Journal of Applied Mathematics and Physics, 2013, 1, 45-48
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15006
Open Access JAMP
An Optimal Inequality for One-Parameter Mean
Hongya Gao, Yanjie Zhang, Tian Wang
College of Mathematics and Computer Science, Hebei University, Baoding, China
Received September 16, 2013; revised October 15, 2013; accepted October 21, 2013
Copyright © 2013 Hongya Gao et al. 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.
ABSTRACT
In the present paper, we answer the question: for 0 < α < 1 fixed, what are the greatest value

p

and the least value

q

such that the inequality







1
,,,
p, q
J
abAab GabJab
 


,
p
holds for all with ? where
for , the one-parameter mean
,0abab
pR
J
ab , arithmetic mean


,
A
ab

and geometric mean of two posi-
tive real numbers and are defined by

,Gab

ab



11
,,
p
p
p
,,
,g,,
,,
log
pp
p
aa
ab
b
Jab abb ab
ab









1,0,
1,
0
b

1
log lo
log
pp
pa b
pa
a
ab
ab
ab




,











,Aab2
ab

 and

,Gab ab

, respectively.
Keywords: Optimal Inequality; One-Parameter Mean; Arithmetic Mean; Geometric Mean
1. Introduction
For , the one-parameter mean pR

,
p
J
ab , arithme-
tic mean

,

A
ab and geometric mean


,Gab of
two positive real numbers and are defined by ab





11
,,
,,1
1
,loglog ,,
,,
log log
pp
pp
p
aa
pa babp
pab
Jab abab abp
ab
ab abp
ab
















,0,
1,
0,
b
(1)

,2
ab
Aab  and

,Gab ab, respectively.
There has been some literature on the one-parameter
mean values

,
p

J
ab , see [1-6]. It is well-known that
the one-parameter mean


,
p
J
ab is continuous and
strictly increases with respect to for fixed
with pR
,ab0ab

. Many means are special cases of
the one-parameter mean, for example:
 
1,
2
ab
J
ab


,A ab

 , the arithmetic mean,

,,
3
aabb
He ab


1/2
J
ab , the Heronian mean,




12 ,,
J
ababG ab
 , the geometric mean, and
 
22
,
ab

,
J
abH ab
ab

, the harmonic mean.
In [1], Gao and Niu found the greatest values
and the least values such that the inequalities 1
,ps
2
,qs








1
,,, ,
pq
,
J
abAab GabHJab
  ab 

and








1 2
1
,1 ,1
,,, ,
ss
GabAab GabHGab
  ,
0
ab


hold for all with , where ,abab


0,1   ,

H. Y. GAO ET AL. 46
and




11
,1 ,
s
ss
s
ab
Gab ab







, as the Gini mean.
In [2], Cheune and Qi proved the logarithmic con-
vexiity of the one-parameter mean values

,
p

J
ab

and
presented the monotonicity of

J
rJr for rR

.
In [3], Wang, Qiu and Chu obtained the greatest v alue
1 and the least value such that the double inequality r2
r
 


1
,,1,
r2
,
r
J
abA abHabJab   
holds for all with
,0ab ab

.
In [4], Hu, T Chu presed u andntethe greatest value
an 1
r
d the least value 2
r such that the double inequality
 
12
,, ,
rr
J
abT abJab holds for all ,0ab
with ab, where
 

2
,
2arctan
ab
Tab ab
ab






denotes the first Seiffert mean. he greatest value and
th
,
q
In [5], Long and Chu found tp
e least value q such that the inequality
 
,,1,
p

J
abA abHab Jab 
holds for all with ,0ab ab

.
In [6], the auths estab Sor lishedchur-convexities of
tw k
purpose of this paper is to answer the question: for
o types of one-parameter mean values in n variables,
and obtained Schur-convexities of some well-nown func-
tions.
The
01

 fixed, what are the greatest value


p

and
alue

qthe least v

such that the inequality
  
1
,,,,
 
pq

J
abab GabJab
 
A
holds for all with ,0ab ab

?
2. A Preliminary Lemma
m of this paper, we need
t > 1, one has
In order to prove the main theore
the following lemma.
Lemma 2.1. For all
 

3
1logtt t
31.
21
mt
t

 (2)
Proof. The logarithmic derivative of is

mt

  

2
log ,
1log
mt
mt tt t



 (3)
where
. (4)
Simple calculations lead to
mt nt


nt



22
1
41log 31,lim0
t
t tt tnt



 
1
5422 lont tt


1
g, lim0
t
t nt
t



  (5)
 
21
41
32log, lim0,
t
ntt nt
tt



 (6)
 
2
3
21 0. t
nt t



 (7)
(2) follows from (3)-(7) an
3. Main Result
is paper is the following theorem.
d the fact

1
lim 1
t
mt

.
The main result of th
Theorem 3.1. Let 01


. Then for any ,ab
w0
ith ab

, we have



 
1
131
22
,, ,.,
J
abA a

b GabJab




 (8)
Moreover, the bounds

1
2
,
J
ab
  and


31
2
,
J
ab
 
ar no loss of generality to assume that
Le
e optimal.
Proof. It isab.
t 21
a
tb

, 13 1 ,p

22






 and



 
2
1212
,1 ,
,1 ,1
p
Jt
ft At G t
 

then


  
 
11
122 22
1
log 11 1
pp
ft gt
ft
ft tttt 



  ,
(9)
where





 

 


44 4224
22
22 212
1
112
211 1
112
211 1
,
pp
p
pp
p
tttp t
pt t
1
1
p
p
1
g
x
xp
px x
hx
 
 
 
 
x




  
 

 

(10)
where 21.xt

 Simple calculations lead to


1
1
lim 0,
x
hx

 (11)








 

2
1
1
1
211211
221
21 1,
1 2
p
p
p
p
hx pxpx
ppx
pp x






 
 
 
(12)
Open Access JAMP

H. Y. GAO ET AL. 47

1
1
lim 0,
t
hx




(13)
(14)
 
2
12
,
p
hxxhx

 
where
 

 
 
2
2
2
2211 1
22 11
1221
121,
p
hxp px
pp
px
ppp







 
 

1p
x


pp

2
1
lim 0,
x
hx



(15)
,
 
23
21hxp xhx

where
 

 
32211
21
221,

1
1
p
p
x

 (16)
hx ppx
pp
pp



 

 



3
1
lim23 1
x
hxp p 

 (17)
where
(20)
 
2
34
21
p
hxp px hx

  (18)
 
421 1hxpx p
   

1   (19)

4
1
lim2 3 1,
x
hx p 


 
421 .hxp

 (21)
We now distinguish betwee n two cases.
Case 1. 31
2
p
 
. We first consider the case
1
3
  since in this case the one-parameter mean

,
p
J
ab has different expression from others. The re-
sult
 
12
33 0
,1 ,1,1
A
tGt Jt
follows from Lemma 2.1 since
 
12
33 3
0
,1,1,1,1 1,At GtJtmt
In the following we assume 1
3


.
From (21) we see that


0x g
0 for all
4
h

implies

is strictly increasin for , which for . From
(20) weat plies
1x
x
. (18) im
4
hx
know th1

4
hx1x

30, for 1,
3






from which we kn is strictly decreasing for
1
0, f 


or 0, 3
1
hx
 

ow

3
hx
1
0, 3 



and strictly increasing for 1,1
3




. This
result together with (17) implies for

30hx
1
0, 3


and 




0hx
3r fo1

,1
3


. The same  
reasoning applies to


211
,,h xhxhxs well,
15), (14), (12), (11), (9) and (8w

1
,hx
 
), we ka and using (no


10gt

for 1 0, 

 and

0t
3
 for 1
g
1,1
3



. (8) implies




10ft
 for all 1t. Thus


1
f
t is strictly increasing for 1t, which together
(22)
implies rignequality
Ca
with

1
1
lim 1
t
ft


ht-hand side i of (8).
se 2. 1 p

2

. From (21) wow e kn


40hx

for all , which implies that
creasing for . By (20) one h
1x

4
4
hx
is strictly in-
as

120h 

1x

 ,
and by ( 19) one has


lim .
xhx
4

Thus there exists 11

 such

40hxthat

for


1
1,x

 and


40hx for

x-
plies

. (18) im
1,



30hx
 for

1
1,x

 and

30hx


for


1,x


 . Thus


3
hx is strictly increasing for


1
1,x


(17) and strictly decreasing for

1,
 x. By


10
3

3x
we kn
h and by
lim h0
x
ow


0x fo 1x. The same reasoning
o
3
hr all
applies t




2211
,,,hxhxhxhx

and


1
g
t
0 fo as well, ng (9)-(16), we have r all
1t. (9) ims
and applyi
plie

1
gt


1
ft
0

, thus

1
f
t
The left-hand side i is st nrictly de- equality of (8) creasing for
(22).
ove thas
1t.
follows from
Next we prt the bound

,
31
2
J
ab and
 


1
2,
J
ab
  are optimal.
r any 0 Fo

 an sufficiently small, d 0t




 
31
21,1
log Jt

1
1,
1 1,1AtGt
 
12
2
22
2
12
3
1
22822
18 40.
8
t
t
tt
ttt
tt
ttt
2
loglog 1
2
432 t
432t


 










 

 
 
 


  

 


Open Access JAMP

H. Y. GAO ET AL.
Open Access JAMP
48
This implies



31
2
1
,1,1,1

J
tAtGt






for t sufficiently close to 1.
For any 0

, since

 
1
21
,1
lim ,
,1 ,1
t
Jt
At Gt




 
then there exists such that 1T



1
2
1
,1,1 ,1

J
tAtGt





For
Thch isof Hebei Prov
(No. A2011201011).
RES
[1] H. Y. Gao and W. J. Niu, “Sharp Inequali Related to
s, Vol. 6, No. 4, 2012, pp. 545-555.
52

One-Parameter Mean and Gini Mean,” Journal of Mathe-
matical Inequalitie
http://dx.doi.org/10.7153/jmi-06-
tT.
4. Acknowledgements
is resear supported by NSF ince
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