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. Pečarić1, 2 and Atiq ur Rehman1
1
2
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
Correspondence should be addressed to Atiq ur Rehman, mathcity@gmail.com
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. Pečarić 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
1/s
xis
<
n
i1
1/r
xir
.
1.1
i1
Moreover, if p p1 , . . . , pn is a positive n-tuples such that pi ≥ 1 i 1, . . . , n, then for
s > r > 0 see 1, page 165, we have
n
1/s
pi xis
<
n
i1
1/r
pi xir
.
In 2, we defined the following function:
⎧
t
n
n
⎪
1
⎪
t
⎪
⎪
pi xi −
pi xi ,
⎨t − 1
i1
i1
Δt Δt x; p n
n
n
⎪
⎪
⎪
⎪
p
x
log
p
x
−
pi xi log xi ,
i
i
i
i
⎩
i1
1.2
i1
i1
i1
t/
1,
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
Δs
t−s s−r
≤ Δr
,
Δt
t−r
1.4
such that, xi ∈ 0, a i 1, . . . , n and
n
pi xi ≥ xj ,
n
for j 1, . . . , n,
i1
pi xi ∈ 0, a.
1.5
i1
We defined the following means.
Definition 1.1. Let x and p be two nonnegative n-tuples n ≥ 2 such that pi ≥ 1 i 1, . . . , n.
Then for t, r, s ∈ R ,
Ast,r x; p
n
1/t−r
s t/s
− ni1 pi xit
r−s
i1 pi xi
,
t − s n pi xs r/s − n pi xr
i1
i1
i
n
t
/ r, r / s, t / s,
i
1/s−r
pi xis − s ni1 pi xis log xi
,
s r/s
− ni1 pi xir
i1 pi xi
n
s r/s
log ni1 pi xis − s ni1 pi xir log xi
1
i1 pi xi
s
Ar,r x; p exp
, s
/ r,
n
s r/s
s−r
s
− ni1 pi xir
i1 pi xi
2
2 n
s
log ni1 pi xis − s2 ni1 pi xis log xi
i1 pi xi
s
As,s x; p exp
n
n
.
n
s
s
s
2s
i1 pi xi log
i1 pi xi − s
i1 pi xi log xi
Ass,r x; p Asr,s x; p r−s
s
s
i1 pi xi
log
n
n
i1
s
/ r,
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
pi xi
≥
n
i1
pi f xi 1−
n
pi f0.
1.7
i1
Remark 1.3. In Theorem 1.2, if f is strictly convex, then 1.7 is strict unless x1 · · · xn and
n
i1 pi 1.
J. Pečarić and A. ur Rehman
3
2. Discrete result
Lemma 2.1. Let
⎧
⎪
⎨
xt
,
ϕt x tt − 1
⎪
⎩x log x,
t/
1,
t 1,
2.1
where t ∈ R . Then ϕt x is strictly convex for x > 0.
Here, we use the notation 0 log 0 : 0.
Proof. Since ϕt x xt−2 > 0 for x > 0, therefore ϕt 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
st
,
2
2.2
if and only if the relation
u2 fs 2uwf
st
2
w2 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 x1 , x2 , x3 of an open interval I for which x1 < x2 <
x3 , then
x3 − x2 f x1 x1 − x3 f x2 x2 − x1 f x3 ≥ 0.
2.4
Lemma 2.4. Let f be log-convex function and if, x1 ≤ y1 , x2 ≤ y2 , x1 /
x2 , y1 /
y2 , then the
following inequality is valid:
1/x2 −x1 1/y2 −y1 f y2
f x2
≤
.
f x1
f y1
2.5
By using the above lemmas and Theorem 1.2, as in 2, we can prove the following
results.
Theorem 2.5. Let x and p be two positive n-tuples and let
Δt Δt (x; p Δt
,
t
2.6
4
Journal of Inequalities and Applications
such that condition 1.5 is satisfied and all xi ’s are not equal. Then Δt is log-convex. Also for r < s < t
where r, s, t ∈ R , we have
Δs
t−r
t−s s−r
≤ Δr
Δt
.
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 pi ≥ 1 i 1, . . . , n, then for
t, r, s ∈ R ,
s
x; p
Bt,r
n
1/t−r
s t/s
− ni1 pi xit
rr − s
i1 pi xi
,
tt − s n pi xs r/s − n pi xr
i1
i
i1
1/s−r
pi xis log ni1 pi xis − s ni1 pi xis log xi
, s/
r,
n
s r/s
− ni1 pi xir
i1 pi xi
n
s r/s
log ni1 pi xis − s ni1 pi xir log xi
2r − s
i1 pi xi
s
Br,r x; p exp −
, s/
r,
n
s r/s
rr − s
s
− ni1 pi xir
i1 pi xi
n
2
2 s
log ni1 pi xis − s2 ni1 pi xis log xi
1
i1 pi xi
s
Bs,s x; p exp − n
.
n
n
s
s
s
s
2s
i1 pi xi log
i1 pi xi − s i1 pi xi log xi
2.8
s
Bs,r
x; p
s
Br,s
x; p
rr − s
s2
n
t/
r, r /
s, t /
s,
i
i1
s
s
s
s
Remark 2.7. Let us note that Bs,r
x; p Br,s
x; p limt → s Bt,r
x; p limt → s Br,t
x; p,
s
s
s
s
Br,r x; p limt → r Bt,r x; p and Bs,s x; p limr → s Br,r x; p.
Theorem 2.8. Let
⎧
t/s
n
n
⎪
1
⎪
s
t
⎪
⎪
pi xi
−
pi xi ,
⎪
⎨ tt − s
i1
i1
s
Θt ⎪
n
n
n
⎪
⎪
s
s
s
⎪1
⎪
pi xi log
pi xi − s pi xi log xi ,
⎩ 2
s
i1
i1
i1
t/
s,
2.9
t s.
then for t, r, u ∈ R and t < r < u, we have
Θsr
u−t
u−r s r−t
≤ Θst
.
Θu
2.10
Theorem 2.9. Let r, t, u, v ∈ R , such that t ≤ v, r ≤ u. Then one has
s
s
(x; p ≤ Bv,u
(x; p.
Bt,r
2.11
J. Pečarić and A. ur Rehman
5
Remark 2.10. From 2.7, we have
Δs
s
t−r
≤
Δr
r
t−s Δt
t
s−r
t−r
⇒ Δs
≤
st−r t−s s−r
.
Δr
Δt
r t−s ts−r
2.12
Since log x is concave, therefore for r < s < t, we have
t − s log r r − t log s s − r log t < 0 ⇒
st−r
> 1.
r t−s ts−r
2.13
This implies that 1.4, which we derived in 2, is better than 2.7.
Also note that
1/t−r
r
Ast,r x; p,
t
1/s−r
1/s−r
r
r
s
s
Br,s
x; p Bs,r
x; p Ass,r x; p Asr,s x; p,
s
s
1
s
Br,r x; p exp −
Asr,r x; p,
r
1
s
Bs,s
x; p exp −
Ass,s x; p.
s
s
Bt,r
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 t0 ∈ a, b be fixed, h be continuous and monotonic with ht0 0, g be a function
of bounded variation and
b
t
dgx,
Gt :
dgx.
3.1
0 ≤ Gt ≤ 1 for t0 ≤ t ≤ b,
3.2
Gt :
a
t
a If
0 ≤ Gt ≤ 1
for a ≤ t ≤ t0 ,
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 ≤ t0 such that
Gt ≤ 0 for t < s,
Gt ≥ 1
for s ≤ t ≤ t0 ,
Gt ≤ 0
for t > t0 ,
3.4
Gt ≥ 1
for t0 < t < s,
Gt ≤ 0
for t ≥ s,
3.5
or there exists an s ≥ t0 such that
Gt ≤ 0 for t < t0 ,
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 t0 ∈ a, b be fixed, h be continuous and monotonic with ht0 0, g be a
function of bounded variation. Choose g such that function Λt is positive valued, where Λt is
defined as follows:
Λt Λt a, b, h, g b
ϕt hx dgx − ϕt
b
a
hxdgx .
3.6
a
Theorem 3.3. Let Λt , defined as above, satisfy condition 3.2. Then Λt is log-convex. Also for r <
s < t, where r, s, t ∈ R , one has
Λs
t−r
t−s s−r
≤ Λr
.
Λt
3.7
Proof. Let fx u2 ϕs x 2uwϕr x w2 ϕt x, where r s t/2 and u, w ∈ R,
2
f x u2 xs−2 2uwxr−2 w2 xt−2 uxs−2/2 wxt−2/2 ≥ 0.
3.8
This implies that fx is convex.
By Theorem 3.1, we have,
b
f ht dgt − f
a
⇒ u
2
b
b
htdgt −
a
ϕs hxdgx − ϕs
b
b
2uw
2w
2
a
ϕr hxdgx − ϕr
a
b
a
hxdgx
a
b
dgt − 1 f0 ≥ 0
ϕt hxdgx − ϕt
a
b
hxdgx
b
a
hxdgx
a
⇒ u2 Λs 2uwΛr w2 Λt ≥ 0.
Now, by Lemma 2.2, we have Λt is log-convex in Jensen sense.
≥0
3.9
J. Pečarić and A. ur Rehman
7
Since limt → 1 Λt Λ1 , this implies that Λt is continuous for all t ∈ R , therefore it is a
log-convex 1, page 6.
Since Λt is log-convex, that is, log Λt is convex, therefore by Lemma 2.3 for r < s < t
and taking f log Λ, we have
t − s log Λr r − t log Λs s − r log Λt ≥ 0,
3.10
which is equivalent to 3.7.
t −Λt such that condition 3.4 or 3.5 is satisfied. Then Λ
t is log-convex.
Theorem 3.4. Let Λ
Also for r < s < t, where r, s, t ∈ R , one has
s
Λ
t−r
t−s s−r
r
t
≤ Λ
.
Λ
3.11
Definition 3.5. Let t0 ∈ a, b be fixed, h be continuous and monotonic with ht0 0, g be a
function of bounded variation. Then for t, r, s ∈ R , one defines
s
Ft,r
a, b, h, g
⎧
⎫1/t−r
⎨ rr − s b ht xdgx − b hxdgx
t/s ⎬
a
a
,
b
r/s ⎭
⎩ tt − s b r
h
xdgx
−
hxdgx
a
a
t
/ r, r / s, t / s,
s
a, b, h, g
Fs,r
s
Fr,s
a, b, h, g
⎫1/s−r
⎧
⎨ rr − s s b hs x log hxdgx − b hs xdgx
log b hs xdgx ⎬
a
a
a
,
b
b s
r/s
⎭
⎩ s2
r
h
xdgx
−
h
xdgx
a
a
s/
r,
s
a, b, h, g
Fr,r
exp
b
b
b
r/s
log a hs xdgx
2r − s s a hr x log hxdgx− a hs xdgx
,
−
b
b
r/s rr − s
s a hr xdgx− a hs xdgx
s/
r,
s
Fs,s
a, b, h, g
exp
b
b
2
b
2 1 s2 a hs x log hx dgx − a hs xdgx log a hs xdgx
− .
b
b
b
s
2s s a hs x log hxdgx − a hs xdgx log a hs xdgx
3.12
s
s
s
Remark 3.6. Let us note that Fs,r
a, b, h, g Fr,s
a, b, h, g limt→s Ft,r
a, b, h, g s
s
s
s
s
a, b,
limt→s Fr,t a, b, h, g, Fr,r a, b, h, g limt→r Ft,r a, b, h, g and Fs,s a, b, h, g limr→s Fr,r
h, g.
8
Journal of Inequalities and Applications
Theorem 3.7. Let r, t, u, v ∈ R , such that t ≤ v, r ≤ u. Then
s
s
a, b, h, g ≤ Fv,u
a, b, h, g.
Ft,r
3.13
Proof. Let
Λt Λt a, b, h, g
b
⎧
b
t ⎪
1
⎪
t
⎪
,
t/
1,
h xdgx −
hxdgx
⎪
⎨ tt − 1
a
a
b
b
b
⎪
⎪
⎪
⎪
⎩ hx log hxdgx − hxdgx log hxdgx, t 1.
a
a
3.14
a
Now, taking x1 r, x2 t, y1 u, y2 v, where r, t, u, v /
1, and ft Λt in Lemma 2.4, we
have
b
t ⎞1/t−r
t
h
xdgx
−
hxdgx
a
⎠
⎝ rr − 1 a
tt − 1 b hr xdgx − b hxdgx
r
⎛
b
a
a
v ⎞1/v−u
hv xdgx − a hxdgx
uu
−
1
a
⎠
≤⎝
.
vv − 1 b hu xdgx − b hxdgx
u
a
a
⎛
b
b
3.15
Since s > 0, by substituting h hs , t t/s, r r/s, u u/s, and v v/s, where r, t, v, u / s,
in above inequality, we get
b t
b
t/s ⎞s/t−r
h xdgx − a hs xdgx
rr
−
s
a
⎠
⎝
b s
r/s
tt − s b r
h xdgx − a h xdgx
a
⎛
⎛
v/s ⎞s/v−u
hv xdgx − a hs xdgx
uu
−
s
a
⎠
≤⎝
.
b s
u/s
vv − s b u
h xdgx − a h xdgx
a
b
b
3.16
By raising power 1/s, we get an inequality 3.13 for r, t, v, u /
s.
From Remark 3.6 , we get 3.13 is also valid for r s or t s or r t or t r s.
Lemma 3.8. Let f ∈ C2 I such that
m ≤ f x ≤ M.
3.17
J. Pečarić and A. ur Rehman
9
Consider the functions φ1 , φ2 defined as
φ1 x Mx2
− fx,
2
3.18
mx2
φ2 x fx −
.
2
Then φi x for i 1, 2 are convex.
Proof. We have that
φ1 x M − f x ≥ 0,
3.19
φ2 x f x − m ≥ 0,
that is, φi for i 1, 2 are convex.
Theorem 3.9. Let t0 ∈ a, b be fixed, h be continuous and monotonic with ht0 0, g be a function
of bounded variation, and f ∈ C2 I such that condition 3.2 is satisfied. Then there exists ξ ∈ I such
that
b
f hx dgx − f
a
f ξ
2
b
b
b
hxdgx −
dgx − 1
a
a
h xdgx −
2
b
a
2 .
hxdgx
3.20
a
Proof. In Theorem 3.1, setting f φ1 and f φ2 , respectively, as defined in Lemma 3.8, we
get the following inequalities:
b
f hx dgx − f
a
b
M
≤
2
b
m
≥
2
hxdgx −
h xdgx −
b
b
dgx − 1
a
2
a
b
a
f hx dgx − f
a
b
3.21
2 b
hxdgx
,
a
b
hxdgx −
dgx − 1
a
h xdgx −
2
a
a
b
2 .
hxdgx
3.22
a
Now, by combining both inequalities, we get
b b
b
2 a f hx dgx − f a hxdgx − a dgx − 1 f0
m≤
≤ M.
b
b
2
2 xdgx −
h
hxdgx
a
a
3.23
10
Journal of Inequalities and Applications
So by condition 3.17, there exists ξ ∈ I such that
b
b
b 2 a f hx dgx − f a hxdgx − a dgx − 1 f0
f ξ,
b
b
2
2
h xdgx − a hxdgx
a
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 t0 ∈ a, b be fixed, h be continuous and monotonic with ht0 0, g be a
function of bounded variation, and f1 , f2 ∈ C2 I such that condition 3.2 is satisfied. Then there
exists ξ ∈ I such that the following equality is true:
b
− f1
f hx dgx
a 1
b f hx dgx
a 2
−
b
b
hxdgx − a dgx − 1 f1 0 f1 ξ
,
b
− a dgx − 1 f2 0 f2 ξ
a
b
f2 a hxdgx
3.25
provided that denominators are nonzero.
Proof. Let a function k ∈ C2 I be defined as
k c1 f1 − c2 f2 ,
3.26
where c1 and c2 are defined as
c1 b
f2 hx dgx − f2
a
c2 b
b
b
hxdgx −
dgx − 1 f2 0,
a
f1 hx dgx − f1
a
b
a
hxdgx −
b
a
dgx − 1 f1 0.
3.27
a
Then, using Theorem 3.9 with f k, we have
0
c1 f1 ξ
−
c2 f2 ξ
b
h xdgx −
2
b
a
2 .
hxdgx
3.28
a
Since
b
a
h xdgx −
2
b
2
hxdgx
a
0,
/
3.29
J. Pečarić and A. ur Rehman
11
therefore, 3.28 gives
c2 f1 ξ
.
c1 f2 ξ
3.30
After putting values, we get 3.25.
Let α be a strictly monotone continuous function, we defined Tα h, g as follows
integral version of quasiarithmetic sum 2:
Tα h, g α−1
b
α hx dgx .
3.31
a
Theorem 3.11. Let α, β, γ ∈ C2 a, b be strictly monotonic continuous functions. Then there exists
η in the image of hx such that
b
α Tα h, g − α Tγ h, g − a dgx − 1 α ◦ γ −1 0 α ηγ η − α ηγ η
b
β Tβ h, g − β Tγ h, g − a dgx − 1 β ◦ γ −1 0 β ηγ η − β ηγ η
3.32
is valid, provided that all denominators are nonzero.
Proof. If we choose the functions f1 and f2 so that f1 α◦γ −1 , f2 β◦γ −1 , and hx → γhx.
Substituting these in 3.25,
b
α Tα h, g − α Tγ h, g − a dgx − 1 α ◦ γ −1 0
b
β Tβ h, g − β Tγ h, g − a dgx − 1 β ◦ γ −1 0
α γ −1 ξ γ γ −1 ξ − α γ −1 ξ γ γ −1 ξ
−1 −1 .
β γ ξ γ γ ξ − β γ −1 ξ γ γ −1 ξ
3.33
Then by setting γ −1 ξ η, we get 3.32.
Corollary 3.12. Let t0 ∈ a, b be fixed, h be continuous and monotonic with ht0 0, g be a
function of bounded variation, and let t, r, s ∈ R . Then
s
a, b, h, g η.
Ft,r
3.34
Proof. If t, r, and s are pairwise distinct, then we put αx xt , βx xr and γx xs 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. Pečarić, 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. Pečarić 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.