Advances in Pure Mathematics, 2011, 1, 63-66
doi:10.4236/apm.2011.13014 Published Online May 2011 (http://www.scirp.org/journal/apm)
Several Ideas on Some Integral Inequalities
Waadallah T. Sulaiman
Department of C om put er E ngineering, College of Engineering, University of Mosul, Mosul, Iraq
Received March 4, 2011; revised March 14, 2011; accepted March 20, 2011
Abstract
Several new integral inequalities are presented via new ideas.
Keywords: Integral Inequality
1. Introduction
In [1] the following result was proved
Theorem 1.1. If 0f is continuous function on [0,1]
such that
 
11
dd, 0,1
xx
ft tttx
 (1)
then
 
11
1
00
dd,0fxxxfxx



 (2)
and the following question was posed.
If f satisfies the above assumption, under what addi-
tional assumption can one claim that
 
11
00
dd,,0fxxxfxx
 


 (3)
The following result as well, was achieved in [2]
Theorem 1.2 If 0f is a continuous function on
[0,b] satisfying
 
dd,0, 0,
bb
xx
f
ttt tbxb


 (4)
then
 
00
dd,0.
bb
fxxxfxx
 


 (5)
In their roles, Zabadan [3] and Hoang [4] generalized the
previous results by introducing the following theorems
respectively.
Theorem 1.3 Suppose


1
,,,,0,
f
gLab fgg
is non-decreasing. If
  
dd,,,
bb
xx
f
ttgttx ab
 (6)
then
 
dd,,0,1
bb
aa
fxxfxgxx
   
 (
7)
Theorem 1.4 Suppose




11
,, ,fL abgCab
,0,fg gis non-decreasing. If
  
dd,,,
bb
xx
f
ttgttx ab

then ,0, 1,



 the following integral ine-
quality holds
 
dd.
bb
aa
fxxfxgxx
  

The object of this paper is to prove some of the above
results via simpler methods as well as to present some
other new results .
2. Results
We start with the following:
Lemma 2.1 Let 0

 is non-decreasing on


,ab ,
If
 
d0, ,,
b
x
tt xab  
 (8)
then

d0.
b
a
xxx  
 (9)
If (8) reverses, then (9) reverses.
Proof.

W. T. SULAIMAN 64
 
  
 
dd
dd
dd.
bb
ax
b
bb
xa
a
bb
aa
xttx
x
ttxx x
att xxx

 
 







 



Therefore
 
 
ddd
d0,
bbb
aax
b
a
x
xxx ttx
att
 








being the sum of two non-negative terms.
Lemma 2.2. Let 0

 is non-increasing, If
 
d0,,,
x
a
tt xab  
 (10)
then

d0.
b
a
xxx  
 (11)
If (10) reverses, then (11) reverses.
Proof.
 
 
dd dd
dd
b
bxx b
aaa a
a
bb
aa
x
ttxxttx xx
btt xxx
 
 






 

Therefore
 

ddd
()d 0,
bbb
aax
b
a
x
xxx ttx
btt
 


  




being the sum of two non-negative terms.
The following two results are similar to theorem 1.4
and proved via short simple methods:
Theorem 2.3 Let ,0fg defined on [a,b], f is
non-decreasin g. If
  
dd,,,
bb
xx
f
ttgttx ab


 (12)
then
 
dd,0,
bb
aa
fxxftgtt
 


 (13)
Proof. The proof follows from Lemma 2.1 by putting






,.
x
fx xfxgx

 
Theorem 2.4 Let ,0fg defined on [a,b], g is
non-decreasing. If (12) is satisfied with

replaced by

, then (13) is satisfied for ,0.

 
Proof. By putting


,
x
gx

 



x
fx

 


g
x

 in Lemma 2.1, we get
 
dd.
bb
aa
f
xgx xgxx
  


(14)
Now making use of the arithmetic-geometric inequality,
we have for ,0.

 
 
,
f
xgxfxgx
 

 



The above inequality, via (14) implies

 
  
 
d
dd
dd
d.
b
a
bb
aa
bb
aa
b
a
fxx
fxgxxg xx
f
xgx xfxgx x
fxgxx

 
 


 
 



 
 





The theorem follows.
The coming two results are new:
Theorem 2.5 Let ,0fg defined on [a,b], f is
non-decreasing. If
 
dd,
bb
xx
ftt gtt
 


(15)
then
 
dd,,0,
bb
aa
fxxftgtt
 



 (16)
Proof. The proof follows from lemma 2.1 by putting








,.
x
fxxfxg x




Theorem 2.6 Let ,0fg defined on [a,b], f is
non-increasing . If
 
dd,
xx
aa
ftt gtt



(17)
then
 
dd,0,
bb
aa
fxxftgtt



 (18)
Proof. The proof follows from lemma 2.2 by putting






,.
x
fx xfxgx

 

W. T. SULAIMAN 65
Theorem 2.7 [4] Suppose ,0,fg
g
is non-de-
creasing. If
 
dd
bb
xx
ft tgtt

(19)
then 1, 0,

  the following integral inequality
holds
 
dd.
bb
aa
fxxfxgxx
  

Proof. On putting
 
1,
x
gx







x
fx  

g
xin lemma 2.1, we obtain
 
1dd.
bb
aa
f
xgx xgx x
 

(20)
By the AG inequality, we have, for 1,


  
1
11 .
f
xgxfxgx
 





The above inequality, via (19) implies for 1,


 
 
 
1
dd
dd
dd0.
bb
aa
bb
aa
bb
aa
fxxgxx
f
xgx xgx x
gxxgxx


















(21)
By the AG inequality again,
  
,
f
xgxfxgx
 

 



which implies
  
 
 
ddd
dd
d.
bbb
aaa
bb
aa
b
a
f
xxf xxf xx
f
xxg xx
fxgxx
 
 


 

 










All of the following results are new:
Theorem 2.8 Let ,0,fg f is non-decreasing such
that (15) is satisfied, then
 
dd,
bb
aa
f
xgxxgx x
  


(22)
for all ,0.

 
Proof. As
 

 


0,fxgxfxg x
  

then
 

 

d
d.
b
a
b
a
g
xfx gx x
f
xfx gx x
 
 






But by Lemma 2.1 by putting



,
x
fx

 






x
fxgx



, we obtain
 

d0,
b
a
fxfxg xx
  


then, the result follows:
Theorem 2.9 Suppose ,0,fg g is non-decreasing.
If (19) satisfied, then the following integral inequality
holds
 
dd,,0,1.
bb
aa
fxxfxgxx
 
 




(23)
Proof. By the AG inequality,
  
.
f
xg xfxgx
 

 



Integrating, we have, via (2 1) ,

 
 
d
d
d
b
a
bb
aa
bb
aa
fxx
f
xgx xgx
f
xgx xfx

 
 

















which implies (23).
Theorem 2.10 Let ,0,fg and defined on [a,b], f
is non-decreasing and g is non-increasing or conversely.
Then
 
  
22
1
ddd
1dd
2
bbb
aaa
bb
aa
f
xg xxfxxg xx
ba
fxx gxx
ba
 





 






(24)
Proof. Concerning the left inequality , we have










0, ,0,fxfygygx




which implies
 

 

dd 0,
bb
aa
fxfy gygxxy




and hence

W. T. SULAIMAN 66
  
  
dd dd
dd
bb bb
aa aa
bb bb
aa aa
f
xgy xyfygx xy
fxgxxyfygy
 
 


 
 
and the above implies
 

()dddd
d
bb bb
aa aa
b
a
f
xgy xyfxgxxy
bafxg xx
 



 

therefore
  
1
ddd
bbb
aaa
f
xgx xfx xgy x
ba



The right inequality follows by opening the inequality

2
dd0.
bb
aa
fxx gxx






Remark. It may be mentioned that many other new
inequalities can be obtained via Lemmas 2.1 and 2.2.
3. References
[1] Q. A. Ngo, D. D. Thang, T. T. Dat and D. A. Tuan,
“Notes on an Integral Inequality,” Journal of Ine-
qualities in Pure and Applied Mathematics, Vol. 7,
No. 4, 2006, Art. 12 0.
[2] W. J. Liu, C. C. Li and J. W. Dong, “On an Problem
Concerning an Integral Inequality,” Journal of Inequali-
ties in Pure and Applied Mathematics, Vol. 8, No. 3,
2007, Art. 74.
[3] G. Zabadan, “Notes on an Open Problem,” Journal of
Inequalities in Pure and Applied Mathematics, Vol. 9, No.
2, 2007, Art. 37
[4] N. S. Hoang, “Notes on an Inequality,” Journal of
Inequalities in Pure and Applied Mathematics, Vol.
9, No. 2, 2009, Art . 42 .