Peˇcari ´c 1, 2 and Atiq ur Rehman 1 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan 2 Faculty of Textile Technology, University of Zagreb, Pierottij
Trang 1Volume 2008, Article ID 305623, 12 pages
doi:10.1155/2008/305623
Research Article
On Logarithmic Convexity for Power Sums and
Related Results II
J Peˇcari ´c 1, 2 and Atiq ur Rehman 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,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
Copyrightq 2008 J Peˇcari´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
Letx be positive n-tuples The well-known inequality for power sums of order s and r, for
s > r > 0see 1, page 164, states that
n
i1
x i s
1/s
<
n
i1
x r i
1/r
Moreover, if p p1, , p n is a positive n-tuples such that p i ≥ 1 i 1, , n, then for
s > r > 0see 1, page 165, we have
n
i1
p i x i s
1/s
<
n
i1
p i x r i
1/r
In2, we defined the following function:
Δt Δtx; 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 ilog
n
i1
p i x i−n
i1
p i x i log x i , t 1.
1.3
Trang 22 Journal of Inequalities and Applications
We introduced the Cauchy means involving power sums Namely, the following results were obtained in2
For r < s < t, where r, s, t∈ R, we have
Δs t −r ≤ Δr t −s
Δt s −r
such that, x i ∈ 0, a i 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,rx; p
r − s
t − s
n
i1p i x s i t/s− n
i1p i x t i
n
i1p i x s i r/s− n
i1p i x r i
1/t−r
, t / r, r / s, t / s,
A s s,r x; p A s
r,sx; p
r − s
s
n
i1p i x s
i1p i x s
i − s n
i1p i x s
i log x i
n
i1p i x s i r/s− n
i1p i x r i
1/s−r
, s / r,
A s
r,rx; p exp
1
s − r
n
i1p i x i s r/s log n
i1p i x s i − s n
i1p i x i r log x i
s n
i1p i x s i r/s− n
i1p i x r i
, s / r,
A s s,sx; p exp
n
i1p i x s i log n
i1p i x i s 2− s2 n
i1p i x s i
log x i 2
2s n
i1p i x s
i1p i x s
i1p 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 result1, 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
Remark 1.3 InTheorem 1.2, if f is strictly convex, then1.7 is strict unless x1 · · · x nand n
i1p i 1
Trang 32 Discrete result
Lemma 2.1 Let
ϕ t x
⎧
⎪
⎪
x t
t t − 1 , t / 1,
x log x, t 1, 2.1
where t∈ R Then ϕ t x is strictly convex for x > 0.
Here, we use the notation 0 log 0 : 0
Proof Since ϕt x x t−2> 0 for x > 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
f sft ≥ f2
s t
2
if and only if the relation
u2f s 2uwf
s t
2
holds for each real u, w and s, t ∈ I.
The following lemma is equivalent to definition of convex function1, page 2
Lemma 2.3 If f is continuous and convex for all x1, x2, x3of an open interval I for which x1 < x2<
x3, then
x3− x2 f
x1 x1− x3 f
x2 x2− x1 f
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:
f x 2
f
x1
1/x2−x1
≤
f y 2
f
y1
1/y2−y1
By using the above lemmas andTheorem 1.2, as in2, 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
Trang 44 Journal of Inequalities and Applications
such that condition1.5 is satisfied and all x i ’s are not equal ThenΔt is log-convex Also for r < s < t where r, s, t∈ R, we have
Δs
t −r
≤ Δr
t −s
Δt
s −r
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,rx; p
r r − s
t t − s
n
i1p i x s i t/s− n
i1p i x t i
n
i1p i x s i r/s− n
i1p i x r i
1/t−r
, t / r, r / s, t / s,
B s,r s x; p B s
r,sx; p
r r − s
s2
n
i1p i x s
i1p i x s
i − s n
i1p i x s
i log x i
n
i1p i x s i r/s− n
i1p i x r i
1/s−r
, s / r,
B s r,rx; p exp− 2r − s
r r − s
n
i1p i x s i r/s log n
i1p i x s i − s n
i1p i x r i log x i
s n
i1p i x i s r/s− n
i1p i x i r
, s / r,
B s,s s x; p exp
−1
s
n
i1p i x s
i1p i x s i
2− s2 n
i1p i x s i
log x i 2 2s n
i1p i x s
i1p i x s
i − s n
i1p i x s
i log x i
.
2.8
Remark 2.7 Let us note that B s
s,r x; p B s
r,sx; p limt → s B s
t,rx; p limt → s B s
r,tx; p,
B s
r,rx; p limt → r B t,r s x; p and B s
s,sx; p limr → s B s
r,rx; p.
Theorem 2.8 Let
Θs
t
⎧
⎪
⎪
⎪
⎪
1
t t − s
n
i1
p i x s i
t/s
−n
i1
p i x t i
, t / s,
1
s2
n
i1
p i x s i
log
n
i1
p i x s i
− sn
i1
p i x s
i log x i
, t s.
2.9
then for t, r, u∈ Rand t < r < u, we have
Θs
r u −t≤ Θs
t
u −r
Θs
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
Trang 5Remark 2.10 From2.7, we have
Δ
s
s
t −r
≤
Δ
r
r
t −sΔ
t
t
s −r
⇒ Δs t −r≤ s t −r
r t −s t s −r
Δr t −s
Δt s −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 ⇒ s t −r
r t −s t s −r > 1. 2.13
This implies that1.4, which we derived in 2, is better than 2.7
Also note that
B s t,rx; p
r t
1/t−r
A s t,r x; p,
B s r,s x; p B s
s,rx; p
r s
1/s−r
A s s,rx; p
r s
1/s−r
A s r,s x; p,
B s r,rx; p exp
−1
r
A s r,r x; p,
B s s,sx; p exp
−1
s
A s s,s x; p.
2.14
Let us note that there are not integral analogs of results from2 Moreover, inSection 3
we will show that previous results have their integral analogs
3 Integral results
The following theorem is very useful for further result1, 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
G t :
t
a
dg x, G t :
b
t
a If
0≤ Gt ≤ 1 for a ≤ t ≤ t0, 0≤ Gt ≤ 1 for t0≤ t ≤ b, 3.2
then for every convex function f : I → R such that hx ∈ I for all x ∈ a, b,
b
a
f
h t dg t ≥ f
b
a
h tdgt
b
a
dg t − 1
Trang 6
6 Journal of Inequalities and Applications
b Ifb
a h tdgt ∈ I and either there exists an s ≤ t0such that
G t ≤ 0 for t < s, G t ≥ 1 for s ≤ t ≤ t0, G t ≤ 0 for t > t0, 3.4
or there exists an s ≥ t0such that
G t ≤ 0 for t < t0, G t ≥ 1 for t0< t < s, G t ≤ 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 in3.3 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 functionΛtis positive valued, whereΛtis defined as follows:
Λt Λt a, b, h, g
b
a
ϕ t
h x dg x − ϕ t
b
a
h xdgx
Theorem 3.3 Let Λ t , defined as above, satisfy condition3.2 Then Λ t is log-convex Also for r <
s < t, where r, s, t∈ R, one has
Λs t −r≤ Λr t −s
Proof Let f x u2ϕ s x 2uwϕ r x w2ϕ t x, where r s t/2 and u, w ∈ R,
fx u2x s−2 2uwx r−2 w2x t−2 ux s−2/2 wx t−2/2 2
This implies that f x is convex.
ByTheorem 3.1, we have,
b
a
f
h t dg t − f
b
a
h tdgt
−
b
a
dg t − 1
f0 ≥ 0
⇒ u2
b a
ϕ s hxdgx − ϕ s
b a
h xdgx
2uw
b a
ϕ r hxdgx − ϕ r
b a
h xdgx
2w2
b
a
ϕ t hxdgx − ϕ t
b
a
h xdgx
≥ 0
⇒ u2Λs 2uwΛ r w2Λt ≥ 0.
3.9
Now, byLemma 2.2, we haveΛtis log-convex in Jensen sense
Trang 7Since limt→ 1Λt Λ1, this implies thatΛt is continuous for all t∈ R, therefore it is a log-convex1, page 6
SinceΛtis log-convex, that is, logΛtis convex, therefore byLemma 2.3for 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 to3.7
Theorem 3.4 Let Λ t −Λt such that condition3.4 or 3.5 is satisfied Then Λ t is log-convex Also for r < s < t, where r, s, t∈ R, one has
Λs
t −r
≤ Λr
t −s Λt
s −r
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
F t,r s a, b, h, g
⎧
⎨
⎩
r r − s
t t − s
b
a h t xdgx − b
a h xdgx t/s
b
a h r xdgx − b
a h xdgx r/s
⎫
⎬
⎭
1/t−r
, t / r, r / s, t / s,
F s,r s a, b, h, g
F s
r,s a, b, h, g
⎧
⎨
⎩
r r − s
s2
sb
a h s x log hxdgx − b
a h s xdgx logb
a h s xdgx
b
a h r xdgx − b
a h s xdgx r/s
⎫
⎬
⎭
1/s−r
, s / r,
F r,r s a, b, h, g
exp
− 2r − s
r r − s
sb
a h r x log hxdgx− b
a h s xdgx r/slog b
a h s xdgx
sb
a h r xdgx− b
a h s xdgx r/s
, s / r,
F s
s,s a, b, h, g
exp
−1
ss2
b
a h s x log hx 2
dg x − b
a h s xdgx log b
a h s xdgx 2 2s
sb
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, g F s
r,s a, b, h, g lim t →s F s
t,r a, b, h, g
limt →s F r,t s a, b, h, g, F s
r,r a, b, h, g lim t →r F t,r s a, b, h, g and F s
s,s a, b, h, g lim r →s F s
r,r a, b,
h, g
Trang 88 Journal of Inequalities and Applications
Theorem 3.7 Let r, t, u, v ∈ R, such that t ≤ v, r ≤ u Then
F t,r s a, b, h, g ≤ F s
Proof Let
Λt Λt a, b, h, g
⎧
⎪
⎪
⎪
⎪
1
t t ư 1
b
a
h t xdgx ư
b
a
h xdgx
t
, t / 1,
b
a
h x log hxdgx ư
b
a
h xdgx log
b
a
h xdgx, t 1.
3.14
Now, taking x1 r, x2 t, y1 u, y2 v, where r, t, u, v / 1, and ft Λ tinLemma 2.4, we have
⎛
⎝rr ư 1
t t ư 1
b
a h t xdgx ư b
a h xdgx t
b
a h r xdgx ư b
a h xdgx r
⎞
⎠
1/tưr
≤
⎛
⎝uu ư 1
v v ư 1
b
a h v xdgx ư b
a h xdgx v
b
a h u xdgx ư b
a h xdgx u
⎞
⎠
1/vưu
.
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 ư s
t t ư s
b
a h t xdgx ư b
a h s xdgx t/s
b
a h r xdgx ư b
a h s xdgx r/s
⎞
⎠
s/ tưr
≤
⎛
⎝uu ư s
v v ư s
b
a h v xdgx ư b
a h s xdgx v/s
b
a h u xdgx ư b
a h s xdgx u/s
⎞
⎠
s/ vưu
.
3.16
By raising power 1/s, we get an inequality3.13 for r, t, v, u / s.
FromRemark 3.6, we get3.13 is also valid for r s or t s or r t or t r s.
Lemma 3.8 Let f ∈ C2I such that
Trang 9Consider the functions φ1, φ2defined as
φ1x Mx2
2 − fx,
φ2x fx − mx2
2 .
3.18
Then φ i x for i 1, 2 are convex.
Proof We have that
φ1x M − fx ≥ 0,
φ2x fx − m ≥ 0, 3.19 that is, φ i 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 ∈ C2I such that condition 3.2 is satisfied Then there exists ξ ∈ I such
that
b
a
f
h x dg x − f
b a
h xdgx
−
b a
dg x − 1
fξ
2
b
a
h2xdgx −
b
a
h xdgx
2
.
3.20
Proof InTheorem 3.1, setting f φ1 and f φ2, respectively, as defined inLemma 3.8, we get the following inequalities:
b
a
f
h x dg x − f
b
a
h xdgx
−
b
a
dg x − 1
≤ M 2
b
a
h2xdgx −
b a
h xdgx
2
,
3.21
b
a
f
h x dg x − fb
a
h xdgx−b
a
dg x − 1
≥ m 2
b
a
h2xdgx −
b
a
h xdgx
2
.
3.22
Now, by combining both inequalities, we get
m≤ 2
b
a f
h x dg x − f b
a h xdgx − b
a dg x − 1 f0
b
a h2xdgx − b
Trang 1010 Journal of Inequalities and Applications
So by condition3.17, there exists ξ ∈ I such that
2b
a f
h x dg x − f b
a h xdgx − b
a dg x − 1 f0
b
a h2xdgx − b
ξ, 3.24
and3.24 implies 3.20
Moreover,3.21 is valid if fis bounded from above and again we have3.20 is valid
Of course3.20 is obvious if fis 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 ∈ C2I such that condition 3.2 is satisfied Then there
exists ξ ∈ I such that the following equality is true:
b
a f1
h x dg x − f1
b
a h xdgx − b
a dg x − 1 f10
b
a f2
h x dg x − f2
b
a h xdgx − b
a dg x − 1 f20
f1ξ
f2ξ , 3.25
provided that denominators are nonzero.
Proof Let a function k ∈ C2I be defined as
where c1and c2are defined as
c1
b
a
f2
h x dg x − f2
b
a
h xdgx
−
b
a
dg x − 1
f20,
c2
b
a
f1
h x dg x − f1
b a
h xdgx
−
b a
dg x − 1
f10.
3.27
Then, usingTheorem 3.9with f k, we have
0 c1f1ξ − c2f2ξ
b
a
h2xdgx −
b
a
h xdgx
2
Since
b
a
h2xdgx −
b
a
h xdgx
2
/
Trang 11therefore,3.28 gives
c2
c1 f1ξ
After putting values, we get3.25
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, g α−1b
a
α
Theorem 3.11 Let α, β, γ ∈ C2a, b be strictly monotonic continuous functions Then there exists
η in the image of h x such that
α
T α h, g − α T γ h, g − b
a dg x − 1 α ◦ γ−10
β
T β h, g − β T γ h, g − b
a dg x − 1 β ◦ γ−10
αηγη − αηγη
βηγη − βηγη 3.32
is valid, provided that all denominators are nonzero.
Proof If we choose the functions f1and f2so that f1 α◦γ−1, f2 β◦γ−1, and hx → γhx.
Substituting these in3.25,
α
T α h, g − α T γ h, g − b
a dg x − 1 α ◦ γ−10
β
T β h, g − β T γ h, g − b
a dg x − 1 β ◦ γ−10
α
γ−1ξ γ
γ−1ξ − α
γ−1ξ γ
γ−1ξ
β
γ−1ξ γ
γ−1ξ − β
γ−1ξ γ
γ−1ξ
3.33
Then by setting γ−1ξ 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
Proof If t, r, and s are pairwise distinct, then we put α x x t , βx x r and γ x x s in
3.32 to get 3.34
For other cases, we can consider limit as inRemark 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
... Trang 44 Journal of Inequalities and Applications
such that condition1.5 is satisfied and all... − b
Trang 1010 Journal of Inequalities and Applications
So by condition3.17, there exists... means of convex type in connection with Power sums For means, we shall use the following result1, page 154
Theorem 1.2 Let x and p be two nonnegative n-tuples such that condition 1.5