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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 163202, 10 pages
doi:10.1155/2008/163202
Research Article
New Means of Cauchy’s Type
Matloob Anwar
1
and J. Pe
ˇ
cari
´
c
1, 2
1
Abdus Salam School of Mathematical Sciences, GC University, Lahore Gulberg 54660, Pakistan
2
Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
Correspondence should be addressed to Matloob Anwar, matloob
Received 30 December 2007; Accepted 7 April 2008
Recommended by Wing-Sum Cheung
We will introduce new means of Cauchy’s type M
s
r,l
f, μ defined, for example, as M
s
r,l
f, μ
ll −s/rr − sM
r
r
f, μ − M
r
s
f, μ/M
l
l
f, μ − M
l
s
f, μ
1/r−l
, in the case when l
/
r
/
s, l, r
/
0.
We will show that this new Cauchy’s mean is monotonic, that is, the following result. Theorem.Let
t, r, u, v ∈
R,suchthatt ≤ v, r ≤ u.ThenforM
s
r,l
f, μ, one has M
s
t,r
≤ M
s
v,u
. We will also give some
related comparison results.
Copyright q 2008 M. Anwar and J. Pe
ˇ
cari
´
c. 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.
1. Introduction
Let Ω be a convex set equipped with a probability measure μ. Then for a strictly monotonic
continuous function f, the integral power mean of order r ∈
R is defined as follows:
M
r
f, μ
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Ω
fu
r
dμu
1/r
,r
/
0,
exp
Ω
log
fu
dμu
,r 0.
1.1
Throughout our present investigation, we tacitly assume, without further comment, that all
the integrals involved in our results exist.
The following inequality for differences of power means was obtained see 1, Remark
8:
rr − s
ll − s
m ≤
M
r
r
f, μ − M
r
s
f, μ
M
l
l
f, μ − M
l
s
f, μ
≤
rr − s
ll − s
M, 1.2
2 Journal of Inequalities and Applications
where r, l, s ∈
R, l
/
r
/
s, r,l
/
0andwherem and M are, respectively, the minimum and the
maximum values of the function x
r−l
on the image of fuu ∈ Ω.
Letusnotethat1.2 was obtained as consequence of the following result see, e.g., 1,
Corollary 1.
Theorem 1.1. Let r, s, l ∈
R, and let Ω be a convex set equipped with a probability measure μ.Then,
M
r
r
f, μ − M
r
s
f, μ
M
l
l
f, μ − M
l
s
f, μ
rr − s
ll − s
η
r−l
1.3
for some η in the image of fuu ∈ Ω, provided that the denominator on the left-hand side of 1.3 is
non-zero.
We can also note that from 1.3 we can get the following form of 1.2:
inf
u∈Ω
fu ≤
ll − s
rr − s
M
r
r
f, μ − M
r
s
f, μ
M
l
l
f, μ − M
l
s
f, μ
1/r−l
≤ sup
u∈Ω
fu, 1.4
where r, l, s ∈
R, r
/
l
/
s, r, l
/
0. Moreover, 1.4 suggests introducing a new mean of Cauchy
type. We will prove in Section 3 a comparison theorem for these means. Finally we will, in
Section 4, give some applications.
2. New Cauchy’s mean
From 1.4, we can define a new mean M
s
r,l
as follows:
M
s
r,l
f, μ
ll − s
rr − s
M
r
r
f, μ − M
r
s
f, μ
M
l
l
f, μ − M
l
s
f, μ
1/r−l
,l
/
r
/
s, l, r
/
0.
2.1
Now by taking lim
l→0
M
s
r,l
f, μ, we will get
M
s
r,0
f, μM
s
0,r
f, μlim
l→0
M
s
r,l
f, μ
s
M
r
r
f, μ − M
r
s
f, μ
rr − s
log M
s
f, μ − log M
0
f, μ
1/r
,r
/
s, r, s
/
0.
2.2
Now by taking lim
r→s
M
s
r,l
f, μ, we will get
lim
r→s
M
s
r,l
f, μM
s
s,l
f, μM
s
l,s
f, μ
ll − s
s
fu
s
log fudμu−M
s
s
f, μ log M
s
f, μ
M
l
l
f, μ−M
l
s
f, μ
1/s−l
,l
/
s, l, s
/
0.
2.3
M. Anwar and J. Pe
ˇ
cari
´
c3
By similar way, we can calculate all the cases for r, s, l ∈
R. Finally, we get the following
definition of M
s
r,l
f, μ:
M
s
r,l
f, μ
ll − s
rr − s
M
r
r
f, μ − M
r
s
f, μ
M
l
l
f, μ − M
l
s
f, μ
1/r−l
,l
/
r
/
s, l, r
/
0;
M
s
r,0
f, μM
s
0,r
f, μ
s
M
r
r
f, μ − M
r
s
f, μ
rr − s
log M
s
f, μ − log M
0
f, μ
1/r
,r
/
s, r, s
/
0;
M
s
s,l
f, μM
s
l,s
f, μ
ll − s
s
fu
s
log fudμu − M
s
s
f, μ log M
s
f, μ
M
l
l
f, μ − M
l
s
f, μ
1/s−l
,
l
/
s, l, s
/
0;
M
s
s,0
f, μM
s
0,s
f, μ
fu
s
log fudμu − M
s
s
f, μ log M
s
f, μ
log M
s
f, μ − log M
0
f, μ
1/s
,s
/
0;
M
0
r,l
f, μ
l
2
M
r
r
f, μ − M
r
0
f, μ
r
2
M
l
l
f, μ − M
l
0
f, μ
1/r−l
,l,r
/
0;
M
0
r,0
f, μM
0
0,r
f, μ
2
M
r
r
f, μ − M
r
0
f, μ
r
2
M
2
2
log f, μ − M
2
1
log f,μ
1/r
,r
/
0;
M
s
t,t
exp
−
2t − s
tt − s
f
t
log fdμu − M
t
s
f, μ log M
s
f, μ
M
t
t
f, μ − M
t
s
f, μ
,t
/
s;
M
0
t,t
exp
−
2
t
f
t
log fdμu − M
t
0
f, μ log M
0
f, μ
M
t
t
f, μ − M
t
0
f, μ
,t
/
0;
M
0
0,0
exp
1
3
log f
3
dμu −
log M
0
f, μ
3
log f
2
dμu −
log M
0
f, μ
2
,
M
s
s,s
exp
−
1
s
f
s
log f
2
dμu − M
s
s
f, μ
log M
s
f, μ
2
2
f
s
log fdμu −
M
s
s
f, μ log M
s
f, μ
,s
/
0;
M
s
0,0
exp
1
s
log f
2
dμu −
log M
s
f, μ
2
2
log fdμu − log M
s
f, μ
,s
/
0.
2.4
3. Monotonicity of new means
In this section, we will prove the monotonicity of 2.4. We need the following lemmas for
log-convex function.
4 Journal of Inequalities and Applications
Lemma 3.1. Let f be log-convex function and if x
1
≤ y
1
,x
2
≤ y
2
,x
1
/
x
2
,y
1
/
y
2
, then the following
inequality is valid:
f
x
2
f
x
1
1/x
2
−x
1
≤
f
y
2
f
y
1
1/y
2
−y
1
. 3.1
Proof. In 2, page 3 we have the following result for convex function f,withx
1
≤ y
1
,x
2
≤
y
2
,x
1
/
x
2
,y
1
/
y
2
:
f
x
2
− f
x
1
x
2
− x
1
≤
f
y
2
− f
y
1
y
2
− y
1
. 3.2
Putting f log f, we get
log
f
x
2
f
x
1
1/x
2
−x
1
≤ log
f
y
2
f
y
1
1/y
2
−y
1
, 3.3
after applying exponential function we get 3.1.
The following two lemmas are proved for functionals in 3Theorem 4 and Lemma 2,
for Lemma 3.2 see also 4,Theorem1.
Lemma 3.2. Let us consider Λ
t
defined as
Λ
t
g,μ
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
M
t
t
g,μ − M
t
1
g,μ
tt − 1
,t
/
0, 1;
log M
1
g,μ − log M
t
0
g,μ,t 0;
g log gμ − M
0
g,μ log M
0
g,μ,t 1.
3.4
Then, Λ
t
is a log-convex function.
Lemma 3.3. Let us consider Λ
t
defined as
Λ
t
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
t
2
M
t
t
f, μ − M
t
0
f, μ
,t
/
0;
1
2
M
2
2
log f, μ − M
2
1
log f, μ
,t 0.
3.5
Then, Λ
t
is a log-convex function.
Theorem 3.4. Let t, r, u, v ∈
R,suchthat,t ≤ v, r ≤ u. Then for 2.4,wehave
M
s
t,r
≤ M
s
v,u
. 3.6
M. Anwar and J. Pe
ˇ
cari
´
c5
Proof
Case 1 s
/
0. Let us consider Λ
t
defined as in Lemma 3.2. Λ
t
is a continuous and log-convex.
So, Lemma 3.1 implies that for t, r, u, v ∈
R, such that, t ≤ v, r ≤ u, t
/
r, v
/
u,wehave
Λ
t
Λ
r
1t−r
≤
Λ
v
Λ
u
1/v−u
. 3.7
For s>0 by substituting g f
s
,t t/s, r r/s, u u/s, v v/s ∈ R, such that, t/s ≤
v/s, r/s ≤ u/s, t
/
r, v
/
u,in3.4,weget
Λ
t,s
f, μ
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
s
2
t1 − s
M
t
t
f, μ − M
t
s
f, μ
,t
/
0,s;
s
log M
s
f, μ − log M
0
f, μ
,t 0;
s
f
s
log f − M
s
0
f, μ log M
0
f, μ
,t s.
3.8
And 3.7 becomes
Λ
t,s
Λ
r,s
1t−r
≤
Λ
v,s
Λ
u,s
1/v−u
. 3.9
From 3.9, we get our required result.
Now when s<0 by substituting g f
s
,t t/s, r r/s, u u/s, v v/s ∈ R, such
that, v/s ≤ t/s, u/s ≤ r/s, t
/
r, v
/
u,in3.4 we get 3.8.
And 3.7 becomes
Λ
v,s
Λ
u,s
s/v−u
≤
Λ
t,s
Λ
r,s
s/t−r
. 3.10
Now s<0, from 3.10, by raising power −s,weget
Λ
t,s
Λ
r,s
1/t−r
≤
Λ
v,s
Λ
u,s
1/v−u
. 3.11
From 3.11, we get our required result.
Case 2 s 0. In this case, we can get our result by taking limit s→0in3.8 and also in this
case we can consider Λ
t
defined as in Lemma 3.3.
Λ
t
is log-convex function. So, Lemma 3.1 implies that for t, r, u, v ∈ R, such that, t ≤
v, r ≤ u, t
/
r, v
/
u,wehave
Λ
t
Λ
r
1/t−r
≤
Λ
v
Λ
u
1/v−u
. 3.12
Therefore, we have for t, r, u, v ∈
R, such that, t ≤ v, r ≤ u, t
/
r, v
/
u:
M
0
t,r
≤ M
0
v,u
, 3.13
which completes the proof.
6 Journal of Inequalities and Applications
4. Further consequences and applications
In this section, we will represent the various applications of our previous definition of a new
Cauchy mean and monotonicity of this above defined a new Cauchy mean.
4.1. Tobey and Stolarsky-Tobey means
Let E
n−1
represent the n − 1-dimensional Euclidean simplex given by
E
n−1
u
1
,u
2
, ,u
n−1
: u
i
≥ 0, 1 ≤ i ≤ n − 1,
n−1
i1
u
i
≤ 1
, 4.1
and set u
n
1 −
n−1
i1
u
i
. Moreover, with u u
1
, ,u
n
, let μu be a probability measure on
E
n−1
. The power mean of order p p ∈ R of the positive n-tuple x x
1
, ,x
n
∈ R
n
,withthe
weights u u
1
, ,u
n
, is defined by
M
p
x, μ
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
n
i1
u
i
x
p
i
1/p
,p
/
0;
n
i1
x
u
i
i
,p 0.
4.2
Then, the Tobey mean L
p,r
x; μ is defined as follows:
L
p,r
x; μM
r
M
p
x, μ; μ
, 4.3
where M
r
g,μ denotes the integral power mean, in which Ω is now the n − 1-dimensional
Euclidean simplex E
n−1
. We note that, since M
p
x, μ isameanwehavemin{x
i
}≤M
p
x, μ ≤
max{x
i
}. Now setting fx, μM
p
x, μ in 2.4 we get
Γ
s
p,r,l
x, μ
ll − s
rr − s
L
r
p,r
x, μ − L
r
p,s
x, μ
L
l
p,l
x, μ − L
l
p,s
x, μ
1/r−l
,l
/
r
/
s, l, r
/
0;
Γ
s
p,r,0
x, μΓ
s
p,0,r
x, μ
s
L
r
p,r
x, μ − L
r
p,s
x, μ
rr − s
log L
p,s
x, μ − log L
p,0
x, μ
1/r
,r
/
s, r, s
/
0;
Γ
s
p,s,l
x, μΓ
s
p,l,s
x, μ
ll − s
s
M
p
x, μ
s
log dμu − L
s
p,s
x, μ log L
p,s
x, μ
L
l
p,l
x, μ − L
l
p,s
x, μ
1/s−l
,
l
/
s, l, s
/
0;
Γ
s
p,s,0
x, μΓ
s
p,0,s
x, μ
M
p
x, μ
s
log M
p
x, μdμu−L
s
p,s
x, μ log L
p,s
x, μ
log L
p,s
x, μ−log L
p,0
x, μ
1/s
,s
/
0;
M. Anwar and J. Pe
ˇ
cari
´
c7
Γ
0
p,r,l
x, μ
l
2
L
r
p,r
x, μ − L
r
p,0
x, μ
r
2
L
l
p,l
x, μ − L
l
p,0
x, μ
1/r−l
,l,r
/
0;
Γ
0
p,r,0
x, μΓ
0
p,0,r
x, μ
2
L
r
p,r
x, μ − L
r
p,0
x, μ
r
2
M
2
2
log
M
p
x, μ,μ
− M
2
1
log
M
p
x, μ,μ
1/r
,r
/
0;
Γ
s
p,t,t
x, μexp
−
2t − s
tt − s
M
p
x, μ
t
log M
p
x, μdμu−L
t
p,s
x, μ log L
p,s
x, μ
L
t
p,t
x, μ−L
t
p,s
x, μ
,t
/
s;
Γ
0
p,t,t
x, μexp
−
2
t
M
p
x, μ
t
log M
p
x, μdμu − L
t
p,0
x, μ log L
p,0
x, μ
L
t
p,t
x, μ − L
t
p,0
x, μ
,t
/
0;
Γ
0
p,0,0
x, μexp
1
3
log
M
p
x, μ
3
dμu −
log L
p,0
x, μ
3
log
M
p
x, μ
2
dμu −
log L
p,0
x, μ
2
,
Γ
s
p,s,s
x, μexp
−
1
s
M
p
x, μ
s
log
M
p
x, μ
2
dμu−L
s
p,s
x, μ
log L
p,s
x, μ
2
2
M
p
x, μ
s
log M
p
x, μdμu−
L
s
p,s
x, μ log L
p,s
x, μ
,s
/
0;
Γ
s
p,0,0
x, μexp
1
s
log
M
p
x, μ
2
dμu −
log L
p,s
x, μ
2
2
log M
p
x, μdμu − log M
s
x, μ
,s
/
0.
4.4
Theorem 4.1. Let t, r, u, v ∈
R,suchthat,t<v, r<u.Then for 4.4,wehave
Γ
s
p,t,r
≤ Γ
s
p,v,u
. 4.5
Proof. It is a simple consequence of Theorem 3.4.
Pe
ˇ
cari
´
cand
ˇ
Simi
´
c see 5, Definition 1 introduced the Stolarsky-Tobey mean ε
p,q
x, μ
defined by
ε
p,q
x, μL
p,q−p
x, νM
q−p
M
p
x, μ; μ
, 4.6
where L
p,r
x, ν is the Tobey mean already introduced above.
For the Stolarsky-Tobey mean and 2.4,wegetthefollowing:
Υ
s
p,r,l
x, μ
ll − s
rr − s
ε
r
p,pr
x, μ − ε
r
p,ps
x, μ
ε
l
p,pl
x, μ − ε
l
p,ps
x, μ
1/r−l
,l
/
r
/
s, l, r
/
0;
Υ
s
p,r,0
x, μΥ
s
p,0,r
x, μ
s
ε
r
p,pr
x, μ − ε
r
p,ps
x, μ
rr − s
log ε
p,ps
x, μ − log ε
p,p
x, μ
1/r
,r
/
s, r, s
/
0;
8 Journal of Inequalities and Applications
Υ
s
p,s,l
x, μΥ
s
p,l,s
x, μ
ll − s
s
M
p
x, μ
s
log dμu − ε
s
p,ps
x, μ log ε
p,ps
x, μ
ε
l
p,pl
x, μ − ε
l
p,ps
x, μ
1/s−l
,
l
/
s, l, s
/
0;
Υ
s
p,s,0
x, μΥ
s
p,0,s
x, μ
M
p
x, μ
s
log M
p
x, μdμu − ε
s
p,ps
x, μ log ε
p,ps
x, μ
log ε
p,ps
x, μ − log ε
p,p
x, μ
1/s
,
s
/
0;
Υ
0
p,r,l
x, μ
l
2
ε
r
p,pr
x, μ − ε
r
p,p
x, μ
r
2
ε
l
p,pl
x, μ − ε
l
p,p
x, μ
1/r−l
,l,r
/
0;
Υ
0
p,r,0
x, μΥ
0
p,0,r
x, μ
2ε
r
p,pr
x, μ − ε
r
p,p
x, μ
r
2
M
2
2
log
M
p
x, μ,μ
− M
2
1
log
M
p
x, μ,μ
1/r
,r
/
0;
Υ
s
p,t,t
x, μexp
−
2t − s
tt − s
M
p
x, μ
t
log M
p
x, μdμu − M
t
s
log ε
p.ps
x, μ
ε
t
p,pt
x, μ − ε
t
p,ps
x, μ
,t
/
s;
Υ
0
p,t,t
x, μexp
−
2
t
M
p
x, μ
t
log M
p
x, μdμu − ε
t
P,p
x, μ log ε
p,p
x, μ
ε
t
p,pt
x, μ − ε
t
p,p
x, μ
,t
/
0;
Υ
0
p,0,0
x, μexp
1
3
log
M
p
x, μ
3
dμu −
log ε
p,p
x, μ
3
log
M
p
x, μ
2
dμu −
log ε
p,p
x, μ
2
,
Υ
s
p,s,s
x, μexp
−
1
s
M
p
x, μ
s
log
M
p
x, μ
2
dμu − ε
s
p,ps
x, μlog ε
p.ps
x, μ
2
2
M
p
x, μ
s
log M
p
x, μdμu −
ε
s
p.ps
x, μ log ε
p,ps
x, μ
,
s
/
0;
Υ
s
p,0,0
x, μexp
1
s
log
M
p
x, μ
2
dμu −
log ε
p,ps
x, μ
2
2
log M
p
x, μdμu − log ε
p.ps
x, μ
,s
/
0.
4.7
Theorem 4.2. Let t, r, u, v ∈
R,suchthat,t<v, r<u.Then for 4.7,wehave
Υ
s
p,t,r
≤ Υ
s
p,v,u
. 4.8
Proof. It is a simple consequence of Theorem 3.4.
M. Anwar and J. Pe
ˇ
cari
´
c9
4.2. The complete symmetric mean
The rth complete symmetric polynomial mean the complete symmetric mean of the positive
real n-tuple x is defined by see 6, pages 332,341
Q
r
n
x
q
r
n
x
1/r
c
r
n
x
nr−1
r
1/r
, 4.9
where c
0
n
x1andc
r
n
x
n
j1
n
i1
x
i
j
i
and the sum is taken over all
nr−1
r
nonnegative
integer n-tuples i
1
, ,i
n
with
n
j1
i
j
r r
/
0. The complete symmetric polynomial mean
can also be written in an integral form as follows:
Q
r
n
E
n−1
n
i1
x
i
u
i
r
dμu
1/r
, 4.10
where μ represents a probability measure such that dμun−1!du
1
···du
n−1
. We can see this
as a special case of the integral power mean M
r
f, μ,wherefu
n
i1
x
i
u
i
,μis a probability
measure as above, and Ω is the above defined n−1-dimensional simplex E
n−1
. Thus from 2.4,
we have the following result:
Θ
s
n,r,l
x, μ
ll − s
rr − s
Q
r
n
r
x, μ −
Q
s
n
r
x, μ
Q
l
n
l
x, μ −
Q
s
n
l
x, μ
1/r−l
,l
/
r
/
s, l, r ∈ N.
4.11
A simple consequence of Theorem 3.4 is the following result.
Theorem 4.3. Let t, r, u, v ∈
N,suchthat,t<v, r<u.Then for 4.11,wehave
Θ
s
n,t,r
≤ Θ
s
n,v,u
. 4.12
4.3. Whiteley means
Let x be a positive real n-tuple, s ∈
R s
/
0 and r ∈ N. Then, the sth function of degree r is
defined by the following generating function see 6, pages 341–344:
∞
r0
t
r,s
n
xt
r
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
n
i1
1 x
i
t
s
,s>0,
n
i1
1 − x
i
t
s
,s<0.
4.13
The Whiteley mean is now defined by
W
r,s
n
x
w
r,s
n
x
1/r
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
t
r,s
n
x
nr
s
1/r
,s>0,
t
r,s
n
x
−1
r
nr
s
1/r
,s<0.
4.14
10 Journal of Inequalities and Applications
For s<0, the Whiteley mean can be further generalized if we slightly change the definition of
t
r,s
n
x and define h
r,σ
n
x as follows:
∞
r0
h
r,σ
n
xt
r
n
i1
1
1 − x
i
t
σ
i
, 4.15
where σ σ
1
, ,σ
n
; σ ∈ R
; i 1, ,n. The following generalization of the Whiteley mean
for s<0 is defined by see 7, Lemma 2.3
H
r,σ
n
x
h
r,σ
n
x
n
i1
σ
i
r−1
r
1/r
. 4.16
If we denote by μ a measure on the simplex Δ
n−1
{u
1
, ,u
n
: u
i
≥ 0,i 1, ,n −
1,
n
i1
u
i
≤ 1} such that
dμu
Γ
n
i1
σ
i
n
i1
Γ
σ
i
n
i1
u
σ
i
−1
i
du
1
···du
n−1
, 4.17
where u
n
1 −
n−1
i−1
, then we have μ as a probability measure and we can also write the mean
H
r,σ
n
x in integral form as follows:
H
r,σ
n
x
Δ
n−1
n
i1
x
i
u
i
r
dμu
1/r
. 4.18
Finally, just as we did above in this investigation, we can develop the following analogous
definition:
H
s
n,r,l
x, μ
⎛
⎜
⎝
ll − s
rr − s
H
r,σ
n
r
x, μ −
H
s,σ
n
r
x, μ
H
l,σ
n
l
x, μ −
H
s,σ
n
l
x, μ
⎞
⎟
⎠
1/r−l
,l
/
r
/
s, l, r ∈ N.
4.19
A simple consequence of Theorem 3.4 is the following result.
Theorem 4.4. Let t, r, u, v ∈
N,suchthat,t<v, r<u.Then for 4.19,wehave
H
s
n,t,r
≤ H
s
n,v,u
. 4.20
References
1 J. Pe
ˇ
cari
´
c, M. R. Lipanovi
´
c, and H. M. Srivastava, “Some mean-value theorems of the Cauchy type,”
Fractional Calculus & Applied Analysis, vol. 9, no. 2, pp. 143–158, 2006.
2 J. Pe
ˇ
cari
´
c, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications,
vol. 187 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1992.
3 M. Anwar and J. Pe
ˇ
cari
´
c, “On logarithmic convexity for differences of power means,” to appear in
Mathematical Inequalities & Applications.
4 S. Simi
´
c, “On logarithmic convexity for differences of power means,” Journal of Inequalities and
Applications, vol. 2007, Article ID 37359, 8 pages, 2007.
5 J. Pe
ˇ
cari
´
candV.
ˇ
Simi
´
c, “Stolarsky-Tobey mean in n variables,” Mathematical Inequalities & Applications,
vol. 2, no. 3, pp. 325–341, 1999.
6 P. S. Bullen, Handbook of Means and Their Inequalities, vol. 560 of Mathematics and Its Applications, Kluwer
Academic Publishers, Dordrecht, The Netherlands, 2003.
7 J. Pe
ˇ
cari
´
c, I. Peri
´
c, and M. R. Lipanovi
´
c, “Generalized Whiteley means and related inequalities,” to
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