Advances in Pure Mathematics, 2011, 1, 77-80
doi:10.4236/apm.2011.13017 Published Online May 2011 (http://www.scirp.org/journal/apm)
New Types of Q-Integral Inequalities
Waadallah T. Sulaiman
Department of C om put er E ngineering, College of Engineering, University of Mosul, Mosul, Iraq
Received January 13, 2011; revised March 7, 2011; accepted March 10, 2011
Abstract
Several new q-integral inequalities are presented. Some of them are new, One concerning double integrals,
and others are generalizations of results of Miao and Qi [1]. A new key lemma is proved as well.
Keywords: Q-Integral Inequalities
1. Introduction
For 01q the q-analog of the derivative, denoted by
q
D is defined by (see [2])
  
,0
q
fx fqx
Df xx
xqx


. (1)
Whenever

0f exists,
 
00
q
Df f

, and as
1q , the q-derivative reduces to the usual derivative.
The q-analog of integration from 0 to a is given by
(see [3])
 

0
0
d1
akk
qk
f
xxa qfaqq




, (2)
provided the sum converges absolutely. On a general
interval


,ab the q-integral is defined by (see [4])
 
00
ddd
bba
qqq
a
f
xx fxxfxx

. (3)
The q-Jackson integral and the q-derivative are related
by the fundamental theorem of quantum calculus, which
can be stated as follows (see [4, p. 73]) :
If F is an anti q-derivative of the function f, namely
q
DF f, continuous at
x
a, then
 
d
b
q
a
f
xxFbFa
. (4)
For any function f, we have
 
d
x
qq
a
Dft tfx
 (5)
For 0b and ,,
n
abqnN we denote



,:0
k
q
abbqk n and


1
,,.
qq
abaq b



(6)
It is not difficult to check the following










qqq
DfxgxfxDgx gqxDfx (7)






 
qq
q
g
xDfxfxDgx fx
Dgx gxgqx
 



 (8)
In [5] the following result was proved
Theorem 1.1. Let f be a function defined on


,q ab
satisfying
 

3
0 and2for,
and 3.
t
qq
f
aDfxtxaxab
t



Then
 
1
dd
t
bb
tqq
aa
fx xfqx x




 (9)
and in [1], the following results were proved
Theorem 1.2. If


f
x is a non-negative and in-
creasing function on


,q ab and satisfies
 
2
21
11
q
fqxDfx fxxa






(10)
for 1

 and 1

 then
 
dd.
bb
qq
aa
fxx fxx






(11)
Theorem 1.3. If


f
x is a non-negative and in-
creasing function on ,
nm
bq b




 and satisfies



2
1
11
m
q
Df xfqxxa





  (12)
on


,q ab and for ,1

  then

W. T. SULAIMAN 78


dd.
bb
m
qq
aa
fxx fqxx





 (13)
Theorem 1.4. If


f
x is a non-negative function on


0, q b and satisfies

dd
bb
qq
xx
f
ttt t



(14)
for


0, q
x
b and 0

 then the inequality
 
dd
bb
qq
xx
f
tttftt
  

(15)
holds for all positive numbers

and

.
Lemma 1.5[5]. Let 1p and

g
x be a nonnega-
tive, monotonic function on


,q ab .Then
 






11
.
ppp
qq q
pgqx DgxDgxpgxDgx


(16)
Remark 1. It may be mentioned that the function g
should be non-decreasing, which is not stated. As well if
g is non-increasing, (16) reverses. If g non-decreasing
and p is such that 01p, then it is not difficult to
show that (14) reverses.
2. Results
We start with the following key lemmas
Lemma 2.1. Let ,0,f

 and both non-decreasing
functions,

is differentiable, f defined on


,q ab .
Then
 


,
qq q
f
qxDfx DfxfxDfx 
  (17)
If f is non-increasing, (15) reverses.
Proof. We have
 

  



 


    
dd
fx fx
fqx fqx
f
xfqxfx fqx
tt fxt
fx fx fqx
 








therefore
  

 
 
  
1
1
q
q
fx fqx
Dfx qx
fx fqx
f
xfxDfx
qx
 








The rest is also similar.
Probably the following lemma is useful in some cases.
Lemma 2.2. Let ,0,f

 and both non-decreasing
functions, f defined on


,q ab Define
  
 
,
q
f
xfqx
Df fx fqx
 


. (18)
Then




,
qq q
DfDfx fxDfx 

 (19)
Proof. We have,
 



 



 
 
 
  
1
,
1
q
qq
fx fqx
Dfx qx
fxfqxfxfqx DfDfx
fx fqxqx


 





By (17),






,
qq qq
DfDf xDfxfxDfx 

 .
All the rest are similar.
Theorem 2.3. Let ,,, 0,,
f
gg

  are both non-
decreasing and defined on,],[q
ba

0ga  . If





  
q
f
xgxDgx  

, (20)
then
 
d
x
q
a
f
tt gx   
. (21)
If g is non-increasing and





 
q
f
xgqxDgx  

, (22)
satisfies, then (21) reverses.
Proof. Set
 
d
x
q
a
F
xfttgx   
.
We have, by lemma 2.1,
  
 
d
0.
x
qq qq
a
q
DF xDfttDgx
fx gxDgx
 
 





 

Therefore, F is non-decreasing, which implies




0.Fx Fa
The result follows.
Corollary 2.4. Let


f
x be a nonnegative and in-
creasing function on


,q ab such that

0fa

. Let
0, 1,2

  . If




1
2
1
(1)
q
fqxDfx
fxxa









 , (23)
is satisfied, then
 
dd
bb
qq
aa
fxx fxx






. (24)

W. T. SULAIMAN 79
Furthermore, if







 
1
2
1
(1)
q
fqxDfx
fqxxa









 (25)
is satisfied, then
 
dd
bb
qq
aa
f
xx fqxx





 (26)
Proof. For


,q
x
ab let
 
,, d
x
q
a
x
xxxgxfxx
 
  

then, we have, via lemma 1.5,
 
 
 
1
1
.
q
x
q
a
x
q
a
fx gxDgx
fxftdtfx
fxf xftdtfxhx

 

 















 






Now,
 
 
 
 



1
1
2
1
2
1
1
1
0.
x
qqq q
a
q
x
q
a
q
Dh xDfxDft dt
fqxDfx
f
tdtfx
fqxDfx
fxxa

 




























 



Therefore,

hx is non-decreasing, but


0ha

,
then

0hx. The result follows by theorem 2.3.
The proof of the second part is similar, and therefore,
it is omitted.
Remark 2. Theorem 1.2 follows from corollary 2.4,
the first part, by putting 1.


Theorem 2.5. Let ,
f
g are non-negative functions
on


,q ab , either f or g is non-decreasing and they sat-
isfies
  
dd,,,
bb
qq
q
xx
f
tt gttxab


 (27)
then the inequality
 
dd
bb
qq
xx
f
tt ftgtt
 
 (28)
holds for all positive numbers

and

.
Proof. Suppose that f is non-decreasing. Using the fact
 
d
b
qq
a
f
bfa Dfxx

we have
 
   
  

 
dd
d
d
d
d.
bbx
qqqq
baa
bb b
qqq q
at a
bb b
qqq q
at a
bx
qq q
aa
b
q
a
f
xxfxDftdtfax
Dftfxdxdtfaf xx
Dftgxdxdtfagxx
gx Dftdtfax
fxgxx
 
 
 















 


Now, suppose g is non-decreasing, then, we have
 

 
  


d
dd
dd d
dd d
dd
d.
b
q
b
bx
qq q
aa
bb b
qqqq
at a
bb b
qqq q
at a
bx
qq q
aa
b
q
a
fxgxx
fx Dgttgax
Dgtfxxtgafxx
Dgtg xxtgag xx
gxDgttfax
gxx


 


 













 
 


(29)
Using the arithmetic geometric inequality yields
  
.
f
xgxfxgx
 

 



Integrating the above inequality and making use of (29)
gives
 
 
dd
dd.
bb
aa
bb
aa
f
xxg xx
fxgxxg xx
 
 

 







The result follows.
Remark 3. Theorem 1.4 follows from theorem 2.5 by
putting


0, .agxx


Corollary 2.6. Let 0f. If

W. T. SULAIMAN 80

 

sin cosd,, π2
x
qq
a
fx fttxa
fx




 (30)
then
  
sind sind
xx
qq
aa
f
tt ftt




(31)
Proof. The proof follows from theorem 2.3 by putting
 
sin ,d ,
x
q
a
x
xxgxftt   

as follows
 
  
sin cos0.
q
x
q
a
fx gxDgx
fx ftdtfx








The following result concerning similar inequality but
for double integrals.
Theorem 2.7. Let 0f non-decreasing in both x
and y,

,0,fay,

  2,

0.

 If
 




1
,
2
,,
1,,, ,
qx
q
fqxyDfxy
x
ayaxyab









(32)
then
 
,dd ,dd
y
xb x
qq qq
ay aa
f
uv uvfuv uv






 (33)
Proof. Set
 
,,dd,dd.
y
xb x
qq qq
ay aa
Fxyfuvu vfuvu v







 
We have via lemma 2.1 and by keeping y fixed,
 

,,
,
,,dd
,dd
xb
qxqxq q
ay
y x
qxq q
aa
DFxyDfuvu v
Dfuvuv










 
,,,d
b
qx q
y
DFxyf xvv


 
 

 
 


1
11
1
1
,dd,d
,
,,
,,
,.
yy
x
qq q
aa a
fuv uvfxv v
fxyby
f
xyxayafxyy a
fxybyfxyxaya
fxykx




 


 















 
 

 
Now,









1
,
2
,,
1
0.
qx q
DkxbyfqxyDfxy
xa ya






 
 

Therefore, k is non-decreasing, as

0ka

then


0kx which implies

,,0. qx
DFxy that is


,
F
xy is non-decreasing in x. But ,0),(

yaF then


,0Fxy.
4. References
[1] Y. Miao and F. Qi, “Several Q-Integral Inequalities,”
Journal of Mathematical Inequalities, Vol. 3, No. 1, 2009,
pp. 115-121.
[2] E. W. Weisstein, “q-derivative,” From Math World - A
Wolfram Web Resource, 2010.
http://math-world.wolfram.com/q-Deriative.html
[3] E. W. Weisstein, “q-Integral,” Math World—A Wolfram
Web Resource, 2010.
http://mathworld. wolfram.com/q-Integral.html
[4] V. Kac and P. Cheung, “Quantum Calculus,” “Univer-
sitex, Springer-Verlag, New York, 2002.
doi:10.1007/978-1-4613-0071-7
[5] K. Brahim, N. Bettaibi and M. Sellemi, “On Some Feng
Qi Type q-Integral Inequalities,” Journal of Inequalities
in Pure and Applied Mathematics, Vol. 9, No. 2, 2008,
Art. 43.