Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 948430, 17 pages
doi:10.1155/2010/948430
Research Article
Generalization of an Inequality for Integral
Transforms with Kernel and Related Results
Sajid Iqbal,1 J. Pečarić,1, 2 and Yong Zhou3
1
Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan
Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
3
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
2
Correspondence should be addressed to Sajid Iqbal, sajid uos2000@yahoo.com
Received 27 March 2010; Revised 2 August 2010; Accepted 27 October 2010
Academic Editor: András Rontó
Copyright q 2010 Sajid Iqbal et al. 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 establish a generalization of the inequality introduced by Mitrinović and Pečarić in 1988.
We prove mean value theorems of Cauchy type for that new inequality by taking its difference.
Furthermore, we prove the positive semidefiniteness of the matrices generated by the difference
of the inequality which implies the exponential convexity and logarithmic convexity. Finally, we
define new means of Cauchy type and prove the monotonicity of these means.
1. Introduction
Let Kx, t be a nonnegative kernel. Consider a function u : a, b → R, where u ∈ Uv, K,
and the representation of u is
ux b
Kx, tvtdt
1.1
a
for any continuous function v on a, b. Throughout the paper, it is assumed that all integrals
under consideration exist and that they are finite.
The following theorem is given in 1 see also 2, page 235.
Theorem 1.1. Let ui ∈ Uv, K i 1, 2 and rt ≥ 0 for all t ∈ a, b. Also let φ : R → R be a
function such that φx is convex and increasing for x > 0. Then
b
u1 x v1 x dx,
dx ≤
rxφ sxφ u2 x v2 x a
a
b
1.2
2
Journal of Inequalities and Applications
where
sx v2 x
b
a
rtKt, x
dt,
u2 t
u2 t / 0.
1.3
The following definition is equivalent to the definition of convex functions.
Definition 1.2 see 2. Let I ⊆ R be an interval, and let φ : I → R be convex on I. Then, for
s1 , s2 , s3 ∈ I such that s1 < s2 < s3 , the following inequality holds:
φs1 s3 − s2 φs2 s1 − s3 φs3 s2 − s1 ≥ 0.
1.4
Let us recall the following definition.
Definition 1.3 see 3, page 373. A function h : a, b → R is exponentially convex if it is
continuous and
n
ξi ξj h xi xj ≥ 0
1.5
i,j1
for all n ∈ N and all choices of ξi ∈ R,xi xj ∈ a, b, i, j 1, . . . , n.
The following proposition is useful to prove the exponential convexity.
Proposition 1.4 see 4. Let h : a, b → R. The following statements are equivalent.
i h is exponentially convex.
ii h is continuous, and
xi xj
ξi ξj h
≥0
2
i,j1
n
1.6
for every n ∈ N,ξi ∈ a, b, and xi ∈ a, b, 1 ≤ i ≤ n.
Corollary 1.5. If h : a, b → R is exponentially convex, then h is log-convex; that is,
1−λ
h λx 1 − λy ≤ hxλ h y
∀x, y ∈ a, b, λ ∈ 0, 1.
1.7
This paper is organized in this manner. In Section 2, we give the generalization
of Mitrinović-Pečarić inequality and prove the mean value theorems of Cauchy type. We
also introduce the new type of Cauchy means. In Section 3, we give the proof of positive
semidefiniteness of matrices generated by the difference of that inequality obtained from the
generalization of Mitrinović-Pečarić inequality and also discuss the exponential convexity. At
the end, we prove the monotonicity of the means.
Journal of Inequalities and Applications
3
2. Main Results
Theorem 2.1. Let ui ∈ Uv, K i 1, 2, and rx ≥ 0 for all x ∈ a, b. Also let I ⊆ R be an
interval, let φ : I → R be convex, and let u1 x/u2 x, v1 x/v2 x ∈ I. Then
b
rxφ
a
b
u1 x
v1 x
dx ≤
dx,
qxφ
u2 x
v2 x
a
2.1
where
qx v2 x
b
a
Proof. Since u1 b
a
rtKt, x
dt,
u2 t
u2 t /
0.
Kx, tv1 tdt and v2 t > 0, we have
b
b
u1 x
1
dx rxφ
rxφ
Kx, tv1 tdt dx
u2 x
u2 x a
a
a
b
b
v1 t
1
dt dx
rxφ
Kx, tv2 t
u2 x a
v2 t
a
b
b
Kx, tv2 t v1 t
dt dx.
rxφ
u2 x
v2 t
a
a
b
2.2
2.3
By Jensen’s inequality, we get
b
b
Kx, tv2 t
v1 t
u1 x
dx ≤
φ
dt dx
rxφ
rx
u2 x
u2 x
v2 t
a
a
a
b b
rxKx, tv2 t
v1 t
φ
dt dx
u2 x
v2 t
a
a
b
b rxKx, t
v1 t
v2 t
dx dt
φ
v2 t
u2 x
a
a
b
b
qtφ
a
2.4
v1 t
dt.
v2 t
Remark 2.2. If φ is strictly convex on I and v1 x/v2 x is nonconstant, then the inequality in
2.1 is strict.
4
Journal of Inequalities and Applications
Remark 2.3. Let us note that Theorem 1.1 follows from Theorem 2.1. Indeed, let the condition
of Theorem 1.1 be satisfied, and let u
i ∈ U|v|, K; that is,
u
1 x b
2.5
Kx, t|v1 t|dt.
a
So, by Theorem 2.1, we have
b
a
b
b
v1 x u
1 x
|v1 x|
dx dx
≥
dx.
qxφ qxφ
rxφ
v x v x
u x
2
2
a
2
a
2.6
On the other hand, φ is increasing function, we have
u
1 x
φ
u2 x
φ
1
u2 x
b
Kx, t|v1 t|dt
a
1 b
≥φ
Kx, tv1 tdt
u2 x a
u1 x |u1 x|
.
φ
φ u2 x
u2 x 2.7
From 2.6 and 2.7, we get 1.2.
If f ∈ Ca, b and α > 0, then the Riemann-Liouville fractional integral is defined by
Iaα fx
1
Γα
x
ftx − tα−1 dt.
2.8
a
We will use the following kernel in the upcoming corollary:
⎧
α−1
⎪
⎨ x − t ,
KI x, t Γα
⎪
⎩0,
a ≤ t ≤ x,
2.9
x < t ≤ b.
Corollary 2.4. Let ui ∈ Ca, b i 1, 2, and rx ≥ 0 for all x ∈ a, b. Also let I ⊆ R be
an interval, let φ : I → R be convex, u1 x/u2 x, Iaα u1 x/Iaα u2 x ∈ I, and u1 x, u2 x have
Riemann-Liouville fractional integral of order α > 0. Then
b
rxφ
a
b Iaα u1 x
u1 t
dx
≤
QI tdt,
φ
Iaα u2 x
u2 t
a
2.10
where
u2 t
QI t Γα
b
t
rxx − tα−1
dx,
Iaα u2 x
Iaα u2 x /
0.
2.11
Journal of Inequalities and Applications
5
Let ACa, b be space of all absolutely continuous functions on a, b. By ACn a, b,
we denote the space of all functions g ∈ Cn a, b with g n−1 ∈ ACa, b.
Let α ∈ R and g ∈ ACn a, b. Then the Caputo fractional derivative see 5, p. 270
of order α for a function g is defined by
α
gt D∗a
1
Γn − α
t
a
g n s
t − sα−n1
2.12
ds,
where n α 1; the notation of α stands for the largest integer not greater than α.
Here we use the following kernel in the upcoming corollary:
⎧
⎪
x − tn−α−1
⎪
⎨
, a ≤ t ≤ x,
Γn − α
KD x, t ⎪
⎪
⎩0,
x < t ≤ b.
2.13
Corollary 2.5. Let ui ∈ ACn a, b i 1, 2, and rx ≥ 0 for all x ∈ a, b. Also let I ⊆ R be an
n
n
α
α
u1 x/D∗a
u2 x ∈ I, and u1 x, u2 x have
interval, let φ : I → R be convex, u1 t/u2 t, D∗a
Caputo fractional derivative of order α > 0. Then
b n α
u1 t
u1 x
D∗a
dx ≤
rxφ
φ
QD tdt,
α
n
D∗a u2 x
a
a
u t
b
2.14
2
where
n
u t
QD t 2
Γn − α
b
t
rxx − tn−α−1
dx,
α
D∗a
u2 x
α
D∗a
u2 x /
0.
2.15
Let L1 a, b be the space of all functions integrable on a, b. For β ∈ R , we say that
β
β−k
f ∈ L1 a, b has an L∞ fractional derivative Da f in a, b if and only if Da f ∈ Ca, b for
β−1
β
k 1, . . . , β 1, Da f ∈ ACa, b, and Da ∈ L∞ a, b.
The next lemma is very useful to give the upcoming corollary 6 see also 5, p. 449.
β
Lemma 2.6. Let β > α ≥ 0, f ∈ L1 a, b has an L∞ fractional derivative Da f in a, b, and
β−k
Da fa 0,
k 1, . . . , β 1.
2.16
Then
1
Daα fs Γ β−α
for all a ≤ s ≤ b.
s
a
β
s − tβ−α−1 Da ftdt
2.17
6
Journal of Inequalities and Applications
Clearly
Daα f is in ACa, b
Daα f is in Ca, b
for β − α ≥ 1,
2.18
for β − α ∈ 0, 1,
hence
Daα f ∈ L∞ a, b,
2.19
Daα f ∈ L1 a, b.
Now we use the following kernel in the upcoming corollary:
⎧
⎪
s − tβ−α−1
⎪
⎨ ,
Γ β−α
KL s, t ⎪
⎪
⎩0,
a ≤ t ≤ s,
2.20
s < t ≤ b.
β
Corollary 2.7. Let β > α ≥ 0, ui ∈ L1 a, b i 1, 2 has an L∞ fractional derivative Da ui in a, b,
β−k
and rx ≥ 0 for all x ∈ a, b. Also let Da ui a 0 for k 1, . . . , β 1 i 1, 2, let φ : I → R
β
β
be convex, and Daα u1 x/Daα u2 x, Da u1 x/Da u2 x ∈ I. Then
b β
Daα u1 x
Da u1 t
QL tdt,
dx ≤
rxφ
φ
β
Daα u2 x
a
a
D u2 t
b
2.21
a
where
β
D u2 t
QL t a
Γ β−α
b
t
rxx − tβ−α−1
dx,
Daα u2 x
Daα u2 x / 0.
2.22
Lemma 2.8. Let f ∈ C2 I, and let I be a compact interval, such that
m ≤ f x ≤ M,
∀x ∈ I.
2.23
Consider two functions φ1 , φ2 defined as
φ1 x Mx2
− fx,
2
mx2
φ2 x fx −
.
2
Then φ1 and φ2 are convex on I.
2.24
Journal of Inequalities and Applications
7
Proof. We have
φ1
x M − f x ≥ 0,
φ2
x f x − m ≥ 0,
2.25
that is φ1 , φ2 are convex on I.
Theorem 2.9. Let f ∈ C2 I, let I be a compact interval, ui ∈ Uv, K i 1, 2, and rx ≥ 0 for
all x ∈ a, b. Also let u1 x/u2 x, v1 x/v2 x ∈ I, v1 x/v2 x be nonconstant, and let qx be
given in 2.2. Then there exists ξ ∈ I such that
b
u1 x
v1 x
− rxf
dx
v2 x
u2 x
a
f ξ b
v1 x 2
u1 x 2
qx
dx.
− rx
2
v2 x
u2 x
a
qxf
2.26
Proof. Since f ∈ C2 I and I is a compact interval, therefore, suppose that m min f , M max f . Using Theorem 2.1 for the function φ1 defined in Lemma 2.8, we have
b
rx
a
M
2
u1 x
u2 x
2
u1 x
−f
u2 x
b
M v1 x 2
v1 x
dx ≤
dx. 2.27
qx
−f
2 v2 x
v2 x
a
From Remark 2.2, we have
b
a
v1 x 2
u1 x 2
qx
dx > 0.
− rx
v2 x
u2 x
2.28
Therefore, 2.27 can be written as
b
2 a qxfv1 x/v2 x − rxfu1 x/u2 x dx
≤ M.
b
2
2
qxv
dx
−
rxu
x/v
x
x/u
x
1
2
1
2
a
2.29
We have a similar result for the function φ2 defined in Lemma 2.8 as follows:
b
2 a qxfv1 x/v2 x − rxfu1 x/u2 x dx
≥ m.
b
2
2
qxv
dx
−
rxu
x/v
x
x/u
x
1
2
1
2
a
2.30
Using 2.29 and 2.30, we have
b
2 a qxfv1 x/v2 x − rxfu1 x/u2 x dx
m≤ ≤ M.
b
qxv1 x/v2 x2 − rxu1 x/u2 x2 dx
a
2.31
8
Journal of Inequalities and Applications
By Lemma 2.8, there exists ξ ∈ I such that
b
qxfv1 x/v2 x
a
b
qxv1 x/v2 x2
a
− rxfu1 x/u2 x dx f ξ
.
2
− rxu1 x/u2 x2 dx
2.32
This is the claim of the theorem.
Let us note that a generalized mean value Theorem 2.9 for fractional derivative was
given in 7. Here we will give some related results as consequences of Theorem 2.9.
Corollary 2.10. Let f ∈ C2 I, let I be a compact interval, ui ∈ Ca, b i 1, 2, and rx ≥ 0
for all x ∈ a, b. Also let u1 x/u2 x, Iaα u1 x/Iaα u2 x ∈ I, let u1 x/u2 x be nonconstant, let
QI t be given in 2.11, and u1 x, u2 x have Riemann-Liouville fractional integral of order α > 0.
Then there exists ξ ∈ I such that
b
α
I u1 x
u1 x
− rxf aα
dx
QI xf
u2 x
Ia u2 x
a
α
f ξ b
Ia u1 x 2
u1 x 2
QI x
dx.
− rx α
2
u2 x
Ia u2 x
a
2.33
Corollary 2.11. Let f ∈ C2 I, let I be compact interval, ui ∈ ACn a, b i 1, 2, and
n
n
n
n
α
α
rx ≥ 0 for all x ∈ a, b. Also let u1 t/u2 t, D∗a
u1 x/D∗a
u2 x ∈ I, let u1 x/u2 x be
nonconstant, let QD t be given in 2.15, and u1 x, u2 x have Caputo derivative of order α > 0.
Then there exists ξ ∈ I such that
b
QD xf
n
a
f ξ
2
n
u1 x
u2 x
⎛
b
α
D∗a
u1 x
− rxf
α
D∗a
u2 x
⎝QD x
a
n
u1 x
2
n
u2 x
dx
α
u1 x
D∗a
− rx
α
D∗a u2 x
2
⎞
2.34
⎠dx.
Corollary 2.12. Let β > α ≥ 0, f ∈ C2 I, let I be a compact interval, ui ∈ L1 a, b i 1, 2 has an
β−k
L∞ fractional derivative, and rx ≥ 0 for all x ∈ a, b. Let Da ui a 0 for k 1, . . . , β 1 i β
β
β
β
1, 2, Daα u1 x/Daα u2 x, Da u1 x/Da u2 x ∈ I, let Da u1 x/Da u2 x be nonconstant, and let
QL t be given in 2.22. Then there exists ξ ∈ I such that
b
QL xf
a
β
Da u1 x
Daα u1 x
− rxf
Daα u2 x
dx
β
Da u2 x
⎛
⎞
β
2
2
α
f ξ b ⎝
Da u1 x ⎠
Da u1 x
QL x
dx.
− rx
α
β
2
D
a u2 x
a
Da u2 x
2.35
Journal of Inequalities and Applications
9
Theorem 2.13. Let f, g ∈ C2 I, let I be a compact interval, ui ∈ Uv, K i 1, 2, and rx ≥ 0
for all x ∈ a, b. Also let u1 x/u2 x, v1 x/v2 x ∈ I, v1 x/v2 x be nonconstant, and let
qx be given in 2.2. Then there exists ξ ∈ I such that
b
a
b
a
qxfv1 x/v2 xdx −
qxgv1 x/v2 xdx −
b
a
rxfu1 x/u2 xdx
a
rxgu1 x/u2 xdx
b
f ξ
.
g ξ
2.36
It is provided that denominators are not equal to zero.
Proof. Let us take a function h ∈ C2 I defined as
hx c1 fx − c2 gx,
2.37
b
v1 x
u1 x
dx − rxg
dx,
qxg
c1 v2 x
u2 x
a
a
b
b
v1 x
u1 x
c2 qxf
dx − rxf
dx.
v2 x
u2 x
a
a
2.38
where
b
By Theorem 2.9 with f h, we have
0
c2
f ξ − g ξ
2
2
c
1
b
v1 x
qx
v2 x
a
2
b
u1 x
dx − rx
u2 x
a
2
dx .
2.39
Since
b
b
v1 x 2
u1 x 2
qx
dx − rx
dx /
0,
v2 x
u2 x
a
a
2.40
c1 f ξ − c2 g ξ 0.
2.41
c2 f ξ
.
c1 g ξ
2.42
so we have
This implies that
This is the claim of the theorem.
Let us note that a generalized Cauchy mean-valued theorem for fractional derivative
was given in 8. Here we will give some related results as consequences of Theorem 2.13.
10
Journal of Inequalities and Applications
Corollary 2.14. Let f, g ∈ C2 I, let I be a compact interval, ui ∈ Ca, b i 1, 2, and rx ≥ 0
for all x ∈ a, b. Also let u1 x/u2 x, Iaα u1 x/Iaα u2 x ∈ I, let u1 x/u2 x be nonconstant,
let QI t be given in 2.11, and u1 x, u2 x have Riemann-Liouville fractional derivative of order
α > 0. Then there exists ξ ∈ I such that
b
a
b
a
QI xfu1 x/u2 xdx −
QI xgu1 x/u2 xdx −
b
a
rxfIaα u1 x/Iaα u2 xdx
a
rxgIaα u1 x/Iaα u2 xdx
b
f ξ
.
g ξ
2.43
It is provided that denominators are not equal to zero.
Corollary 2.15. Let f, g ∈ C2 I, let I be a compact interval, ui ∈ ACn a, b i 1, 2, and
n
n
n
n
α
α
rx ≥ 0 for all x ∈ a, b. Also let u1 t/u2 t, D∗a
u1 x/D∗a
u2 x ∈ I, let u1 x/u2 x
be nonconstant, let QD t be given in 2.15, and u1 x, u2 x have Caputo fractional derivative of
order α > 0. Then there exists ξ ∈ I such that
b
n
n
α
α
QD xf u1 x/u2 x dx − a rxfD∗a
u1 x/D∗a
u2 xdx f ξ
.
b
b
n
n
α
α
QD xg u1 x/u2 x dx − a rxgD∗a
u1 x/D∗a
u2 xdx g ξ
a
b
a
2.44
It is provided that denominators are not equal to zero.
Corollary 2.16. Let β > α ≥ 0, f, g ∈ C2 I, let I be a compact interval, ui ∈ L1 a, b i 1, 2 has
β
β−k
an L∞ fractional derivative Da ui in a, b, and rx ≥ 0 for all x ∈ a, b. Also let Da ui a 0 for
β
β
β
β
k 1, . . . , β 1 i 1, 2, Daα u1 x/Daα u2 x, Da u1 x/Da u2 x ∈ I, let Da u1 x/Da u2 x be
nonconstant, and let QL t be given in 2.22. Then there exists ξ ∈ I such that
β
b
β
dx − a rxfDaα u1 x/Daα u2 xdx f ξ
Q
u
u
D
xf
x/D
x
L
1
2
a
a
a
.
β
b
b
β
QL xg Da u1 x/Da u2 x dx − a rxgDaα u1 x/Daα u2 xdx g ξ
a
b
2.45
It is provided that denominators are not equal to zero.
Corollary 2.17. Let I ⊆ R , let I be a compact interval, ui ∈ Uv, K i 1, 2, and rx ≥ 0 for
all x ∈ a, b. Let u1 x/u2 x, v1 x/v2 x ∈ I, let v1 x/v2 x be nonconstant, and let qx be
given in 2.2. Then, for s, t ∈ R \ {0, 1} and s /
t, there exists ξ ∈ I such that
⎛
⎞1/t−s
b
t
t
qxv
dx
−
rxu
dx
x/v
x
x/u
x
ss − 1 a
1
2
1
2
a
⎠
.
ξ⎝
tt − 1 b qxv1 x/v2 xs dx − b rxu1 x/u2 xs dx
b
a
a
2.46
Journal of Inequalities and Applications
11
s, s, t /
0, 1. By Theorem 2.13, we have
Proof. We set fx xt and gx xs , t /
b
qxv1 x/v2 xt dx
a
b
qxv1 x/v2 xs dx
a
−
b
a
rxu1 x/u2 xt dx
a
rxu1 x/u2 xs dx
tt − 1ξt−2
.
ss − 1ξs−2
2.47
b
b
t
t
ss − 1 a qxv1 x/v2 x dx − a rxu1 x/u2 x dx
.
b
tt − 1 qxv1 x/v2 xs dx − b rxu1 x/u2 xs dx
2.48
−
b
This implies that
ξ
t−s
a
a
This implies that
⎛
⎞1/t−s
b
b
t
t
qxv
dx
−
rxu
dx
x/v
x
x/u
x
ss
−
1
1
2
1
2
a
a
⎠
ξ⎝
.
tt − 1 b qxv1 x/v2 xs dx − b rxu1 x/u2 xs dx
a
2.49
a
Remark 2.18. Since the function ξ → ξt−s is invertible and from 2.46, we have
⎞1/t−s
b
b
t
t
qxv
dx
−
rxu
dx
x/v
x
x/u
x
ss
−
1
1
2
1
2
a
a
⎠
m≤⎝
≤ M.
tt − 1 b qxv1 x/v2 xs dx − b rxu1 x/u2 xs dx
⎛
a
2.50
a
Now we can suppose that f /g is an invertible function, then from 2.36 we have
ξ
f g −1
⎛
⎞
b
b
qxv1 x/v2 xdx − a rxu1 x/u2 xt dx
⎝ a
⎠.
b
b
s
qxv
rxu
dx
−
x/v
xdx
x/u
x
1
2
1
2
a
a
2.51
We see that the right-hand side of 2.49 is mean, then for distinct s, t ∈ R it can be written as
Ms,t
1/t−s
t
s
2.52
12
Journal of Inequalities and Applications
as mean in broader sense. Moreover, we can extend these means, so in limiting cases for
s, t /
0, 1,
limMs,t
t→s
Ms,s
⎛
exp⎝
b
a
⎞
b
qxAxs log Axdx − a rxBxs log Bxdx
2s − 1 ⎠
−
,
b
b
s
s
ss − 1
qxAx
dx
−
rxBx
dx
a
a
lim Ms,s
s→0
⎛
b
b
2
⎞
2
qxlog Axdx − a rxlog Bxdx
⎜
⎟
M0,0 exp⎝ a
1⎠,
b
b
2 a qx log Axdx − a rx log Bxdx
lim Ms,s
s→1
M1,1
⎛
⎞
b
2
2
qxAxlog
Axdx
−
rxBxlog
Bxdx
⎜
⎟
a
exp⎝ a
− 1⎠,
b
b
2 a qxAx log Axdx − a rxBx log Bxdx
b
2.53
limMs,t
t→0
⎛
⎜
Ms,0 ⎝ b
s
b
⎞1/s
qxAx dx − a rxBx dx
⎟
⎠
b
qx log Axdx − a rx log Bxdx ss − 1
a
b
a
s
,
limMs,t
t→1
Ms,1
⎛ ⎜
⎝
b
a
⎞1/1−s
b
qxAx log Axdx − a rxBx log Bxdx ss − 1
⎟
,
⎠
b
b
s
s
qxAx dx − a rxBx dx
a
where Ax v1 x/v2 x and Bx u1 x/u2 x.
Remark 2.19. In the case of Riemann-Liouville fractional integral of order α > 0, we well use
the notation Ms,t instead of Ms,t and we replace vi x with ui x, ui x with Iaα ui x, and
qx with QI x.
Journal of Inequalities and Applications
13
Remark 2.20. In the case of Caputo fractional derivative of order α > 0, we well use the
α
s,t instead of Ms,t and we replace vi x with un x, ui x with D∗a
ui x, and
notation M
i
qx with QD x.
s,t instead of
Remark 2.21. In the case of L∞ fractional derivative, we will use the notation M
β
α
Ms,t and we replace vi x with Da ui x, ui x with Da ui x, and qx with QL x.
3. Exponential Convexity
Lemma 3.1. Let s ∈ R, and let ϕs : R → R be a function defined as
ϕs x :
⎧
xs
⎪
⎪
,
⎪
⎪
⎪
⎨ ss − 1
s/
0, 1,
− log x,
⎪
⎪
⎪
⎪
⎪
⎩x log x,
3.1
s 0,
s 1.
Then ϕs is strictly convex on R for each s ∈ R.
Proof. Since ϕ
s x xs−2 > 0 for all x ∈ R , s ∈ R, therefore, ϕ is strictly convex on R for each
s ∈ R.
Theorem 3.2. Let ui ∈ Uv, K i 1, 2, ui x, vi x > 0 i 1, 2, rx ≥ 0 for all x ∈ a, b,
let qx be given in 2.2, and
t
b
qxϕt
a
b
v1 x
u1 x
dx − rxϕt
dx.
v2 x
u2 x
a
3.2
Then the following statements are valid.
a For n ∈ N and si ∈ R, i 1, . . . , n, the matrix n
si sj /2 i,j1
is a positive semidefinite
matrix. Particularly
"k
!
det
b The function s →
c The function s →
s < t < ∞:
si sj /2
≥0
for k 1, . . . n.
3.3
i,j1
s
is exponentially convex on R.
s
is log-convex on R, and the following inequality holds, for −∞ < r <
t−r
s
≤
t−s
s−r
r
t
.
3.4
14
Journal of Inequalities and Applications
Proof. a Here we define a new function μ,
μx k
ai aj ϕsij x,
3.5
i,j1
for k 1, . . . , n, ai ∈ R, sij ∈ R, where sij si sj /2,
μ x n
ai aj x
sij −2
2
n
si /2−1
ai x
≥ 0.
i,j1
3.6
i1
This shows that μx is convex for x ≥ 0. Using Theorem 2.1, we have
k
ai aj
sij
i,j1
≥ 0.
3.7
n
si sj /2 i,j1
From the above result, it shows that the matrix is a positive semidefinite matrix.
Specially, we get
det
k
si sj /2 i,j1
≥0
∀k 1, . . . n.
3.8
b Since
lim
s→1
lim
s→0
s
1
,
3.9
s
0
,
it follows that s is continuous for s ∈ R. Then, by using Proposition 1.4, we get the
exponential convexity of the function s → s .
c Since s is continuous for s ∈ R and using Corollary 1.5, we get that s is log
convex. Now by Definition 1.2 with ft log t and r, s, t ∈ R such that r < s < t, we
get
log
which is equivalent to 3.4.
t−r
s
≤ log
t−s
r
log
s−r
t
,
3.10
Journal of Inequalities and Applications
15
Corollary 3.3. Let ui ∈ Ca, b i 1, 2, and rx ≥ 0 for all x ∈ a, b. Also let u1 x/u2 x,
Iaα u1 x/Iaα u2 x ∈ R , u1 x, u2 x have Riemann-Liouville fractional integral of order α > 0, let
QI t be given in 2.11, and
t
b
QI xϕt
a
Then the statement of Theorem 3.2 with
b
α
Ia u1 x
u1 x
dx − rxϕt α
dx.
u2 x
Ia u2 x
a
t
instead of
t
3.11
is valid.
Corollary 3.4. Let ui ∈ ACn a, b i 1, 2, and rx ≥ 0 for all x ∈ a, b. Also let
n
n
α
α
u1 t/u2 t, D∗a
u1 x/D∗a
u2 x ∈ R , u1 x, u2 x have Caputo fractional derivative of order
α > 0, let QD t be given in 2.15, and
t
b
QD xϕt
a
n
u1 x
n
u2 x
dx −
b
rxϕt
a
α
u1 x
D∗a
dx.
α
D∗a
u2 x
3.12
Then the statement of Theorem 3.2 with t instead of t is valid.
Corollary 3.5. Let β > α ≥ 0, ui ∈ L1 a, b i 1, 2 has L∞ fractional derivative, and rx ≥ 0
β−k
for all x ∈ a, b. Also let Da ui a 0 for k 1, . . . , β 1 i 1, 2, Daα u1 x/Daα u2 x,
β
β
Da u1 x/Da u2 x ∈ R , let QL t be given in 2.22, and
t
b
a
QL xϕt
β
Da u1 x
β
Da u2 x
dx −
b
rxϕt
a
Daα u1 x
dx.
Daα u2 x
3.13
Then the statement of Theorem 3.2 with # t instead of t is valid.
2.52.
In the following theorem, we prove the monotonicity property of Ms,t defined in
Theorem 3.6. Let the assumption of Theorem 3.2 be satisfied, also let t be defined in 3.2, and
t, s, u, v ∈ R such that s ≤ v, t ≤ u. Then the following inequality is true:
Ms,t ≤ Mv,u .
3.14
16
Journal of Inequalities and Applications
Proof. For a convex function ϕ, using the Definition 1.2, we get the following inequality:
ϕx2 − ϕx1 ϕ y2 − ϕ y1
≤
x2 − x1
y2 − y1
3.15
with x1 ≤ y1 , x2 ≤ y2 , x1 /
x2 , and y1 /
y2 . Since by Theorem 3.2 we get that t is log-convex.
We set ϕt log t , x1 s, x2 t, y1 v, y2 u, s / t, and v / u. Terefore, we get
log
− log
t−s
t
s
≤
log
− log
u−v
u
v
,
1/t−s
1/u−v
t
log ≤ log u
,
s
3.16
v
which is equivalent to 3.14 for s /
t, v /
u.
For s t, v u, we get the required result by taking limit in 3.16.
Corollary 3.7. Let ui ∈ Ca, b i 1, 2, and let the assumption of Corollary 3.3 be satisfied, also
let t be defined by 3.11. For t, s, u, v ∈ R such that s ≤ v, t ≤ u, then the following inequality
holds:
Ms,t ≤ Mv,u .
3.17
Corollary 3.8. Let ui ∈ ACn a, b i 1, 2 and let the assumption of Corollary 3.4 be satisfied,
also let be defined by 3.12. For t, s, u, v ∈ R such that s ≤ v, t ≤ u, then the following inequality
t
holds:
v,u .
s,t ≤ M
M
3.18
Corollary 3.9. Let β > α ≥ 0, ui ∈ L1 a, b i 1, 2 and the assumption of Corollary 3.5 be
satisfied, also let # t be defined by 3.13. For t, s, u, v ∈ R such that s ≤ v, t ≤ u. Then following
inequality holds
v,u .
s,t ≤ M
M
3.19
Journal of Inequalities and Applications
17
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