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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 646034, 14 pages
doi:10.1155/2010/646034
Research Article
Improvement and Reversion of Slater’s Inequality
and Related Results
M. Adil Khan
1
and J. E. Pe
ˇ
cari
´
c
1, 2
1
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan
2
Faculty of Textile Technology, University of Zagreb, Zagreb 10002, Croatia
Correspondence should be addressed to M. Adil Khan,
Received 6 March 2010; Accepted 2 June 2010
Academic Editor: Kunquan Lan
Copyright q 2010 M. Adil Khan and J. E. 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.
We use an inequality given by Mati
´
candPe
ˇ
cari
´
c 2000 and obtain improvement and reverse of
Slater’s and related inequalities.
1. Introduction
In 1981 Slater has proved an interesting companion inequality to Jensen’s inequality 1.
Theorem 1.1. Suppose that φ : I ⊆ R → R is increasing convex function on interval I,for
x
1
,x
2
, ,x
n
∈ I
◦
(where I
◦
is the interior of the interval I) and for p
1
,p
2
, ,p
n
≥ 0 with
P
n
n
i1
p
i
> 0,if
n
i1
p
i
φ
x
i
> 0,then
1
P
n
n
i1
p
i
φ
x
i
≤ φ
n
i1
p
i
φ
x
i
x
i
n
i1
p
i
φ
x
i
.
1.1
When φ is strictly convex on I, inequality 1.1 becomes equality if and only if x
i
c for some c ∈ I
◦
and for all i with p
i
> 0.
It was noted in 2 that by using the same proof the following generalization of Slater’s
inequality 1981 can be given.
2 Journal of Inequalities and Applications
Theorem 1.2. Suppose that φ : I ⊆ R → R is convex function on interval I,forx
1
,x
2
, ,x
n
∈ I
◦
(where I
◦
is the interior of the interval I) and for p
1
,p
2
, ,p
n
≥ 0 with P
n
n
i1
p
i
> 0.Let
n
i1
p
i
φ
x
i
/
0,
n
i1
p
i
φ
x
i
x
i
n
i1
p
i
φ
x
i
∈ I
◦
,
1.2
then inequality 1.1 holds.
When φ is strictly convex on I, inequality 1.1 becomes equality if and only if x
i
c for some
c ∈ I
◦
and for all i with p
i
> 0.
Remark 1.3. For multidimensional version of Theorem 1.2 see 3.
Another companion inequality to Jensen’s inequality is a converse proved by
Dragomir and Goh in 4.
Theorem 1.4. Let φ : I ⊆ R → R be differentiable convex function defined on interval I.Ifx
i
∈
I,i 1, 2, ,n n ≥ 2 are arbitrary members and p
i
≥ 0 i 1, 2, ,n with P
n
n
i1
p
i
> 0,
and let
x
1
P
n
n
i1
p
i
x
i
,
y
1
P
n
n
i1
p
i
φ
x
i
.
1.3
Then the inequalities
0 ≤
y − φ
x
≤
1
P
n
n
i1
p
i
φ
x
i
x
i
− x
1.4
hold.
In the case when φ is strictly convex, one has equalities in 1.4 if and only if there is some
c ∈ I such that x
i
c holds for all i with p
i
> 0.
Mati
´
candPe
ˇ
cari
´
cin5 proved more general inequality from which 1.1 and 1.4
can be obtained as special cases.
Theorem 1.5. Let φ : I ⊆ R → R be differentiable convex function defined on interval I and let
x
i
,p
i
,P
n
, x, and y be stated as in Theorem 1.4.Ifd ∈ I is arbitrary chosen number, then one has
y
≤ φ
d
1
P
n
n
i1
p
i
x
i
− d
φ
x
i
.
1.5
Also, when φ is strictly convex, one has equality in 1.5 if and only if x
i
d holds for all i with
p
i
> 0.
Remark 1.6. If φ, x
i
,p
i
,P
n
, and x are stated as in Theorem 1.4 and we let
n
i1
p
i
φ
x
i
/
0,
also if
x
n
i1
p
i
x
i
φ
x
i
/
n
i1
p
i
φ
x
i
∈ I, then by setting d x in 1.5, we get Slater’s
inequality 1.1 and similarly by setting d
x in 1.5,weget1.4.
Journal of Inequalities and Applications 3
The following refinement of 1.4 is also valid 5.
Theorem 1.7. Let φ : I ⊆ R → R be strictly convex differentiable function defined on interval I and
let x
i
,p
i
,P
n
, x, and y be stated as in Theorem 1.4 and d φ
−1
1/P
n
n
i1
p
i
φ
x
i
, then the
inequalities
y ≤ φ
d
1
P
n
n
i1
p
i
φ
x
i
x
i
− d
,
1.6
0 ≤
y − φ
x
≤ φ
d
1
P
n
n
i1
p
i
φ
x
i
x
i
− d
− φ
x
≤
1
P
n
n
i1
p
i
φ
x
i
x
i
− x
1.7
hold.
The equalities hold in 1.6 and in 1.7 if and only if x
1
x
2
··· x
n
.
Remark 1.8. In 6 Dragomir has also proved Theorem 1.7.
In this paper, we use an inequality given in 5 and derive two mean value
theorems, exponential convexity, log-convexity, and Cauchy means. As applications, such
results are also deduce for related inequality. We use some log-convexity criterion and
prove improvement and reverse of Slater’s and related inequalities. We also prove some
determinantal inequalities.
2. Mean Va lue Theorems
Theorem 2.1. Let φ ∈ C
2
I,whereI is closed interval in R, and let P
n
n
i1
p
i
, p
i
> 0, x
i
,d ∈ I
with x
i
/
d i 1, 2, ,n and
y 1/P
n
n
i1
p
i
φx
i
. Then there exists ξ ∈ I such that
φ
d
1
P
n
n
i1
p
i
x
i
− d
φ
x
i
−
y
φ
ξ
2P
n
n
i1
p
i
x
i
− d
2
.
2.1
Proof. Since φ
x is continuous on I, m ≤ φ
x ≤ M for x ∈ I, where m min
x∈I
φ
x and
M max
x∈I
φ
x.
Consider the functions φ
1
, φ
2
defined as
φ
1
x
Mx
2
2
− φ
x
,
φ
2
x
φ
x
−
mx
2
2
.
2.2
Since
φ
1
x
M − φ
x
≥ 0,
φ
2
x
φ
x
− m ≥ 0,
2.3
φ
i
x for i 1, 2 are convex.
4 Journal of Inequalities and Applications
Now by applying φ
1
for φ in inequality 1.5, we have
Md
2
2
− φ
d
1
P
n
n
i1
p
i
x
i
− d
Mx
i
− φ
x
i
−
1
P
n
n
i1
p
i
Mx
2
i
2
− φ
x
i
≥ 0.
2.4
From 2.4 we get
φ
d
1
P
n
n
i1
p
i
x
i
− d
φ
x
i
−
y ≤
M
2P
n
n
i1
p
i
x
i
− d
2
,
2.5
and similarly by applying φ
2
for φ in 1.5,weget
φ
d
1
P
n
n
i1
p
i
x
i
− d
φ
x
i
−
y
≥
m
2P
n
n
i1
p
i
x
i
− d
2
.
2.6
Since
n
i1
p
i
x
i
− d
2
> 0asx
i
/
d, p
i
> 0
i 1, 2, ,n
,
2.7
by combining 2.5 and 2.6, we have
m ≤
2P
n
φ
d
1/P
n
n
i1
p
i
x
i
− d
φ
x
i
−
y
n
i1
p
i
x
i
− d
2
≤ M.
2.8
Now using the fact that for m ≤ ρ ≤ M there exists ξ ∈ I such that φ
ξρ,weget2.1.
Corollary 2.2. Let φ ∈ C
2
I,whereI is closed interval in R, and let x
i
, x, y, and P
n
be stated as
in Theorem 1.4 with p
i
> 0 and x
i
/
x i 1, 2, ,n. Then there exists ξ ∈ I such that
φ
x
1
P
n
n
i1
p
i
x
i
−
x
φ
x
i
−
y
φ
ξ
2P
n
n
i1
p
i
x
i
−
x
2
.
2.9
Proof. By setting d
x in Theorem 2.1,weget2.9.
Theorem 2.3. Let φ, ψ ∈ C
2
I,whereI is closed interval in R, and let P
n
n
i1
p
i
, p
i
> 0 and
x
i
,d ∈ I with x
i
/
d i 1, 2, ,n. Then there exists ξ ∈ I such that
φ
ξ
ψ
ξ
φ
d
1/P
n
n
i1
p
i
x
i
− d
φ
x
i
−
1/P
n
n
i1
p
i
φ
x
i
ψ
d
1/P
n
n
i1
p
i
x
i
− d
ψ
x
i
−
1/P
n
n
i1
p
i
ψ
x
i
, 2.10
provided that the denominators are nonzero.
Journal of Inequalities and Applications 5
Proof. Let the function k ∈ C
2
I be defined by
k c
1
φ − c
2
ψ, 2.11
where c
1
and c
2
are defined as
c
1
ψ
d
1
P
n
n
i1
p
i
x
i
− d
ψ
x
i
−
1
P
n
n
i1
p
i
ψ
x
i
,
c
2
φ
d
1
P
n
n
i1
p
i
x
i
− d
φ
x
i
−
1
P
n
n
i1
p
i
φ
x
i
.
2.12
Then, using Theorem 2.1 with φ k, we have
0
c
1
φ
ξ
2P
n
−
c
2
ψ
ξ
2P
n
n
i1
p
i
x
i
− d
2
,
2.13
because kd1/P
n
n
i1
p
i
x
i
− dk
d − 1/P
n
n
i1
p
i
kx
i
0.
Since 1/P
n
n
i1
p
i
x
i
− d
2
> 0asx
i
/
d and p
i
> 0 i 1, 2, ,n, therefore, 2.13
gives us
c
2
c
1
φ
ξ
ψ
ξ
.
2.14
After putting the values of c
1
and c
2
,weget2.10 .
Corollary 2.4. Let φ, ψ ∈ C
2
I,whereI is closed interval in R, and P
n
n
i1
p
i
, p
i
> 0 and let
x
i
∈ I, x 1/P
n
n
i1
p
i
x
i
with x
i
/
x i 1, 2, ,n. Then there exists ξ ∈ I such that
φ
ξ
ψ
ξ
φ
x
1/P
n
n
i1
p
i
x
i
− x
φ
x
i
−
1/P
n
n
i1
p
i
φ
x
i
ψ
x
1/P
n
n
i1
p
i
x
i
− x
ψ
x
i
−
1/P
n
n
i1
p
i
ψ
x
i
, 2.15
provided that the denominators are nonzero.
Proof. By setting d
x in Theorem 2.3,weget2.15.
Corollary 2.5. Let x
i
,d ∈ I with x
i
/
d and P
n
n
i1
p
i
, p
i
> 0 i 1, 2, ,n. Then for u, v ∈
R \{0, 1}, u
/
v,thereexistsξ ∈ I,whereI is positive closed interval, such that
ξ
u−v
v
v − 1
d
u
u/P
n
n
i1
p
i
x
i
− d
x
u−1
i
−
1/P
n
n
i1
p
i
x
u
i
u
u − 1
d
v
v/P
n
n
i1
p
i
x
i
− d
x
v−1
i
−
1/P
n
n
i1
p
i
x
v
i
. 2.16
Proof. By setting φxx
u
and ψxx
v
, x ∈ I,inTheorem 2.3,weget2.16 .
6 Journal of Inequalities and Applications
Corollary 2.6. Let x
i
∈ I, P
n
n
i1
p
i
, p
i
> 0 i 1, 2, ,n, and x 1/P
n
n
i1
p
i
x
i
with
x
i
/
x. Then for u, v ∈ R \{0, 1}, u
/
v,thereexistsξ ∈ I,whereI is positive closed interval, such
that
ξ
u−v
v
v − 1
x
u
u/P
n
n
i1
p
i
x
i
− x
x
u−1
i
−
1/P
n
n
i1
p
i
x
u
i
u
u − 1
x
v
v/P
n
n
i1
p
i
x
i
− x
x
v−1
i
−
1/P
n
n
i1
p
i
x
v
i
. 2.17
Proof. By setting φxx
u
and ψxx
v
, x ∈ I,in2.15,weget2.17.
Remark 2.7. Note that we can consider the interval I m
x
,M
x
, where m
x
min
i
{x
i
,d},
M
x
max
i
{x
i
,d}.
Since the function ξ → ξ
u−v
with u
/
v is invertible, then from 2.16 we have
m
x
≤
v
v − 1
d
u
u/P
n
n
i1
p
i
x
i
− d
x
u−1
i
−
1/P
n
n
i1
p
i
x
u
i
u
u − 1
d
v
v/P
n
n
i1
p
i
x
i
− d
x
v−1
i
−
1/P
n
n
i1
p
i
x
v
i
1/u−v
≤ M
x
.
2.18
We will say that the expression in the middle is a mean of x
i
,d.
From 2.17 we have
min
i
{
x
i
}
≤
v
v − 1
x
u
u/P
n
n
i1
x
i
− x
x
u−1
i
−
1/P
n
n
i1
p
i
x
u
i
u
u − 1
x
v
v/P
n
n
i1
p
i
x
i
− x
x
v−1
i
−
1/P
n
n
i1
p
i
x
v
i
1/u−v
≤ max
i
{
x
i
}
.
2.19
The expression in the middle of 2.19 is a mean of x
i
.
In fact similar results can also be given for 2.10 and 2.15. Namely, suppose that
φ
/ψ
has inverse function, then from 2.10 and 2.15 we have
ξ
φ
ψ
−1
φ
d
1/P
n
n
i1
p
i
x
i
− d
φ
x
i
−
1/P
n
n
i1
p
i
φ
x
i
ψ
d
1/P
n
n
i1
p
i
x
i
− d
ψ
x
i
−
1/P
n
n
i1
p
i
ψ
x
i
.
ξ
φ
ψ
−1
φ
x
1/P
n
n
i1
p
i
x
i
− x
φ
x
i
−
1/P
n
n
i1
p
i
φ
x
i
ψ
x
1/P
n
n
i1
p
i
x
i
− x
ψ
x
i
−
1/P
n
n
i1
p
i
ψ
x
i
.
2.20
So, we have that the expression on the right-hand side of 2.20 is also means.
3. Improvements and Related Results
Definition 3.1 see 7, page 2.Afunctionφ : I → R is convex if
φ
s
1
s
3
− s
2
φ
s
2
s
1
− s
3
φ
s
3
s
2
− s
1
≥ 0 3.1
holds for every s
1
<s
2
<s
3
, s
1
,s
2
,s
3
∈ I.
Journal of Inequalities and Applications 7
Lemma 3.2 see 8. Let one define the function
ϕ
t
x
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
x
t
t
t − 1
,t
/
0, 1,
− log x, t 0,
x log x, t 1.
3.2
Then ϕ
t
xx
t−2
, that is, ϕ
t
is convex for x>0.
Definition 3.3 see 9.Afunctionφ : I → R is exponentially convex if it is continuous and
n
k,l1
a
k
a
l
φ
x
k
x
l
≥ 0,
3.3
for all n ∈ N, a
k
∈ R, and x
k
∈ I, k 1, 2, ,nsuch that x
k
x
l
∈ I, 1 ≤ k, l ≤ n, or equivalently
n
k,l1
a
k
a
l
φ
x
k
x
l
2
≥ 0.
3.4
Corollary 3.4 see 9. If φ is exponentially convex function, then
det
φ
x
k
x
l
2
n
k,l1
≥ 0
3.5
for every n ∈ N x
k
∈ I, k 1, 2, ,n.
Corollary 3.5 see 9. If φ : I → 0, ∞ is exponentially convex function, then φ is a log-convex
function that is
φ
λx
1 − λ
y
≤ φ
λ
x
φ
1−λ
y
, ∀x, y ∈ I, λ ∈
0, 1
.
3.6
Theorem 3.6. Let x
i
,p
i
,d ∈ R
i 1, 2, ,n, P
n
n
i1
p
i
. Consider Γ
t
to be defined by
Γ
t
ϕ
t
d
1
P
n
n
i1
p
i
x
i
− d
ϕ
t
x
i
−
1
P
n
n
i1
p
i
ϕ
t
x
i
.
3.7
Then
i for every m ∈ N and for every s
k
∈ R,k∈{1, 2, 3, ,m}, the matrix Γ
s
k
s
l
/2
m
k,l1
is a
positive semidefinite matrix; particularly
det
Γ
s
k
s
l
/2
m
k,l1
≥ 0;
3.8
ii the function t → Γ
t
is exponentially convex;
8 Journal of Inequalities and Applications
iii if Γ
t
> 0, then the function t → Γ
t
is log-convex, that is, for −∞ <r<s<t<∞, one has
Γ
s
t−r
≤
Γ
r
t−s
Γ
t
s−r
.
3.9
Proof. i Let us consider the function defined by
μ
x
m
k,l1
a
k
a
l
ϕ
s
kl
x
,
3.10
where s
kl
s
k
s
l
/2,a
k
∈ R for all k ∈{1, 2, 3, ,m},x>0
Then we have
μ
x
m
k,l1
a
k
a
l
x
s
kl
−2
m
k1
a
k
x
s
k
−2/2
2
≥ 0.
3.11
Therefore, μx is convex function for x>0. Using μx in inequality 1.5,weget
m
k,l1
a
k
a
l
Γ
s
kl
≥ 0,
3.12
so the matrix Γ
s
k
s
l
/2
m
k,l1
is positive semi-definite.
ii Since lim
t → 0
Γ
t
Γ
0
and lim
t → 1
Γ
t
Γ
1
,soΓ
t
is continuous for all t ∈ R,x>0, and
we have exponentially convexity of the function t → Γ
t
.
iii Let Γ
t
> 0, then by Corollary 3.5 we have that Γ
t
is log-convex, that is, t → log Γ
t
is convex, and by 3.1 for −∞ <r<s<t<∞ and taking φtlog Γ
t
,weget
t − s
log Γ
r
r − t
log Γ
s
s − r
log Γ
t
≥ 0, 3.13
which is equivalent to 3.9.
Corollary 3.7. Let x
i
,p
i
∈ R
i 1, 2, ,n, P
n
n
i1
p
i
and x 1/P
n
n
i1
p
i
x
i
. Consider
Γ
t
to be defined by
Γ
t
ϕ
t
x
1
P
n
n
i1
p
i
x
i
−
x
ϕ
t
x
i
−
1
P
n
n
i1
p
i
ϕ
t
x
i
.
3.14
Then
i for every m ∈ N and for every s
k
∈ R,k∈{1, 2, 3, ,m}, the matrix
Γ
s
k
s
l
/2
m
k,l1
is a
positive semi-definite matrix. Particularly
det
Γ
s
k
s
l
/2
m
k,l1
≥ 0,
3.15
Journal of Inequalities and Applications 9
ii the function t →
Γ
t
is exponentially convex;
iii if
Γ
t
> 0, then the function t →
Γ
t
is log-convex, that is, for −∞ <r<s<t<∞, one has
Γ
s
t−r
≤
Γ
r
t−s
Γ
t
s−r
.
3.16
Proof. To get the required results, set d
x in Theorem 3.6.
Let x x
1
,x
2
, ,x
n
be positive n-tuple and p
1
,p
2
, ,p
n
positive real numbers, and
let P
n
n
i1
p
i
.LetM
t
x denote the power mean of order t t ∈ R, defined by
M
t
x
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1
P
n
n
i1
p
i
x
t
i
1/t
,t
/
0,
n
i1
x
p
i
i
1/P
n
,t 0.
3.17
Let us note t hat M
1
xx.
By 2.18 we can give the following definition of Cauchy means.
Let x
i
,d ∈ I with x
i
/
d, I is positive closed interval, and P
n
n
i1
p
i
, p
i
> 0 i
1, 2, ,n,
M
u,v
Γ
u
Γ
v
1/u−v
3.18
for −∞ <u
/
v<∞ are means of x
i
,d. Moreover we can extend these means to the other
cases.
So by limit we have
M
u,u
exp
P
n
d
u
log d
u − 1
n
i1
p
i
x
u
i
log x
i
P
n
M
u
u
x
− d
u
n
i1
p
i
x
u−1
i
log x
i
P
n
M
u−1
u−1
x
P
n
d
u
u − 1
M
u
u
x
− duM
u−1
u−1
x
−
2u − 1
u
u − 1
,u
/
0, 1,
M
0,0
exp
P
n
log
2
d − P
n
M
2
2
log x
2P
n
log M
0
x
− 2d
n
i1
p
i
x
−1
i
log x
i
2P
n
log d − log M
0
x
1 − dM
−1
−1
x
1
,
M
1,1
exp
P
n
d log
2
d 2
n
i1
p
i
x
i
log x
i
− dP
n
M
2
2
log x
− 2logM
0
x
2
P
n
d
log d − 1
P
n
x − dP
n
log M
0
x
− 1
,
3.19
where log x log x
1
, log x
2
, ,log x
n
.
10 Journal of Inequalities and Applications
Theorem 3.8. Let t, s, u, v ∈ R such that t ≤ u, s ≤ v, then the following inequality is valid:
M
t,s
≤ M
u,v
. 3.20
Proof. For convex function φ it holds that 7, page 2
φ
x
2
− φ
x
1
x
2
− x
1
≤
φ
y
2
− φ
y
1
y
2
− y
1
3.21
with x
1
≤ y
1
, x
2
≤ y
2
, x
1
/
x
2
, y
1
/
y
2
. Since by Theorem 3.6, Γ
t
is log-convex, we can set in
3.21: φxlog Γ
x
, x
1
t, x
2
s, y
1
u,andy
2
v, then we get
log Γ
s
− log Γ
t
s − t
≤
log Γ
v
− log Γ
u
v − u
.
3.22
From 3.22 we get 3.20 for s
/
t and u
/
v.
For s t and u v we have limiting case.
Similarly by 2.19 we can give the following definition of Cauchy type means.
Let x
i
∈ I with x
i
/
x, I is positive closed interval, and P
n
n
i1
p
i
,p
i
> 0 i
1, 2, ,n,
M
u,v
Γ
u
Γ
v
1/u−v
3.23
for −∞ <u
/
v<∞ are means of x
i
. Moreover we can extend these means to the other cases.
So by limit we have
M
u,u
exp
P
n
x
u
log x
u − 1
n
i1
p
i
x
u
i
log x
i
P
n
M
u
u
x
−
x
u
n
i1
p
i
x
u−1
i
log x
i
P
n
M
u−1
u−1
x
P
n
x
u
u − 1
M
u
u
x
−
xuM
u−1
u−1
x
−
2u − 1
u
u − 1
,u
/
0, 1,
M
0,0
exp
P
n
log
2
x − P
n
M
2
2
log x
2P
n
log M
0
x
− 2
x
n
i1
p
i
x
−1
i
log x
i
2P
n
log
x − log M
0
x
1 −
xM
−1
−1
x
1
,
M
1,1
exp
P
n
x log
2
x 2
n
i1
p
i
x
i
log x
i
− xP
n
M
2
2
log x
2logM
0
x
2
P
n
x
log x − 1
P
n
x − xP
n
log M
0
x
− 1
,
3.24
where log x log x
1
, log x
2
, ,log x
n
.
Journal of Inequalities and Applications 11
Theorem 3.9. Let t, s, u, v ∈ R such that t ≤ u, s ≤ v, then the following inequality is valid:
M
t,s
≤
M
u,v
.
3.25
Proof. The proof is similar to the proof of Theorem 3.8.
Let M
t
x be stated as above, define d
t
as
d
t
n
i1
p
i
x
i
ϕ
t
x
i
n
i1
p
i
ϕ
t
x
i
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
M
t
t
x
M
t−1
t−1
x
,t
/
0, 1,
M
−1
x
,t 0,
P
n
x
n
i1
p
i
x
i
log x
i
P
n
1 log M
0
x
,t 1.
3.26
The following improvement and reverse of Slater’s inequality are valid.
Theorem 3.10. Let x
i
,p
i
,d
t
∈ R
i 1, 2, ,n, P
n
n
i1
p
i
.LetF
t
be defined by
F
t
ϕ
t
d
t
−
1
P
n
n
i1
p
i
ϕ
t
x
i
.
3.27
Then
i
F
t
≥
H
s; t
t−r/s−r
H
r; t
s−t/s−r
,
3.28
for −∞ <r<s<t<∞ and −∞ <t<r<s<∞.
ii
F
t
≤
H
s; t
t−r/s−r
H
r; t
s−t/s−r
,
3.29
for −∞ <r<t<s<∞.
where,
H
s; t
ϕ
s
d
t
1
P
n
n
i1
p
i
x
i
− d
t
ϕ
s
x
i
−
1
P
n
n
i1
p
i
ϕ
s
x
i
.
3.30
12 Journal of Inequalities and Applications
Proof. i By setting d d
t
in 3.7, Γ
t
becomes F
t
, and for −∞ <r<s<t<∞, setting d d
t
in 3.9,weget
ϕ
s
d
t
1
P
n
n
i1
p
i
x
i
− d
t
ϕ
s
x
i
−
1
P
n
n
i1
p
i
ϕ
s
x
i
t−r
≤
ϕ
r
d
t
1
P
n
n
i1
p
i
x
i
− d
t
ϕ
r
x
i
−
1
P
n
n
i1
p
i
ϕ
r
x
i
t−s
F
t
s−r
,
3.31
that is,
F
t
s−r
≥
ϕ
s
d
t
1
P
n
n
i1
p
i
x
i
− d
t
ϕ
s
x
i
−
1
P
n
n
i1
p
i
ϕ
s
x
i
t−r
×
ϕ
r
d
t
1
P
n
n
i1
p
i
x
i
− d
t
ϕ
r
x
i
−
1
P
n
n
i1
p
i
ϕ
r
x
i
s−t
.
3.32
From 3.32 we get 3.28, and similarly for −∞ <t<r<s<∞ 3.9 becomes
Γ
r
s−t
≤
Γ
t
s−r
Γ
s
r−t
;
3.33
by the same process we can get 3.28.
ii For −∞ <r<t<s<∞ 3.9 becomes
Γ
s
t−r
≤
Γ
r
t−s
Γ
t
s−r
;
3.34
setting d d
t
in 3.34,weget3.29.
Theorem 3.11. Let x
i
,p
i
,d
t
∈ R
i 1, 2, ,n, P
n
n
i1
p
i
.
Then for every m ∈ N and for every s
k
∈ R,k ∈{1, 2, 3, ,m}, the matrices Hs
k
s
l
/2,s
1
m
k,l1
, Hs
k
s
l
/2, s
1
s
2
/2
m
k,l1
are positive semi-definite matrices. Particularly
det
H
s
k
s
l
2
,s
1
m
k,l1
≥ 0,
3.35
det
H
s
k
s
l
2
,
s
1
s
2
2
m
k,l1
≥ 0,
3.36
where Hs, t is defined by 3.30.
Proof. By setting d d
s
1
and d d
s
1
s
2
/2
in Theorem 3.6i, we get the required results.
Remark 3.12. We note that Ht, tF
t
. So by setting m 2in3.35, we have special case of
3.28 for t s
1
,s s
2
,andr s
1
s
2
/2ifs
1
<s
2
and for t s
1
,r s
2
,ands s
1
s
2
/2
if s
2
<s
1
. Similarly by setting m 2in3.36, we have special case of 3.29 for r s
1
,s
s
2
,ts
1
s
2
/2ifs
1
<s
2
and for r s
2
,s s
1
,ts
1
s
2
/2ifs
2
<s
1
.
Journal of Inequalities and Applications 13
Let M
t
x be stated as above, define d
t
as
d
t
ϕ
t
−1
1
P
n
n
i1
p
i
ϕ
t
x
i
M
t−1
x
,t∈ R.
3.37
The following improvement and reverse of inequality 1.6 are also valid.
Theorem 3.13. Let x
i
,p
i
, d
t
∈ R
for all i 1, 2, ,n, P
n
n
i1
p
i
.LetG
t
be defined by
G
t
ϕ
t
d
t
1
P
n
n
i1
p
i
x
i
− d
t
ϕ
t
x
i
−
1
P
n
n
i1
p
i
ϕ
t
x
i
.
3.38
Then
i
G
t
≥
K
s; t
t−r/s−r
K
r; t
s−t/s−r
,
3.39
for −∞ <r<s<t<∞ and −∞ <t<r<s<∞.
ii
G
t
≤
K
s; t
t−r/s−r
K
r; t
s−t/s−r
,
3.40
for −∞ <r<t<s<∞,
where
K
s; t
ϕ
s
d
t
1
P
n
n
i1
p
i
x
i
−
d
t
ϕ
s
x
i
−
1
P
n
n
i1
p
i
ϕ
s
x
i
.
3.41
Proof. i By setting d
d
t
in 3.9,weget3.39 for −∞ <r<s<t<∞, and similarly we can
get 3.39 for the case −∞ <t<r<s<∞.
ii For −∞ <r<t<s<∞ 3.9 becomes
Γ
s
t−r
≤
Γ
r
t−s
Γ
t
s−r
;
3.42
setting d
d
t
in 3.42,weget3.40.
14 Journal of Inequalities and Applications
Theorem 3.14. Let x
i
,p
i
, d
t
∈ R
i 1, 2, ,n, P
n
n
i1
p
i
.
Then for every m ∈ N and for every s
k
∈ R,k∈{1, 2, 3, ,m}, the matrices Ks
k
s
l
/2,s
1
m
k,l1
, Ks
k
s
l
/2, s
1
s
2
/2
m
k,l1
are positive semi-definite matrices. Particularly
det
K
s
k
s
l
2
,s
1
m
k,l1
≥ 0,
3.43
det
K
s
k
s
l
2
,
s
1
s
2
2
m
k,l1
≥ 0,
3.44
where Ks, t is defined by 3.41.
Proof. By setting d
d
s
1
and d d
s
1
s
2
/2
in Theorem 3.6i, we get the required results.
Remark 3.15. We note that Kt, tG
t
. So by setting m 2in3.43, we have special case of
3.39 for t s
1
,s s
2
,rs
1
s
2
/2ifs
1
<s
2
and for t s
1
,r s
2
,ands s
1
s
2
/2if
s
2
<s
1
. Similarly by setting m 2in3.44, we have special case of 3.40 for r s
1
,s s
2
,
and t s
1
s
2
/2ifs
1
<s
2
and for r s
2
,s s
1
,andt s
1
s
2
/2ifs
2
<s
1
.
Acknowledgments
The research of the first and second authors was funded by Higher Education Commission,
Pakistan. The research of the second author was supported by the Croatian Ministry of
Science, Education, and Sports under the Research Grant 117-1170889-0888.
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