Applied Mathematics, 2011, 2, 899-903
doi:10.4236/am.2011.27120 Published Online July 2011 (http://www.SciRP.org/journal/am)
Refinements to Hadamard’s Inequality for Log-Convex
Functions
Waadallah T. Sulaiman
Department of C om put er Engineering, College of Engineering, University of Mosul, Mosul, Iraq
Received April 25, 2011; revised May 25, 2011; accepted May 28, 2011
Abstract
In this paper we show that a log-convex function satisfies Hadamard’s inequality, as well as we give an ex-
tension for this result in several directions.
Keywords: Log-Convex Functions, Hadamard’s Inequality, Integral Inequality
1. Introduction
Let be a convex mapping of the interval :fI
I
of real numbers and a, bI

with a < b. The fol-
lowing double inequality
  
1d
22
b
a
f
afb
ab
ffxx
ba






  (1.1)
is known in the literature as Hadamard’s inequality. In
[1], Fejer generalized the inequality (1.1) by proving that
if


:,gab is nonnegative, integrable and symmet-
ric to 2
ab
x, and if
f
is convex on [a,b], then
 

dd
2
d.
2
bb
aa
b
a
ab
fgxxfxgxx
fa fb
g
xx

 






(1.2)
A positive function f is log-convex on a real interval


,ab if for all


,,
x
yab and


0,1 ,   we have
   
1
1
f
xyfxf



 y. (1.3)
If the above inequality reversed, then f is termed
log-concave. We define for ,0xy

,
ln ln ,
,
xy
x
y
xy Lxy
x
xy






In [2] the following result is achieved:
Theorem 1.1. Let f be a positive log-convex function
on [a, b]. The n
  
1d,
b
a

f
atLfafa
ba  . (1.4)
For f a positive log-concave function, the inequa lity is
reversed.
2. Lemmas
The following lemmas are needed for our aim.
Lemma 2.1. Let 0t1,

 then the following ine-
quality holds


1
11
t
t
tt

2.
(2.1)
Proof. Set
 


1
ln1ln1 2.
t
t
ftt t 
We have




lnln10,ft tt


 for 12.t

11 0.
1
ft tt




Therefore f attains its minimum at 12t which is
12 Hence


0ft which implies , and (2.1)
follows.

0
ft
e
Lemma 2.2. For 0,0ab t1,

 the following
inequality holds
1
1
,12
,12
tt
tt
abt
ab abt






,
(2.2)
and for , the following inequality
holds 0, 0, 01ab t
11
2.
tt tt
aba babab


 (2.3)

W. T. SULAIMAN 900
Proof. For 12,t we have
 
1/2 t
ba ba, which
implies 1tt
aba b

, and for 12,t


1/2 , t
ab ba
which implies 1.
tt
abab  We also have
2
11
2222 0
tt tt
abab









, which implies
11
2tt tt aba bab

.
Set Then, on keeping b
fixed, we have

11
.
tt tt
fta babab

 
 
  
11
21 1
110,
11.
tt tt
tt tt
fatabtabforab
fattabt ttab
 
 
 
  
.
,
As , f attains its mini-
mum at which is 0, therefore and
(2.3) is satisfied.
 
1
21 0
ab
fat ta

  

ab

0,fa
Although some of the coming results (Lemma 2.3 and
theorem 3.1) are known, but we prove them by new sim-
ple method.
Lemma 2.3. Let then the following inequal-
ity holds ,0ab
,
ln ln2
ab ab
ab ab


 (2.4)
Proof. Left inequality. Let us assume that Set .ba

1212 ln ,1.fxxxxx 
 
(2.5)

1232 1
11 0
22
fx xx x


as

2
14 3412 321
02
x
xxx
 
x

.
Therefore f is non-decreasing, and that implies


f
x
.
The result follows by putting

10f
x
ba

in
(2.5).
Right inequality. Let and let ,ab.
x
ab Set

1
ln 2,1.
1
x
fx xx
x




(2.6)
We have
 
2
140
1
fx xx



 as
 
2
2
14
10
1
xxx .
Then f is non-decreasing, and hence



10.fx f
The result follows by putting
x
ab in (2.6).
Lemma 2.4. The function

1,
ln 1
x
f
xx
x


 (2.7)
is non-decreasing.
Proof.
 



1
22
2
ln 1.
ln ln
110,
g
x
xx
fx
x
x
gx xx



 
therefore g is non-decreasing. Since

10g,

then


0,gx and hence


0,fx
 that is f is non-de-
creasing.
3. Theorems
Theorem 3.1. Let f be a positive log-convex function on
[a,b], then f satisfies (1.1).
Proof. This can be achieved immediately as the
log-convex function is convex which follows from the
fact that “Every increasing convex function of a convex
function is convex” which implies that


ln
f
x
fx e is
convex. Or the proof can be achieved by following the
definition:
Making use of lemma 2.2, we have

 
    
  
12 12
12 12
11
00
11
1 1
0 0
11
dd
22
1dd
11 1
d1d1d
22
dd
22
bb
aa
bb
aa
b
a
tt tt
ab abxx
ffxfabxfxx
ba ba
fabxxfx x
ba
f xxftatbtftatb t
ba
.
2
f
afbfaf bfafbfafb
tt


 

 

 










 



The following giving a refinement to theorem 3.1.
Theorem 3.2. Let f be a log convex function. Then the following inequality holds
 




 


2
11
dd
2lnln
bb
aa 2
f
bfafafb
ab
ffxxfxx
babafb fa
 




 


 
(3.1)
Proof.

W. T. SULAIMAN901
 
   
1/ 21/ 2
11
dd
22 2
11
()dd dd,
bb
aa
bbb
aaa
ab ababxx
ffxfx
ba ba
1
b
a
f
abx fxxfabxxfxxfxx
ba baba
 
 

 

 

 







which implies
  





  
2
2
11
dd
2ln
bb
aa
ln2
f
bfafafb
ab
ffxxfxx
bafb fa
ba
 




 
 

in view of [2] and Lemma 2.3.
The following presents an extension to Fejer’s gener-
alization (1.2) for log-convex functions
Theorem 3.3. Let f be log convex, g is positive, inte-
grable and symmetric to

2xab Then the fol-
lowing inequality holds
 
 
 
22
ddd
2
dd
ln ln2
bb b
aa a
bb
aa
ab
fgxxfxgxxfxgxx
fb fafafb
bagxx bagxx
fb fa


 





 

 

.
(3.2)
Proof.
 
   
   
1/2 1/2
12 12
1/21/21/2 1/2
12 12
dd()d
22
() ddd
ddd d
bb b
aa a
bbb
aaa
bbbb
aaaa
ab abxx
fgxxfgxxfabxfx gxx
fabxgabx fxgxxfabxgabxxfxgxx
f xgxxfxgxxxf xgxx

 

 
 





 


 







12
12
ba

which implies
  
22
dd
2
bb b
aa a
ab d
f
gx xfxgxxbafxgxx


 




 
Now, for 012t , we have
       
  
 
 
11
1
00
1
0
d11d1d
()()1dd,
ln ln
btt
a
b
a
fxgxxbafta tbgtatbtbafafbgtatbt
fb fa
bafafbgtatbtgxx
fb fa

 

 
 

in view of Lemmas 2.2 and 2.3. Also, we have for 12,t
       
 
 
 
11
11
00
1
0
d11d 1
1dd,
ln ln
btt
a
b
a
fxgxxbaftatbgtatbtbafafbgtatbt
fb fa
bafafb gtatbtgxx
fb fa



 
 

d

W. T. SULAIMAN 902
in view of Lemmas 2.2 and 2.3. Consequently, we obtain, by Lemma 2.2,
 


 




dd
ln ln2
bb
aa
fb fafafbd.
b
a
f
xgxxgxx gxx
fb fa




This completes the proof of the theorem.
The following is another refinement of theorem 3.1.
Theorem 3.4. Assume that be an increas-
ing log-convex function. Then for all
:fI


0,t1, we have
 




 


,d
2lnln
b
a
,
2
f
afb fafb
ab
fwabfxxWt
fa fb




 
 
  (3.3)
where

3
,
44
aba b
wab ff




3
,


(3.4)
 










 
 

11
1.
ln 1 lnlnln 1
fta tbfafbftatb
Wt tt
f
tatbfafb ftatb
 
 
  (3.5)
Proof. We have via Lemmas 2.3 and 2.4
 
  
 
2
2
(1 )
2
(1 )
2
1313332
()d d
224244 4
211 1
ddd()dd
2
1
(1)( )
1
ab b
ab
a
ab tat b
bb b
ab
aaatatb
ababa baba b
f
fffwtfx
ba
fx xfxxfx xfxxfx x
baba ba
tfx
tba





 
 

 
 
 

 

 

 

 

 
xfxx

 
 
 

 


 
 
 
 
 
(1 )
(1 )
1
dd
()
1() 1
(1)( )
ln1ln()lnln1
()
(1 ).
ln ln()ln lnln ln2
tat bb
atatb
xt fxx
tb a
ftatbfafbfta tb
ttWt
ftatbfafbftatb
fb fafb fafb fafa fb
tt
fb fafb fafb fa


 







 
 
 

 


Theorem 3.5. Let f is log-convex and g is
non-negative, integrable,

1111,ppq, then the following inequality holds
  
 
11
dd
ln ln
pq
pp
bb
q
aa
fa fb
ba
f
xgx xgx x
pfa fb












. (3.6)
Proof. We have, via Holder’s inequality
     
  

1
111
1
0
11
1 1
(1 )
0 0
ddd1d d
ddd()d
q
ppq
bbb b
pq pq
aaa a
pt
pq
bb
ptptqpq
aa
fxgxxfxxgxxba f tatbtgxx
fa
bafafbtg xxbafbtgxx
fb

 

 
 


 

 
 


 


 
 
 
 
1
11
()d
ln ln
q
pq
pp bq
a
fa fb
ba gxx
pfa fb













W. T. SULAIMAN903
4. References
r die Fourierreihen, II,” Math. Naturwiss
Anz. Ungar. Akad. Wiss, Hungarian, Vol. 24, 1906, pp.
369-390.
[2] P. M. Gill, C. E. M. Pearse and J. Pečarić, “Hadamard’s
for R-Convex Functions,” Mathematical Ine-
[1] L. Fejér, “UbeInequality
qualities & Applications, Vol. 215, No. 2, 1997, pp. 461-470.