Applied Mathematics, 2011, 2, 465-469
doi:10.4236/am.2011.24059 Published Online April 2011 (http://www.SciRP.org/journal/am)
A Study on New q-Integral Inequalities
Waad T. Sulaiman
Department of C om put er E ngineering, College of Engineering, University of Mosul, Mosul, Iraq
Received February 12, 2011; revised March 4, 2011; accepted March 7, 2011
Abstract
A q-analog, also called a q-extension or q-generalization is a mathematical expression parameterized by a
quantity q that generalized a known expression and reduces to the known expression in the limit 1q

.
There are q-analogs for the fractional, binomial coefficient, derivative, Integral, Fibonacci numbers and so
on. In this paper, we give several results, some of them are new and others are generalizations of the main
results of [1]. As well as we give a generalization to the key lemma ([2], Lemma 1.3).
Keywords: q-Integral Inequlalities, Integral Inequalities
1. Introduction
For 01q, the q-analog of the derivative of the func-
tion

f
x, denoted b y


q
Df x is defined (see [5]) by
 



,0
1
q
fx fqx
Df xx
qx


 (1.1)
If

0f exists, then


00.
q
Df f

 The q-deri-
vative reduces to the usual derivative as 1q.
The q-analog of inte grat i o n may be given (see [6]) by
 

1
0
0
d1 kk
qk
f
xxq fqq



, (1.2)
which reduces to

1
0
d
f
xx
 as 1q.
The q-Jackson integral from 0 to a, for a more
general case, can be defined (see [2,3]) by
 

0
0
d1
akk
qk
f
xxa qfaqq




, (1.3)
provided the sum converges absolutely. The q-Jackson
integral on a general interval may be defined (see [2,3])
by
 
00
ddd
bba
qqq
a
f
xx fxxfxx

. (1.4)
The q-Jackson integral and q-derivative are related by
the “fundamental theorem of quantum calculus” which
can be restated ([3, p.73]) as follows: If
F
is an anti
q-derivative of the function f, that is q DF f, conti-
nuous at
x
a, then
 
d.
b
q
a
f
xxFbFa
 (1.5)
For any function f, we have
 
d
x
qq
a
Dfttfx




. (1.6)
It is not difficult to check that the q-analog of Leib-
niz’s rule is










qqq
Df xgxf xDgxgqxDf x . (1.7)
For 0b and n
abq with n, we den ote






1
,:0 and ,,
k
qq
q
a bbqkna baqb


 

. (1.8)
Some applications of q-integrals:
11 1
00 0
11ln
d, d, lnd,
11
1
n
xq q
n
qqq
xxx xxx
qq
q

 


 

2
1
0
1
2
d
1
q
a
qq
xx
xa 









, where

qa
 is a doubly pe-
riodic function. If 1q

, the integral reduces to

1
0
d
1sin
a
xx
x
a





.
In [4], the authors proved the following resu lts:
Theorem 1.1. If


f
x is a non-negative and in-
creasing function on


,q ab and satisfies
 
2
21
11
q
fqxDfx fxxa






(1.9)

W. T. SULAIMAN 466
for 1

 and 1

, then
 
dd
bb
qq
aa
f
xx fxx






. (1.10)
Theorem 1.2. If


f
x is a non-negative and increa-
sing function on ,
nm
q
bq b



for ,mn and satisfies



2
1
11
m
q
Df xfqxxa





 (1.11)
on


,q ab and for

, 1

, then


dd.
bb
m
qq
aa
fxx fqxx





 (1.12)
Theorem 1.3. If

f
x is a non-negative function on


0, q b and satisfies

dd
bb
qq
xx
f
ttt t



(1.13)
for


0, q
x
b and 0

, then the inequality
 
00
dd
bb
qq
f
tt tftt
  

(1.14)
holds for all positive numbers

and

.
In the coming section, we start with Lemma 2.1,
which represent a generalization of ([1], Lemma 1.3).
Theorems 2.3 and 2.4 are generalizations of Theorems
1.1 and 1.2 respectively, while Theorem 2.5 gives a ge-
neralization for more than one direction to Theorem 1.3.
Other new results are also given.
2. Results
Lemma 2.1. Let
 
,,
f
xfxgx
 be nonnegative,
nondecreasing functions defined on


,q ab , and let
1.p Then







    
1
1.
pp
qq
pq
pfgqxfgqxD gxDfgx
pfgxfgx Dgx




 (2.1)
Proof. We have
 






 




   
 




 

    
1
1
1
1
1
d
1
d
1
1
.
pp
p
q
xp
qx
px
qx
p
pq
f
gxfgqx
Df gxqx
p
f
gtfgtg tt
qx
pfg xfg xgt t
qx
pfg xfg x
g
xgqx
qx
pfgxfgxDgx






 





 


Also,




  
 
 



1
1
d
1
.
px
p
qqx
pq
pfg qxfg qx
Dfgxg tt
qx
pfgqxfgqx Dgx







Lemma 2.2. Define







 
,
q
f
gx fgqx
Dfg gx gqx

. (2.2)
Then



  
,.
qqq
Df gxDfg Dgx (2.3)
Proof. It is follows as
 


  

   



 
1
1
,.
q
qq
fgx fgqx
Df gxqx
fgx fgqx
g
xgqx
gx gqxqx
DfgDgx





Theorem 2.3. Let


,,
f
xfxgx
 be nonnega-
tive, nondecreasing functions defined on


,q ab satis-
fies





   

   




 
1
1
12
2
1
1
q
q
q
f
gqxfgqxDgx
fgxxaDgx
f
gxxaDgx













 

 
(2.4)
for 1

 , 2

, then




dd.
bb
qq
aa
f
gx xfgx x






(2.5)
Proof. Let
    
dd,
xx
qq
aa
M
xfgttfgtt







and
   
d.
x
q
a
hgxf gtt


Then via Lemma 2.1, we hav e








      
 





 










1
1
1
:.
qq
q
q
q
DMxfg xDhgx
fgxh gxhgxDgx
fgxhgxfgx Dgx
f
gxfgxhgx Dgx
fgxFgx

















As







 
,hgxfgxxa

 then

W. T. SULAIMAN 467
    

 


       
        
     
        
  



    
1
12 1
112
2
112
2
1
1
1
qqqq
qqqq
qq
q
qq
q
q
DfgxDfgxD hgx Dgx
Dfgxhgx DgxDhgx Dgqx
fgqxfgqxDgxhgxDgx
hgxhgxDgx
fgqxfgqxDgxhgxDgx
hgxhgxDgx
fgqxfgqxD
 
 
 

 




 

 



 
 

 



 

 



 





   




 
1
12
2
1
1
q
q
g
xf gxxaDgx
fgxxaDgx











 
The above implies

 
0
q
DFg x, by (2.4), which
implies
 
0Fgx  and hence

0,
q
DM x  so that

0Mx.
Theorem 2.4. Let



,,
f
xfxgx
 be nonnegative,
nondecreasing functions defined on ,
nm
q
bq b



satisfies
  

  

 
 




 
1
1
12
2
1
1,
q
mq
mq
fg qxfgqxDg x
fgqxxaDgx
f
gqxxaDgx













 

 
(2.6)
for 1, 2,

  then
 
 
dd
bb
m
qq
aa
f
gxxf gqtt





 (2.7)
Proof. Let
  
 
dd,
xx
m
qq
aa
Nxfgt tfgqt t







and




d.
xmq
a
hgxf gqtt


Then via Lemma 2.1, we hav e
    











 
 


 
 






 










1
1 1
1:.
qq q
mqq
q
DNxf gxDhgxf gxhgxhgxDgx
f
gxhgxfgqxDgxfgxhgxfgx Dgx
fgxfgxhgxDgxfgx Fgx

 
 




 


 
 

As


 

,
m
hgxfgqxxa

 then










       
 
      
     
 



 


 
 


  
1
12 1
112
2
112
2
1
1
1
qqqq
qqqq
qq
q
qq
mq
DfgxDfgxD hgx Dgx
Dfgxhgx DgxDhgx Dgqx
fgqxfgqxDg xhg xDg x
hgxhgxDgx
fgqxfgqxDg xhg xDg x
hgxhgqxDgx
fgqxfgq
 
 
 

 




 

 



 
 

 



 

 



 







 
 


 
 
1
12
2
1
1
m
qq
mq
x
Dg xfgq xxaDg x
fgqxxaDgx











 
The above implies

 
0
q
DFg x, by (2.4), which
implies



0Fgx  and hence

0
q
DN x , so that

0Nx.
Theorem 2.5. Let ,,
f
gh be nonnegative functions,
h defined on


,q ab and ,
f
g are defined on


Rh, g,
h are nondecreasing with
 
0ga ha. If





dd,,,
bb
qq
q
xs
f
httg htttab



then
     
dd
bb
qq
aa
f
hx xfhxghx x
  
 (2.8)

W. T. SULAIMAN 468
for all ,0

 .
Proof. Since by Lemma 2.2,






d,d
xx
qqqqq
aa
g
hxDghttD g hDhtt
 

 ,
then, we have
   
 



  

  
 


 
d
,dd
,dd
,dd
d,dd
d.
b
q
a
bx
qqqq
aa
bb
qq qq
at
bb
qq qq
at
bx
qq qqq
aa
b
q
a
fhxghx x
fhx DghDhttx
DghDhtf hxxt
DghDhtg hxxt
g
hxxD g hDhttx
ghxx
























(2.9)
Using the arithmetic-geometric inequality yields




   
.
f
hxg hx
fhxghx
 


 




Integrating the above inequality yields
   




dd
d,
bb
qq
aa
b
q
a
f
hx xghx x
fhxghx x
 


 






which implies via (2.9),
     




dd
d.
bb
qq
aa
b
q
a
f
hxghxxf hxx
fhxghx x
 










and hence
     
dd
bb
qq
aa
f
hx xfhxghx x
  
 .
Theorem 2.6. Let ,,
f
gh be nonnegative functions,
h defined on


,q ab and
f
,
g
are defined on


Rh,
,
g
h are nondecreasing with
 
0ga ha. If
    
dd,,,
bb
qq
q
xs
f
httghtttab

then
     
dd
bb
qq
aa
f
hx xfhxghx x
  
 (2.10)
for all 1, 0.

 
Proof. For 1

, we have
   
 

 






   
 

 
 
1
1
1
1
1
d
,dd
,dd
,dd
,dd
d.
b
q
a
bx
qqqq
aa
bb
qq qq
at
bb
qq qq
at
bx
qqqq
aa
b
q
a
fhx ghxx
fhxDghD httx
DghDhtfhxx t
DghDhtghxxt
ghxD ghDhttx
ghxx


























(2.11)
Now, applying the AG inequality, we have for 1

,
       
1
11
f
hxghxf hxghx
 




 .
On integrating the above inequality with the using of
(2.11) leads to
     
   
1
dd
11
dd,
bb
qq
aa
bb
qq
aa
g
hxxf hxghxx
g
hx xghx x










which implies
   
dd
bb
qq
aa
g
hx xfhx x



. (2.12)
Again, by the AG inequality, and via integration






 
dd
d,
bb
qq
aa
b
q
a
f
hxghxxf hxx
ghxx
 











and the above implies, by (2.12),






dd.
bb
qq
aa
f
hxghxxf hxx
  


The other way round direction may be obtained from
the following.
Theorem 2.7. Let f, g, h be nonnegative functions, h
defined on


,q ab and f, g are defined on


Rh, g, h
are nondecreasing with


0ga ha. If

W. T. SULAIMAN 469
    
dd,,,
bb
qq
q
xs
f
httghtttab



then
     
dd
bb
qq
aa
f
hx xfhxghx x
  

 (2.13)
for all 0

 .
Proof. We have, via Lemma 2.2,
   

 


 


 


 

 
d
,dd
,dd
,dd
d,dd
d.
b
q
a
bx
qqqq
aa
bb
qq qq
at
bb
qq qq
at
bx
qq qqq
aa
b
q
a
fhxg hxx
fhx DghDhttx
DghDhtf hxxt
DghDhtghxxt
g
hxxD ghDhttx
ghxx





























(2.14)
Making use of the AG i neq uality,




   
.
f
hxg hx
fhxg hx
 



 





Integrating the above inequality, with making use of
(2.14) yield




  
  
  
  
dd
d
d
d
d.
bb
qq
aa
b
q
a
b
q
a
b
q
a
b
q
a
f
hx xghx x
fxghx x
fxghx x
fxghx x
fxghx x
 





 




















3. References
[1] Y. Miao and F. Qi, “Several q-Integral Inequalities,”
Journal of Mathematical Inequalities, Vol. 3, No. 1, 2009,
pp. 115-121.
[2] K. Brahim, N. Bettaibi and M. Sellemi, “On Some Feng
Qi Type q-Intagral Inequlities,” Pure Applied Mathemat-
ics, Vol. 9, No. 2, 2008, Art. 43.
[3] E. W. Weisstein, “q-Derivative,” Math World-A Wol-
fram Web Resource,” 2010.
http://mathword. Wolfram .com/q-Derivative.html
[4] E. W. Weisstein, “q-Integral,” Math World-A Wolfram
Web Resource,” 2010.
http://mathword. Wolfram .com/q-integral.html
[5] F. H. Jackson, “On q-Definite Integrals,” Pure Applied
Mathematics, Vol. 41, No. 15, 1910, pp. 193-203.
[6] V. Kac and P. Cheung, “Quantum Calculus,” Universitext,
Springer-Verlag, New York, 2003.