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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 305623, 12 pages
doi:10.1155/2008/305623
Research Article
On Logarithmic Convexity for Power Sums and
Related Results II
J. Pe
ˇ
cari
´
c
1, 2
andAtiqurRehman
1
1
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
Correspondence should be addressed to Atiq ur Rehman,
Received 14 October 2008; Accepted 4 December 2008
Recommended by Wing-Sum Cheung
In the paper “On logarithmic convexity for power sums and related results” 2008, we introduced
means by using power sums and increasing function. In this paper, we will define new means of
convex type in connection to power sums. Also we give integral analogs of new means.
Copyright q 2008 J. Pe
ˇ
cari
´
c and A. ur Rehman. 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 and preliminaries
Let x be positive n-tuples. The well-known inequality for power sums of order s and r,for
s>r>0 see 1, page 164, states that
n
i1
x
s
i
1/s
<
n
i1
x
r
i
1/r
.
1.1
Moreover, if p p
1
, ,p
n
is a positive n-tuples such that p
i
≥ 1 i 1, ,n, then for
s>r>0 see 1, page 165, we have
n
i1
p
i
x
s
i
1/s
<
n
i1
p
i
x
r
i
1/r
.
1.2
In 2, we defined the following function:
Δ
t
Δ
t
x; p
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1
t − 1
n
i1
p
i
x
i
t
−
n
i1
p
i
x
t
i
,t
/
1,
n
i1
p
i
x
i
log
n
i1
p
i
x
i
−
n
i1
p
i
x
i
log x
i
,t 1.
1.3
2 Journal of Inequalities and Applications
We introduced the Cauchy means involving power sums. Namely, the following results were
obtained in 2.
For r<s<t,where r, s, t ∈ R
, we have
Δ
s
t−r
≤
Δ
r
t−s
Δ
t
s−r
, 1.4
such that, x
i
∈ 0,ai 1, ,n and
n
i1
p
i
x
i
≥ x
j
, for j 1, ,n,
n
i1
p
i
x
i
∈ 0,a. 1.5
We defined the following means.
Definition 1.1. Let x and p be two nonnegative n-tuples n ≥ 2 such that p
i
≥ 1 i 1, ,n.
Then for t, r, s ∈ R
,
A
s
t,r
x; p
r − s
t − s
n
i1
p
i
x
s
i
t/s
−
n
i1
p
i
x
t
i
n
i1
p
i
x
s
i
r/s
−
n
i1
p
i
x
r
i
1/t−r
,t
/
r, r
/
s, t
/
s,
A
s
s,r
x; pA
s
r,s
x; p
r − s
s
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
s
i
log x
i
n
i1
p
i
x
s
i
r/s
−
n
i1
p
i
x
r
i
1/s−r
,s
/
r,
A
s
r,r
x; pexp
1
s − r
n
i1
p
i
x
s
i
r/s
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
r
i
log x
i
s
n
i1
p
i
x
s
i
r/s
−
n
i1
p
i
x
r
i
,s
/
r,
A
s
s,s
x; pexp
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
2
− s
2
n
i1
p
i
x
s
i
log x
i
2
2s
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
s
i
log x
i
.
1.6
In this paper, we introduce new Cauchy means of convex type in connection with
Power sums. For means, we shall use the following result 1, page 154.
Theorem 1.2. Let x and p be two nonnegative n-tuples such that condition 1.5 is valid. If f is a
convex function on 0,a,then
f
n
i1
p
i
x
i
≥
n
i1
p
i
f
x
i
1 −
n
i1
p
i
f0. 1.7
Remark 1.3. In Theorem 1.2,iff is strictly convex, then 1.7 is strict unless x
1
··· x
n
and
n
i1
p
i
1.
J. Pe
ˇ
cari
´
c and A. ur Rehman 3
2. Discrete result
Lemma 2.1. Let
ϕ
t
x
⎧
⎪
⎨
⎪
⎩
x
t
tt − 1
,t
/
1,
x log x, t 1,
2.1
where t ∈ R
.Thenϕ
t
x is strictly convex for x>0.
Here,weusethenotation0log0: 0.
Proof. Since ϕ
t
xx
t−2
> 0forx>0, therefore ϕ
t
x is strictly convex for x>0.
Lemma 2.2 see 3. A positive function f is log convex in Jensen sense on an open interval I, that
is, for each s, t ∈ I
fsft ≥ f
2
s t
2
, 2.2
if and only if the relation
u
2
fs2uwf
s t
2
w
2
ft ≥ 0, 2.3
holds for each real u, w and s, t ∈ I.
The following lemma is equivalent to definition of convex function 1, page 2.
Lemma 2.3. If f is continuous and convex for all x
1
, x
2
, x
3
of an open interval I for which x
1
<x
2
<
x
3
,then
x
3
− x
2
f
x
1
x
1
− x
3
f
x
2
x
2
− x
1
f
x
3
≥ 0. 2.4
Lemma 2.4. 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
. 2.5
By using the above lemmas and Theorem 1.2,asin2, we can prove the following
results.
Theorem 2.5. Let x and p be two positive n-tuples and let
Δ
t
Δ
t
(x; p
Δ
t
t
, 2.6
4 Journal of Inequalities and Applications
such that condition 1.5 is satisfied and all x
i
’s are not equal. Then Δ
t
is log-convex. Also for r<s<t
where r, s, t ∈ R
, we have
Δ
s
t−r
≤
Δ
r
t−s
Δ
t
s−r
. 2.7
Moreover, we can use 2.7 to obtain new means of Cauchy type involving power
sums.
Let us introduce the following means.
Definition 2.6. Let x and p be two nonnegative n-tuples such that p
i
≥ 1 i 1, ,n, then for
t, r, s ∈ R
,
B
s
t,r
x; p
rr − s
tt − s
n
i1
p
i
x
s
i
t/s
−
n
i1
p
i
x
t
i
n
i1
p
i
x
s
i
r/s
−
n
i1
p
i
x
r
i
1/t−r
,t
/
r, r
/
s, t
/
s,
B
s
s,r
x; pB
s
r,s
x; p
rr − s
s
2
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
s
i
log x
i
n
i1
p
i
x
s
i
r/s
−
n
i1
p
i
x
r
i
1/s−r
,s
/
r,
B
s
r,r
x; pexp
−
2r − s
rr − s
n
i1
p
i
x
s
i
r/s
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
r
i
log x
i
s
n
i1
p
i
x
s
i
r/s
−
n
i1
p
i
x
r
i
,s
/
r,
B
s
s,s
x; pexp
−
1
s
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
2
− s
2
n
i1
p
i
x
s
i
log x
i
2
2s
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
s
i
log x
i
.
2.8
Remark 2.7. Let us note that B
s
s,r
x; pB
s
r,s
x; plim
t → s
B
s
t,r
x; plim
t → s
B
s
r,t
x; p,
B
s
r,r
x; plim
t → r
B
s
t,r
x; p and B
s
s,s
x; plim
r → s
B
s
r,r
x; p.
Theorem 2.8. Let
Θ
s
t
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1
tt − s
n
i1
p
i
x
s
i
t/s
−
n
i1
p
i
x
t
i
,t
/
s,
1
s
2
n
i1
p
i
x
s
i
log
n
i1
p
i
x
s
i
− s
n
i1
p
i
x
s
i
log x
i
,t s.
2.9
then for t, r, u ∈ R
and t<r<u, we have
Θ
s
r
u−t
≤
Θ
s
t
u−r
Θ
s
u
r−t
. 2.10
Theorem 2.9. Let r, t, u, v ∈ R
, such that t ≤ v, r ≤ u. Then one has
B
s
t,r
(x; p ≤ B
s
v,u
(x; p. 2.11
J. Pe
ˇ
cari
´
c and A. ur Rehman 5
Remark 2.10. From 2.7, we have
Δ
s
s
t−r
≤
Δ
r
r
t−s
Δ
t
t
s−r
⇒
Δ
s
t−r
≤
s
t−r
r
t−s
t
s−r
Δ
r
t−s
Δ
t
s−r
. 2.12
Since log x is concave, therefore f or r<s<t, we have
t − s log r r − t log s s − r log t<0 ⇒
s
t−r
r
t−s
t
s−r
> 1. 2.13
This implies that 1.4, which we derived in 2, is better than 2.7.
Also note that
B
s
t,r
x; p
r
t
1/t−r
A
s
t,r
x; p,
B
s
r,s
x; pB
s
s,r
x; p
r
s
1/s−r
A
s
s,r
x; p
r
s
1/s−r
A
s
r,s
x; p,
B
s
r,r
x; pexp
−
1
r
A
s
r,r
x; p,
B
s
s,s
x; pexp
−
1
s
A
s
s,s
x; p.
2.14
Let us note that there are not integral analogs of results from 2. Moreover, in Section 3
we will show that previous results have their integral analogs.
3. Integral results
The following theorem is very useful for further result 1, page 159.
Theorem 3.1. Let t
0
∈ a, b be fixed, h be continuous and monotonic with ht
0
0, g be a function
of bounded variation and
Gt :
t
a
dgx, Gt :
b
t
dgx. 3.1
a If
0 ≤ Gt ≤ 1 for a ≤ t ≤ t
0
, 0 ≤ Gt ≤ 1 for t
0
≤ t ≤ b, 3.2
then for every convex function f : I → R such that hx ∈ I for all x ∈ a, b,
b
a
f
ht
dgt ≥ f
b
a
htdgt
b
a
dgt − 1
f0. 3.3
6 Journal of Inequalities and Applications
b If
b
a
htdgt ∈ I and either there exists an s ≤ t
0
such that
Gt ≤ 0 for t < s, Gt ≥ 1 for s ≤ t ≤ t
0
, Gt ≤ 0 for t>t
0
, 3.4
or there exists an s ≥ t
0
such that
Gt ≤ 0 for t<t
0
, Gt ≥ 1 for t
0
<t<s, Gt ≤ 0 for t ≥ s, 3.5
then for every convex function f : I → R such that hx ∈ I for all x ∈ a, b, the reverse of the
inequality in 3.3 holds.
To define the new means of Cauchy involving integrals, we define the following
function.
Definition 3.2. Let t
0
∈ a, b be fixed, h be continuous and monotonic with ht
0
0, g be a
function of bounded variation. Choose g such that function Λ
t
is positive valued, where Λ
t
is
defined as follows:
Λ
t
Λ
t
a, b, h, g
b
a
ϕ
t
hx
dgx − ϕ
t
b
a
hxdgx
. 3.6
Theorem 3.3. Let Λ
t
, defined as above, satisfy condition 3.2.ThenΛ
t
is log-convex. Also for r<
s<t,wherer, s, t ∈ R
, one has
Λ
s
t−r
≤
Λ
r
t−s
Λ
t
s−r
. 3.7
Proof. Let fxu
2
ϕ
s
x2uwϕ
r
xw
2
ϕ
t
x, where r s t/2andu, w ∈ R,
f
xu
2
x
s−2
2uwx
r−2
w
2
x
t−2
ux
s−2/2
wx
t−2/2
2
≥ 0. 3.8
This implies that fx is convex.
By Theorem 3.1, we have,
b
a
f
ht
dgt − f
b
a
htdgt
−
b
a
dgt − 1
f0 ≥ 0
⇒ u
2
b
a
ϕ
s
hxdgx − ϕ
s
b
a
hxdgx
2uw
b
a
ϕ
r
hxdgx − ϕ
r
b
a
hxdgx
2w
2
b
a
ϕ
t
hxdgx − ϕ
t
b
a
hxdgx
≥ 0
⇒ u
2
Λ
s
2uwΛ
r
w
2
Λ
t
≥ 0.
3.9
Now, by Lemma 2.2, we have Λ
t
is log-convex in Jensen sense.
J. Pe
ˇ
cari
´
c and A. ur Rehman 7
Since lim
t → 1
Λ
t
Λ
1
,thisimpliesthatΛ
t
is continuous for all t ∈ R
, therefore it is a
log-convex 1, page 6.
Since Λ
t
is log-convex, that is, log Λ
t
is convex, therefore by Lemma 2.3 for r<s<t
and taking f log Λ, we have
t − s log Λ
r
r − t log Λ
s
s − r log Λ
t
≥ 0, 3.10
which is equivalent to 3.7.
Theorem 3.4. Let
Λ
t
−Λ
t
such that condition 3.4 or 3.5 is satisfied. Then
Λ
t
is log-convex.
Also for r<s<t,wherer, s, t ∈ R
, one has
Λ
s
t−r
≤
Λ
r
t−s
Λ
t
s−r
. 3.11
Definition 3.5. Let t
0
∈ a, b be fixed, h be continuous and monotonic with ht
0
0, g be a
function of bounded variation. Then for t, r, s ∈ R
, one defines
F
s
t,r
a, b, h, g
⎧
⎨
⎩
rr − s
tt − s
b
a
h
t
xdgx −
b
a
hxdgx
t/s
b
a
h
r
xdgx −
b
a
hxdgx
r/s
⎫
⎬
⎭
1/t−r
,t
/
r, r
/
s, t
/
s,
F
s
s,r
a, b, h, g
F
s
r,s
a, b, h, g
⎧
⎨
⎩
rr − s
s
2
s
b
a
h
s
x log hxdgx −
b
a
h
s
xdgx
log
b
a
h
s
xdgx
b
a
h
r
xdgx −
b
a
h
s
xdgx
r/s
⎫
⎬
⎭
1/s−r
,s
/
r,
F
s
r,r
a, b, h, g
exp
−
2r − s
rr − s
s
b
a
h
r
x log hxdgx−
b
a
h
s
xdgx
r/s
log
b
a
h
s
xdgx
s
b
a
h
r
xdgx−
b
a
h
s
xdgx
r/s
,s
/
r,
F
s
s,s
a, b, h, g
exp
−
1
s
s
2
b
a
h
s
x
log hx
2
dgx −
b
a
h
s
xdgx
log
b
a
h
s
xdgx
2
2s
s
b
a
h
s
x log hxdgx −
b
a
h
s
xdgx
log
b
a
h
s
xdgx
.
3.12
Remark 3.6. Let us note that F
s
s,r
a, b, h, gF
s
r,s
a, b, h, glim
t→s
F
s
t,r
a, b, h, g
lim
t→s
F
s
r,t
a, b, h, g, F
s
r,r
a, b, h, glim
t→r
F
s
t,r
a, b, h, g and F
s
s,s
a, b, h, glim
r→s
F
s
r,r
a, b,
h, g.
8 Journal of Inequalities and Applications
Theorem 3.7. Let r, t, u, v ∈ R
, such that t ≤ v, r ≤ u.Then
F
s
t,r
a, b, h, g ≤ F
s
v,u
a, b, h, g. 3.13
Proof. Let
Λ
t
Λ
t
a, b, h, g
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1
tt − 1
b
a
h
t
xdgx −
b
a
hxdgx
t
,t
/
1,
b
a
hx log hxdgx −
b
a
hxdgx log
b
a
hxdgx,t 1.
3.14
Now, taking x
1
r, x
2
t, y
1
u, y
2
v, where r, t, u, v
/
1, and ftΛ
t
in Lemma 2.4,we
have
⎛
⎝
rr − 1
tt − 1
b
a
h
t
xdgx −
b
a
hxdgx
t
b
a
h
r
xdgx −
b
a
hxdgx
r
⎞
⎠
1/t−r
≤
⎛
⎝
uu − 1
vv − 1
b
a
h
v
xdgx −
b
a
hxdgx
v
b
a
h
u
xdgx −
b
a
hxdgx
u
⎞
⎠
1/v−u
.
3.15
Since s>0, by substituting h h
s
, t t/s, r r/s, u u/s, and v v/s, where r, t, v, u
/
s,
in above inequality, we get
⎛
⎝
rr − s
tt − s
b
a
h
t
xdgx −
b
a
h
s
xdgx
t/s
b
a
h
r
xdgx −
b
a
h
s
xdgx
r/s
⎞
⎠
s/t−r
≤
⎛
⎝
uu − s
vv − s
b
a
h
v
xdgx −
b
a
h
s
xdgx
v/s
b
a
h
u
xdgx −
b
a
h
s
xdgx
u/s
⎞
⎠
s/v−u
.
3.16
By raising power 1/s, we get an inequality 3.13 for r, t, v, u
/
s.
From Remark 3.6 ,weget3.13 is also valid for r s or t s or r t or t r s.
Lemma 3.8. Let f ∈ C
2
I such that
m ≤ f
x ≤ M. 3.17
J. Pe
ˇ
cari
´
c and A. ur Rehman 9
Consider the functions φ
1
, φ
2
defined as
φ
1
x
Mx
2
2
− fx,
φ
2
xfx −
mx
2
2
.
3.18
Then φ
i
x for i 1, 2 are convex.
Proof. We have that
φ
1
xM − f
x ≥ 0,
φ
2
xf
x − m ≥ 0,
3.19
that is, φ
i
for i 1, 2 are convex.
Theorem 3.9. Let t
0
∈ a, b be fixed, h be continuous and monotonic with ht
0
0, g be a function
of bounded variation, and f ∈ C
2
I such that condition 3.2 is satisfied. Then there exists ξ ∈ I such
that
b
a
f
hx
dgx − f
b
a
hxdgx
−
b
a
dgx − 1
f
ξ
2
b
a
h
2
xdgx −
b
a
hxdgx
2
.
3.20
Proof. In Theorem 3.1, setting f φ
1
and f φ
2
, respectively, as defined in Lemma 3.8,we
get the following inequalities:
b
a
f
hx
dgx − f
b
a
hxdgx
−
b
a
dgx − 1
≤
M
2
b
a
h
2
xdgx −
b
a
hxdgx
2
,
3.21
b
a
f
hx
dgx − f
b
a
hxdgx
−
b
a
dgx − 1
≥
m
2
b
a
h
2
xdgx −
b
a
hxdgx
2
.
3.22
Now, by combining both inequalities, we get
m ≤
2
b
a
f
hx
dgx − f
b
a
hxdgx
−
b
a
dgx − 1
f0
b
a
h
2
xdgx −
b
a
hxdgx
2
≤ M. 3.23
10 Journal of Inequalities and Applications
So by condition 3.17, t here exists ξ ∈ I such that
2
b
a
f
hx
dgx − f
b
a
hxdgx
−
b
a
dgx − 1
f0
b
a
h
2
xdgx −
b
a
hxdgx
2
f
ξ, 3.24
and 3.24 implies 3.20.
Moreover, 3.21 is valid if f
is bounded from above and again we have 3.20 is valid.
Of course 3.20 is obvious if f
is not bounded from above and below as well.
Theorem 3.10. Let t
0
∈ a, b be fixed, h be continuous and monotonic with ht
0
0, g be a
function of bounded variation, and f
1
,f
2
∈ C
2
I such that condition 3.2 is satisfied. Then there
exists ξ ∈ I such that the following equality is true:
b
a
f
1
hx
dgx − f
1
b
a
hxdgx
−
b
a
dgx − 1
f
1
0
b
a
f
2
hx
dgx − f
2
b
a
hxdgx
−
b
a
dgx − 1
f
2
0
f
1
ξ
f
2
ξ
, 3.25
provided that denominators are nonzero.
Proof. Let a function k ∈ C
2
I be defined as
k c
1
f
1
− c
2
f
2
, 3.26
where c
1
and c
2
are defined as
c
1
b
a
f
2
hx
dgx − f
2
b
a
hxdgx
−
b
a
dgx − 1
f
2
0,
c
2
b
a
f
1
hx
dgx − f
1
b
a
hxdgx
−
b
a
dgx − 1
f
1
0.
3.27
Then, using Theorem 3.9 with f k, we have
0
c
1
f
1
ξ − c
2
f
2
ξ
b
a
h
2
xdgx −
b
a
hxdgx
2
. 3.28
Since
b
a
h
2
xdgx −
b
a
hxdgx
2
/
0, 3.29
J. Pe
ˇ
cari
´
c and A. ur Rehman 11
therefore, 3.28 gives
c
2
c
1
f
1
ξ
f
2
ξ
. 3.30
After putting values, we get 3.25.
Let α be a strictly monotone continuous function, we defined T
α
h, g as follows
integral version of quasiarithmetic sum 2:
T
α
h, gα
−1
b
a
α
hx
dgx
. 3.31
Theorem 3.11. Let α, β, γ ∈ C
2
a, b be strictly monotonic continuous functions. Then there exists
η in the image of hx such that
α
T
α
h, g
− α
T
γ
h, g
−
b
a
dgx − 1
α ◦ γ
−1
0
β
T
β
h, g
− β
T
γ
h, g
−
b
a
dgx − 1
β ◦ γ
−1
0
α
ηγ
η − α
ηγ
η
β
ηγ
η − β
ηγ
η
3.32
is valid, provided that all denominators are nonzero.
Proof. If we choose the functions f
1
and f
2
so that f
1
α◦γ
−1
, f
2
β◦γ
−1
,andhx → γhx.
Substituting these in 3.25,
α
T
α
h, g
− α
T
γ
h, g
−
b
a
dgx − 1
α ◦ γ
−1
0
β
T
β
h, g
− β
T
γ
h, g
−
b
a
dgx − 1
β ◦ γ
−1
0
α
γ
−1
ξ
γ
γ
−1
ξ
− α
γ
−1
ξ
γ
γ
−1
ξ
β
γ
−1
ξ
γ
γ
−1
ξ
− β
γ
−1
ξ
γ
γ
−1
ξ
.
3.33
Then by setting γ
−1
ξη,weget3.32.
Corollary 3.12. Let t
0
∈ a, b be fixed, h be continuous and monotonic with ht
0
0, g be a
function of bounded variation, and let t, r, s ∈ R
.Then
F
s
t,r
a, b, h, gη. 3.34
Proof. If t, r, and s are pairwise distinct, then we put αxx
t
, βxx
r
and γxx
s
in
3.32 to get 3.34.
For other cases, we can consider limit as in Remark 3.6.
Acknowledgments
This research was partially funded by Higher Education Commission, Pakistan. The research
of the first author was supported by the Croatian Ministry of Science, Education and Sports
under the research Grant 117-1170889-0888.
12 Journal of Inequalities and Applications
References
1 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.
2 J. Pe
ˇ
cari
´
c and A. U. Rehman, “On logarithmic convexity for power sums and related results,” Journal
of Inequalities and Applications, vol. 2008, Article ID 389410, 9 pages, 2008.
3 S. Simic, “On logarithmic convexity for differences of power means,” Journal of Inequalities and
Applications, vol. 2007, Article ID 37359, 8 pages, 2007.
