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Volume 2011, Article ID 787939, 7 pagesdoi:10.1155/2011/787939 Research Article Fractional Quantum Integral Inequalities Department of Mathematics, Faculty of Science and Arts, Kocatepe

Volume 2011, Article ID 787939, 7 pages

doi:10.1155/2011/787939

Research Article

Fractional Quantum Integral Inequalities

Department of Mathematics, Faculty of Science and Arts, Kocatepe University,

03200 Afyonkarahisar, Turkey

Correspondence should be addressed to Umut Mutlu ¨Ozkan,umut ozkan@aku.edu.tr

Received 10 November 2010; Revised 19 January 2011; Accepted 16 February 2011

Academic Editor: J Szabados

Copyrightq 2011 H ¨O˘g ¨unmez and U M ¨Ozkan 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  t0q n , 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 q-calculus in the literature and the fractional q-q-calculus on a time scale Ìt0  {t : t  t0q n ,

n a nonnegative integer} ∪ {0}, where t0∈Ê, and 0 < q < 1.

Belarbi and Dahmani2 gave the following integral inequality, using the

Riemann-Liouville fractional integral: if f and g are two synchronous functions on 0, ∞, then

J α

fg

t ≥ Γα  1

t α J α ftJ α gt, 1.1

for all t > 0, α > 0.

Moreover, the authors2 proved a generalized form of 1.1, namely that if f and g

are two synchronous functions on0, ∞, then

t α

Γα  1 J β



fg

t  t β

Γβ  1 J α



fg

t ≥ J α ftJ β gt  J β ftJ α gt, 1.2

for all t > 0, α > 0, and β > 0.

Furthermore, the authors2 pointed out that if f ii1,2, ,n are n positive increasing

functions on0, ∞, then

J α

 n



i1

f i



t ≥J α f11−nn

i1

for any t > 0, α > 0.

In this paper, we have obtained fractional q-integral inequalities, which are quantum

versions of inequalities1.1, 1.2, and 1.3, on the specific time scale Ìt0  {t : t  t0q n ,

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 numbers3

Many authors have studied the fractional integral inequalities and applications For example, we refer the reader to4 6

To the best of our knowledge, this paper is the first one that focuses on fractional

q-integral inequalities

Let t0∈Êand define

Ìt0t : t  t0q n , n a nonnegative integer

∪ {0}, 0 < q < 1. 2.1

If there is no confusion concerning t0, we will denoteÌt0byÌ For a function f :Ì → Ê, the

nabla q-derivative of f is

∇q ft  f



qt

− ft



for all t∈Ì\ {0} The q-integral of f is

t

0fs∇s 1− qt∞

i0

q i f

tq i

The fundamental theorem of calculus applies to the q-derivative and q-integral; in particular,

∇q

t

0

and if f is continuous at 0, then

t

0

∇q fs∇s  ft − f0. 2.5

Let Ìt1, Ìt2 denote two time scales Let f : Ìt1 → Ê be continuous let g : Ìt1 → Ìt2 be

q-differentiable, strictly increasing, and g0  0 Then for b ∈Ìt1,

b

0

ft∇ q gt∇t 

gb

0

f ◦ g−1

The q-factorial function is defined in the following way: if n is a positive integer, then

t − s n  t − st − qst − q2s

· · ·t − q n−1 s

If n is not a positive integer, then

t − s n  t n∞

k0

1− s/tq k

The q-derivative of the q-factorial function with respect to t is

∇q t − s n 1− q n

and the q-derivative of the q-factorial function with respect to s is

∇q t − s n −1− q n

1− q



t − qsn−1 2.10

The q-exponential function is defined as

e q t ∞ k0

1− q k t

Define the q-Gamma function by

Γq ν  1

1− q

1

0

t

1− q

ν−1

e q

qt

∇t, ν ∈Ê

Note that

Γq ν  1  ν qΓq ν, ν ∈Ê

, where ν q: 1− q ν

The fractional q-integral is defined as

∇−ν

q ft  Γ 1

q ν

t

0



t − qsν−1 fs∇s. 2.14

Note that

∇−ν

q 1 Γ 1

q ν

q − 1

q ν− 1t ν

1

Γq ν  1 t ν . 2.15 More results concerning fractional q-calculus can be found in1,7 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 onÌt0 Then for all t > 0, ν > 0, we have

∇−ν

q 

fg

t ≥ Γq ν  1

t ν ∇−ν

q ft∇ −ν

Proof Since f and g are synchronous functions onÌt0, we get



fs − fρ

gs − gρ

for all s > 0, ρ > 0 By3.2, we write

fsgs  fρ

g

ρ

≥ fsgρ

 fρ

Multiplying both side of3.3 by t − qs ν−1 /Γ q ν, we have



t − qsν−1

Γq ν fsgs 



t − qsν−1

Γq ν f



ρ

g

ρ

≥



t − qsν−1

Γq ν fsgρ





t − qsν−1

Γq ν f



ρ

gs.

3.4

Integrating both sides of3.4 with respect to s on 0, t, we obtain

1

Γq ν

t

0



t − qsν−1 fsgs∇s Γ 1

q ν

t

0



t − qsν−1 f

ρ

g

ρ

∇s

≥ Γ 1

q ν

t

0



t − qsν−1 fsgρ

∇s Γ 1

q ν

t

0



t − qsν−1 f

ρ

gs∇s.

3.5

∇−ν

q



fg

t  fρ

g

ρ 1

Γq ν

t

0



t − qsν−1 ∇s

≥ g



ρ

Γq ν

t

0



t − qsν−1 fs∇s  f



ρ

Γq ν

t

0



t − qsν−1 gs∇s.

3.6

Hence, we have

∇−ν

q 

fg

t  fρ

g

ρ

∇−ν

q 1 ≥ gρ

∇−ν

q 

f

t  fρ

∇−ν

q 

g

Multiplying both side of3.7 by t − qρ ν−1 /Γ q ν, we obtain



t − qρν−1

Γq ν ∇−ν q



fg

t 



t − qρν−1

Γq ν f



ρ

g

ρ

∇−ν q 1

≥



t − qρν−1

Γq ν g



ρ

∇−ν

q ft 



t − qρν−1

Γq ν f



ρ

∇−ν

q gt.

3.8

Integrating both side of3.8 with respect to ρ on 0, t, we get

∇−ν

q



fg

t

t

0



t − qρν−1

Γq ν ∇ρ 

∇−ν

q 1

Γq ν

t

0

f

ρ

g

ρ

t − qρν−1 ∇ρ

≥ ∇

−ν

q ft

Γq ν

t

0



t − qρν−1 g

ρ

∇ρ ∇

−ν

q gt

Γq ν

t

0



t − qρν−1 f

ρ

∇ρ.

3.9

Obviously,

∇−ν

q



fg

t ≥ ∇−ν1

q 1∇−ν q ft∇ −ν

q gt  Γq ν  1

t ν ∇−ν

q ft∇ −ν

q gt 3.10

and the proof is complete

The following result may be seen as a generalization ofTheorem 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

t ν

Γq ν  1∇

−μ

q 

fg

t  t μ

Γq

μ  1∇−ν q



fg

t ≥ ∇ −ν

q ft∇ −μ q gt  ∇ −μ q ft∇ −ν

q gt. 3.11

Proof By making similar calculations as inTheorem 3.1we have



t − qρμ−1

Γq

μ ∇−ν

q 

fg

t  ∇ −ν

q 1



t − qρμ−1

Γq

μ f

ρ

g

ρ

≥



t − qρμ−1

Γq

μ g

ρ

∇−ν

q ft 



t − qρμ−1

Γq

μ f

ρ

∇−ν

q gt.

3.12

Integrating both side of3.12 with respect to ρ on 0, t, we obtain

∇−ν

q 

fg

t

t

0



t − qρμ−1

Γq

μ ∇ρ  ∇−ν q 1

Γq

μ

t

0

f

ρ

g

ρ

t − qρμ−1 ∇ρ

≥ ∇−ν q ft

Γq

μ

t

0



t − qρμ−1 g

ρ

∇ρ ∇−ν q gt

Γq

μ

t

0



t − qρμ−1 f

ρ

∇ρ.

3.13

Thus,3.11 holds for all t > 0, ν > 0, μ > 0, so the proof is complete.

Remark 3.3 The inequalities3.1 and 3.11 are reversed if the functions are asynchronous

onÌt0i.e., fx − fygx − gy ≤ 0, for any x, y ∈Ìt0

Theorem 3.4 Let f ii1, ,n be n positive increasing functions onÌt0 Then for any t > 0, ν > 0 we have

∇−ν

q

 n



i1

f i



t ≥∇−ν

q 11−n n

i1

∇−ν

Proof We prove this theorem by induction.

Clearly, for n 1, we have

∇−ν q 

f1



t ≥ ∇ −ν q 

f1



for all t > 0, ν > 0.

For n 2, applying 3.1, we obtain

∇−ν

q 

f1f2

t ≥∇−ν

q 1−1∇−ν

q 

f1

t∇ −ν

q 

f2

for all t > 0, ν > 0.

Suppose that

∇−ν

q

n−1



i1

f i



t ≥∇−ν

q 12−n n−1

i1

∇−ν

q f i t, t > 0, ν > 0. 3.17

Sincef ii1, ,n are positive increasing functions, thenn−1 i1 f i t is an increasing function.

Hence, we can applyTheorem 3.1to the functionsn−1

i1 f i  g, f n  f We obtain

∇−ν

q

 n



i1

f i



t  ∇ −ν

q 

fg

t ≥∇−ν

q 1−1∇−ν

q

n−1



i1

f i



t∇ −ν

q 

f n

t. 3.18 Taking into account the hypothesis3.17, we obtain

∇−ν

q

n

i1

f i



t ≥∇−ν

q 1−1

∇−ν

q 12−n

n−1

i1

∇−ν

q f i



t



∇−ν

q



f n

t 3.19 and this ends the proof

Acknowledgment

The authors thank referees for suggestions which have improved the final version of this paper

References

1 F M Atıcı and P W Eloe, “Fractional q-calculus on a time scale,” Journal of Nonlinear Mathematical Physics, vol 14, no 3, pp 341–352, 2007.

2 S Belarbi and Z Dahmani, “On some new fractional integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol 10, no 3, article 86, 5 pages, 2009.

3 M Bohner and A Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2001.

4 Z Denton and A S Vatsala, “Fractional integral inequalities and applications,” Computers & Mathe-matics with Applications, vol 59, no 3, pp 1087–1094, 2010.

5 G A Anastassiou, “Multivariate fractional Ostrowski type inequalities,” Computers & Mathematics with Applications, vol 54, no 3, pp 434–447, 2007.

6 G A Anastassiou, “Opial type inequalities involving fractional derivatives of two functions and

applications,” Computers & Mathematics with Applications, vol 48, no 10-11, pp 1701–1731, 2004.

7 R P Agarwal, “Certain fractional q-integrals and q-derivatives,” Proceedings of the Cambridge Philosoph-ical Society, vol 66, pp 365–370, 1969.

8 W A Al-Salam, “Some fractional q-integrals and q-derivatives,” Proceedings of the Edinburgh Mathemat-ical Society Series II, vol 15, pp 135–140, 1966.

9 P M Rajkovi´c, S D Marinkovi´c, and M S Stankovi´c, “A generalization of the concept of q-fractional integrals,” Acta Mathematica Sinica (English Series), vol 25, no 10, pp 1635–1646, 2009.

∇q fs∇s  ft − f0. 2.5

Let Ìt1,... qsν−1 fs∇s. 2.14

Note that

∇−ν

q...

ρ

gs∇s.

3.5

∇−ν

q