq ν
0
t
∇−ν
ν−1 ν−1 q gt
t − qρ
t − qρ
g ρ ∇ρ f ρ ∇ρ.
Γq ν 0
3.9
Obviously,
∇−ν
q fg t ≥
1
∇−ν ft∇−ν
q gt
−ν
∇q 1 q
Γq ν 1
tν
−ν
∇−ν
q ft∇q gt
3.10
and the proof is complete.
The following result may be seen as a generalization of Theorem 3.1.
Theorem 3.2. Let f and g be as in Theorem 3.1. Then for all t > 0, ν > 0, μ > 0 we have
μ
t
tν
−μ −μ
−μ
−ν
−ν
∇q fg t ∇−ν
q fg t ≥ ∇q ft∇q gt ∇q ft∇q gt.
Γq ν 1
Γq μ 1
3.11
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Journal of Inequalities and Applications
Proof. By making similar calculations as in Theorem 3.1 we have
μ−1
μ−1
t − qρ
t − qρ
−ν −ν
∇q fg t ∇q 1
f ρ g ρ
Γq μ
Γq μ
μ−1
μ−1
t − qρ
t − qρ
−ν
≥
g ρ ∇q ft f ρ ∇−ν
q gt.
Γq μ
Γq μ
3.12
Integrating both side of 3.12 with respect to ρ on 0, t, we obtain
∇−ν
q
fg t
t 0
∇−ν
q ft
≥
Γq μ
μ−1
t
∇−ν
t − qρ
μ−1
q 1
∇ρ f ρ g ρ t − qρ
∇ρ
Γq μ
Γq μ
0
t
0
∇−ν
μ−1 q gt
g ρ ∇ρ t − qρ
Γq μ
t
3.13
μ−1 f ρ ∇ρ.
t − qρ
0
Thus, 3.11 holds for all t > 0, ν > 0, μ > 0, so the proof is complete.
Remark 3.3. The inequalities 3.1 and 3.11 are reversed if the functions are asynchronous
on Ìt0 i.e., fx − fygx − gy ≤ 0, for any x, y ∈ Ìt0 .
Theorem 3.4. Let fi i1,...,n be n positive increasing functions on Ìt0 . Then for any t > 0, ν > 0 we
have
∇−ν
q
n n
1−n fi t ≥ ∇−ν
∇−ν
q 1
q fi t.
i1
3.14
i1
Proof. We prove this theorem by induction.
Clearly, for n 1, we have
−ν ∇−ν
q f1 t ≥ ∇q f1 t,
3.15
for all t > 0, ν > 0.
For n 2, applying 3.1, we obtain
−1
−ν
−ν ∇−ν
∇−ν
q f1 f2 t ≥ ∇q 1
q f1 t∇q f2 t,
3.16
n−1
n−1
2−n fi t ≥ ∇−ν
∇−ν
1
q
q fi t,
3.17
for all t > 0, ν > 0.
Suppose that
∇−ν
q
i1
i1
t > 0, ν > 0.
Journal of Inequalities and Applications
7
Since fi i1,...,n are positive increasing functions, then n−1
i1 fi t is an increasing function.
f
g, fn f. We obtain
Hence, we can apply Theorem 3.1 to the functions n−1
i1 i
∇−ν
q
n n−1 −1
−ν
−ν
−ν
fi t ∇q fg t ≥ ∇q 1 ∇q
fi t∇−ν
q fn t.
i1
3.18
i1
Taking into account the hypothesis 3.17, we obtain
∇−ν
q
n
n−1
−1 2−n −ν
−ν
−ν
fi t ≥ ∇q 1
∇q fi t ∇−ν
∇q 1
q fn t
i1
3.19
i1
and this ends the proof.
Acknowledgment
The authors thank referees for suggestions which have improved the final version of this
paper.
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