Báo cáo hóa học: " Research Article Generalization of an Inequality for Integral Transforms with Kernel and Related Results" potx

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 differen

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 Pe ˇcari ´c,1, 2 and Yong Zhou3

1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan

2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia

3 School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China

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: Andr´as Ront ´o

Copyrightq 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 Mitrinovi´c and Peˇcari´c 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

u x 

b

a

K x, tvtdt 1.1

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 in1 see also 2, page 235

function such that φ x is convex and increasing for x > 0 Then

b

a

r xφ

u1x

u2x



dx≤

b

a

s xφ

v1x

v2x



dx, 1.2

s x  v2x

b

a

r tKt, x

u2t dt, u2t / 0. 1.3 The following definition is equivalent to the definition of convex functions

Definition 1.2see 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:

φ s1s3− s2  φs2s1− s3  φs3s2− s1 ≥ 0. 1.4 Let us recall the following definition

Definition 1.3see 3, page 373 A function h : a, b → R is exponentially convex if it is continuous and

n



i,j1

ξ i ξ j h

x i  x j



for all n ∈ N and all choices of ξ i ∈ R,x i  x j ∈ a, b, i, j  1, , n.

The following proposition is useful to prove the exponential convexity

i h is exponentially convex.

ii h is continuous, and

n



i,j1

ξ i ξ j h

x

i  x j

2



for every n ∈ N,ξ i ∈ a, b, and x i ∈ a, b, 1 ≤ i ≤ n.

h

λx  1 − λy≤ hx λ h

y1−λ

∀x, y ∈ a, b, λ ∈ 0, 1. 1.7

This paper is organized in this manner In Section 2, we give the generalization

of Mitrinovi´c-Peˇcari´c 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 Mitrinovi´c-Peˇcari´c inequality and also discuss the exponential convexity At the end, we prove the monotonicity of the means

2 Main Results

interval, let φ : I → R be convex, and let u1x/u2x, v1x/v2x ∈ I Then

b

a

r xφ



u1x

u2x



dx≤

b

a

q xφ



v1x

v2x



where

q x  v2x

b

a

r tKt, x

u2t dt, u2t / 0. 2.2

Proof Since u1 b

a K x, tv1tdt and v2t > 0, we have

b

a

r xφ



u1x

u2x



dx

b

a

r xφ 1

u2x

b

a

K x, tv1tdt

dx



b

a

r xφ 1

u2x

b

a

K x, tv2t v1t

v2t dt

dx



b

a

r xφ b

a

K x, tv2t

u2x

v1t

v2t dt

dx.

2.3

By Jensen’s inequality, we get

b

a

r xφ



u1x

u2x



dx≤

b

a

r x b

a

K x, tv2t

u2x φ



v1t

v2t



dt

dx



b

a

b a

r xKx, tv2t

u2x φ



v1t

v2t



dt

dx



b

a φ



v1t

v2t



v2t b

a

r xKx, t

u2x dx

dt



b

a

q tφ



v1t

v2t



dt.

2.4

Remark 2.2 If φ is strictly convex on I and v1x/v2x is nonconstant, then the inequality in

2.1 is strict

Remark 2.3 Let us note thatTheorem 1.1follows fromTheorem 2.1 Indeed, let the condition

ofTheorem 1.1be satisfied, and letu i ∈ U|v|, K; that is,

u1x 

b

a

K x, t|v1t|dt. 2.5

So, byTheorem 2.1, we have

b

a

q xφ

v1x

v2x



dx

b

a

q xφ

|v

1x|

v2x



dx≥

b

a

r xφ

u

1x

u2x



dx. 2.6

On the other hand, φ is increasing function, we have

φ

u

1x

u2x



 φ 1

u2x

b

a

K x, t|v1t|dt

≥ φ 1

u2x







b

a

K x, tv1tdt





 φ

|u

1x|

u2x



 φ

u1x

u2x



.

2.7

From2.6 and 2.7, we get 1.2

If f ∈ Ca, b and α > 0, then the Riemann-Liouville fractional integral is defined by

I a α f x  1

Γα

x

a

f tx − t α−1dt. 2.8

We will use the following kernel in the upcoming corollary:

K I x, t 

⎧

⎪

⎪

x − t α−1

Γα , a ≤ t ≤ x,

0, x < t ≤ b.

2.9

an interval, let φ : I → R be convex, u1x/u2x, I α u1x/I α u2x ∈ I, and u1x, u2x have

Riemann-Liouville fractional integral of order α > 0 Then

b

a

r xφ

I α u

1x

I α u2x



dx≤

b

a φ



u1t

u2t



Q I tdt, 2.10

where

Q I t  u2t

Γα

b

t

r xx − t α−1

I α u2x dx, I a α u2x / 0. 2.11

Let ACa, b be space of all absolutely continuous functions on a, b By AC n a, b,

we denote the space of all functions g ∈ C n a, b with g n−1 ∈ ACa, b.

Let α∈ Rand g ∈ AC n a, b Then the Caputo fractional derivative see 5, p 270

of order α for a function g is defined by

D α ∗a g t  1

Γn − α

t

a

g n s

t − s α −n1 ds, 2.12

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:

K D x, t 

⎧

⎪

⎪

x − t n −α−1

Γn − α , a ≤ t ≤ x,

0, x < t ≤ b.

2.13

interval, let φ : I → R be convex, u n1 t/u n2 t, D α

∗a u1x/D α

∗a u2x ∈ I, and u1x, u2x have

Caputo fractional derivative of order α > 0 Then

b

a

r xφ



D α

∗a u1x

D α

∗a u2x



dx≤

b

a

n

1 t

u n2 t

Q D tdt, 2.14

where

Q D t  u

n

2 t

Γn − α

b

t

r xx − t n −α−1

D ∗a α u2x dx, D α ∗a u2x / 0. 2.15

Let L1a, b be the space of all functions integrable on a, b For β ∈ R, we say that

f ∈ L1a, b has an L∞fractional derivative D β a f in a, b if and only if D β a −k f ∈ Ca, b for

k  1, , β  1, D β−1

a f ∈ ACa, b, and D β

a ∈ L∞a, b.

The next lemma is very useful to give the upcoming corollary6 see also 5, p 449

D a β −k f a  0, k  1, ,β

Then

D α a f s  1

Γβ − α

s

a

s − t β −α−1 D a β f tdt 2.17

for all a ≤ s ≤ b.

D α f is in AC a, b for β − α ≥ 1,

D α a f is in C a, b for β − α ∈ 0, 1, 2.18

hence

D a α f ∈ L∞a, b,

D α f ∈ L1a, b. 2.19

Now we use the following kernel in the upcoming corollary:

K L s, t 

⎧

⎪

⎪

s − t β −α−1

Γβ − α  , a ≤ t ≤ s,

0, s < t ≤ b.

2.20

and r x ≥ 0 for all x ∈ a, b Also let D β −k

a u i a  0 for k  1, , β  1 i  1, 2, let φ : I → R

be convex, and D α u1x/D α u2x, D β

a u1x/D β

a u2x ∈ I Then

b

a

r xφ

D α u

1x

D α u2x



dx≤

b

a

β

a u1t

D a β u2t

Q L tdt, 2.21

where

Q L t  D

β

a u2t

Γβ − α

b

t

r xx − t β −α−1

D α u2x dx, D a α u2x / 0. 2.22

m ≤ f x ≤ M, ∀x ∈ I. 2.23

Consider two functions φ1, φ2defined as

φ1x  Mx2

2 − fx,

φ2x  fx − mx2

2 .

2.24

Then φ1and φ2are convex on I.

Proof We have

φ1x  M − f x ≥ 0,

φ2x  f x − m ≥ 0, 2.25 that is φ1, φ2are convex on I.

all x ∈ a, b Also let u1x/u2x, v1x/v2x ∈ I, v1x/v2x be nonconstant, and let qx be

given in2.2 Then there exists ξ ∈ I such that

b

a



q xf



v1x

v2x



− rxf



u1x

u2x



dx

 f ξ

2

b

a

q x



v1x

v2x

2

− rx



u1x

u2x

2

dx.

2.26

Proof Since f ∈ C2I and I is a compact interval, therefore, suppose that m  min f , M 

max f UsingTheorem 2.1for the function φ1defined inLemma 2.8, we have

b

a

r x M

2



u1x

u2x

2

− f



u1x

u2x



dx≤

b

a

q x M

2



v1x

v2x

2

− f



v1x

v2x



dx. 2.27

FromRemark 2.2, we have

b

a

q x



v1x

v2x

2

− rx



u1x

u2x

2

Therefore,2.27 can be written as

2 b

a



q xfv1x/v2x − rxfu1x/u2xdx

b

a



q xv1x/v2x2− rxu1x/u2x2

dx ≤ M. 2.29

We have a similar result for the function φ2defined inLemma 2.8as follows:

2 b

a



q xfv1x/v2x − rxfu1x/u2xdx

b

a



q xv1x/v2x2− rxu1x/u2x2

dx ≥ m. 2.30 Using2.29 and 2.30, we have

m≤ 2

b

a



q xfv1x/v2x − rxfu1x/u2xdx

b

a



q xv1x/v2x2− rxu1x/u2x2

dx ≤ M. 2.31

ByLemma 2.8, there exists ξ∈ I such that

b

a



q xfv1x/v2x − rxfu1x/u2xdx

b

a



q xv1x/v2x2− rxu1x/u2x2

dx  f ξ

2 . 2.32

This is the claim of the theorem

Let us note that a generalized mean valueTheorem 2.9for fractional derivative was given in7 Here we will give some related results as consequences ofTheorem 2.9

for all x ∈ a, b Also let u1x/u2x, I α u1x/I α u2x ∈ I, let u1x/u2x be nonconstant, let

Q I t be given in 2.11, and u1x, u2x have Riemann-Liouville fractional integral of order α > 0.

Then there exists ξ ∈ I such that

b

a



Q I xf



u1x

u2x



− rxf



I a α u1x

I α u2x



dx

 f ξ

2

b

a

Q I x



u1x

u2x

2

− rx



I α u1x

I α u2x

2

dx.

2.33

r x ≥ 0 for all x ∈ a, b Also let u n1 t/u n2 t, D α

∗a u1x/D α

∗a u2x ∈ I, let u n1 x/u n2 x be

nonconstant, let Q D t be given in 2.15, and u1x, u2x have Caputo derivative of order α > 0.

Then there exists ξ ∈ I such that

b

a

Q D xf u

n

1 x

u n2 x

− rxf

D α

∗a u1x

D α

∗a u2x



dx

 f ξ

2

b

a

⎛

⎝Q D x u

n

1 x

u n2 x

2

− rx



D α

∗a u1x

D ∗a α u2x

2⎞

⎠dx.

2.34

L∞fractional derivative, and r x ≥ 0 for all x ∈ a, b Let D β −k

a u i a  0 for k  1, , β  1 i 

1, 2, D α u1x/D α u2x, D β

a u1x/D β

a u2x ∈ I, let D β

a u1x/D β

a u2x be nonconstant, and let

Q L t be given in 2.22 Then there exists ξ ∈ I such that

b

a

Q L xf D

β

a u1x

D a β u2x

− rxf

D α u

1x

D α u2x



dx

 f ξ

2

b

a

⎛

⎝Q L x D

β

a u1x

D a β u2x

2

− rx



D α u1x

D α u2x

2⎞

⎠dx.

2.35

Theorem 2.13 Let f, g ∈ C2I, let I be a compact interval, u i ∈ Uv, K i  1, 2, and rx ≥ 0

for all x ∈ a, b Also let u1x/u2x, v1x/v2x ∈ I, v1x/v2x be nonconstant, and let

q x be given in 2.2 Then there exists ξ ∈ I such that

b

a q xfv1x/v2xdx − b

a r xfu1x/u2xdx

b

a q xgv1x/v2xdx − b

a r xgu1x/u2xdx 

f ξ

g ξ . 2.36

It is provided that denominators are not equal to zero.

Proof Let us take a function h ∈ C2I defined as

h x  c1f x − c2g x, 2.37 where

c1

b

a

q xg



v1x

v2x



dx−

b

a

r xg



u1x

u2x



dx,

c2

b

a

q xf



v1x

v2x



dx−

b

a

r xf



u1x

u2x



dx.

2.38

ByTheorem 2.9with f  h, we have

0c1

2f ξ − c2

2g ξ b

a

q x



v1x

v2x

2

dx−

b

a

r x



u1x

u2x

2

dx

. 2.39

Since

b

a

q x



v1x

v2x

2

dx−

b

a

r x



u1x

u2x

2

dx /  0, 2.40

so we have

c1f ξ − c2g ξ  0. 2.41 This implies that

c2

c1  f ξ

This is the claim of the theorem

Let us note that a generalized Cauchy mean-valued theorem for fractional derivative was given in8 Here we will give some related results as consequences ofTheorem 2.13

Corollary 2.14 Let f, g ∈ C2I, let I be a compact interval, u i ∈ Ca, b i  1, 2, and rx ≥ 0

for all x ∈ a, b Also let u1x/u2x, I α u1x/I α u2x ∈ I, let u1x/u2x be nonconstant,

let Q I t be given in 2.11, and u1x, u2x have Riemann-Liouville fractional derivative of order

α > 0 Then there exists ξ ∈ I such that

b

a Q I xfu1x/u2xdx − b

a r xfI α u1x/I α u2xdx

b

a Q I xgu1x/u2xdx − b

a r xgI α u1x/I α u2xdx 

f ξ

g ξ . 2.43

It is provided that denominators are not equal to zero.

r x ≥ 0 for all x ∈ a, b Also let u n1 t/u n2 t, D α

∗a u1x/D α

∗a u2x ∈ I, let u n1 x/u n2 x

be nonconstant, let Q D t be given in 2.15, and u1x, u2x have Caputo fractional derivative of

order α > 0 Then there exists ξ ∈ I such that

b

a Q D xfu n1 x/u n2 xdx− b

a r xfD α

∗a u1x/D α

∗a u2xdx

b

a Q D xgu n1 x/u n2 xdx− b

a r xgD α

∗a u1x/D α

∗a u2xdx 

f ξ

g ξ . 2.44

It is provided that denominators are not equal to zero.

an L∞fractional derivative D a β u i in a, b, and rx ≥ 0 for all x ∈ a, b Also let D β −k

a u i a  0 for

k  1, , β  1 i  1, 2, D α u1x/D α u2x, D β

a u1x/D β

a u2x ∈ I, let D β

a u1x/D β

a u2x be

nonconstant, and let Q L t be given in 2.22 Then there exists ξ ∈ I such that

b

a Q L xfD β a u1x/D β

a u2xdx− b

a r xfD α u1x/D α u2xdx

b

a Q L xgD β a u1x/D β

a u2xdx− b

a r xgD α u1x/D α u2xdx 

f ξ

g ξ . 2.45

It is provided that denominators are not equal to zero.

all x ∈ a, b Let u1x/u2x, v1x/v2x ∈ I, let v1x/v2x be nonconstant, and let qx be

given in2.2 Then, for s, t ∈ R \ {0, 1} and s / t, there exists ξ ∈ I such that

ξ

⎛

⎝ss − 1

t t − 1

b

a q xv1x/v2x t dx− b

a r xu1x/u2x t dx

b

a q xv1x/v2x s dx− b

a r xu1x/u2x s dx

⎞

⎠

1/t−s

. 2.46

Proof We set f x  x t and gx  x s , t /  s, s, t / 0, 1 ByTheorem 2.13, we have

b

a q xv1x/v2x t dx− b

a r xu1x/u2x t dx

b

a q xv1x/v2x s dx− b

a r xu1x/u2x s dx  t t − 1ξ t−2

s s − 1ξ s−2. 2.47

This implies that

ξ t −s s s − 1

t t − 1

b

a q xv1x/v2x t dx− b

a r xu1x/u2x t dx

b

a q xv1x/v2x s dx− b

a r xu1x/u2x s dx . 2.48

This implies that

ξ

⎛

⎝ss − 1

t t − 1

b

a q xv1x/v2x t dx− b

a r xu1x/u2x t dx

b

a q xv1x/v2x s dx− b

a r xu1x/u2x s dx

⎞

⎠

1/t−s

. 2.49

Remark 2.18 Since the function ξ → ξ t −sis invertible and from2.46, we have

m≤

⎛

⎝ss − 1

t t − 1

b

a q xv1x/v2x t dx− b

a r xu1x/u2x t dx

b

a q xv1x/v2x s dx− b

a r xu1x/u2x s dx

⎞

⎠

1/t−s

≤ M. 2.50

Now we can suppose that f /g is an invertible function, then from2.36 we have

ξf

g

−1⎛

⎝

b

a q xv1x/v2xdx − b

a r xu1x/u2x t dx

b

a q xv1x/v2xdx − b

a r xu1x/u2x s dx

⎞

⎠. 2.51

We see that the right-hand side of2.49 is mean, then for distinct s, t ∈ R it can be written as

M s,t

 

t



s

1/t−s

2.52

as mean in broader sense Moreover, we can extend these means, so in limiting cases for

s, t /  0, 1,

lim

t → s M s,t

 M s,s

 exp

⎛

⎝

b

a q xAx slogAxdx − b

a r xBx slogBxdx

b

a q xAx s dx− b

a r xBx s dx − 2s− 1

s s − 1

⎞

⎠,

lim

s→ 0M s,s

 M 0,0 exp

⎛

⎜ a b q xlog2Axdx − b

a r xlog2Bxdx

2 b

a q x log Axdx − b

a r x log Bxdx  1

⎞

⎟

⎠,

lim

s→ 1M s,s

 M 1,1

 exp

⎛

⎜ a b q xAxlog2Axdx − b

a r xBxlog2Bxdx

2 b

a q xAx log Axdx − b

a r xBx log Bxdx − 1

⎞

⎟

⎠,

lim

t→ 0M s,t

 M s,0

⎛

⎜ a b q xAx s dx− b

a r xBx s dx

 b

a q x log Axdx − b

a r x log Bxdxs s − 1

⎞

⎟

1/s

,

lim

t→ 1M s,t

 M s,1



⎛

⎜ b

a q xAx log Axdx − b

a r xBx log Bxdxs s − 1

b

a q xAx s dx− b

a r xBx s dx

⎞

⎟

1/1−s

,

2.53

whereAx  v1x/v2x and Bx  u1x/u2x.

Remark 2.19 In the case of Riemann-Liouville fractional integral of order α > 0, we well use

the notation M s,t instead of M s,t and we replace v i x with u i x, u i x with I α u i x, and

q x with Q I x.

as mean in broader sense Moreover, we can extend these means, so in limiting cases for< /p>

s,...

. 2.46