Generalization of an Inequality for Integral Transforms with Kernel and

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 ˇ c, ´ 1, 2 and Yong Zhou3 Sajid Iqbal,1 J. Pecari 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 [email protected] Received 27 March 2010; Revised 2 August 2010; Accepted 27 October 2010 Academic Editor: András Ronto´ 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 Peˇcarić 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-Peˇcarić 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-Peˇcarić inequality and also discuss the exponential convexity. At the end, we prove the monotonicity of the means.


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


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


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


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


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


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


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


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


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.


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
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


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


17

References 1 D. S. Mitrinović and J. E. Peˇcarić, “Generalizations of two inequalities of Godunova and Levin,” Bulletin of the Polish Academy of Sciences, vol. 36, no. 9-10, pp. 645–648, 1988. 2 J. E. Peˇcarić, 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. 3 D. S. Mitrinović, J. E. Peˇcarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. 4 M. Anwar, N. Latif, and J. Peˇcarić, “Positive semidefinite matrices, exponential convexity for majorization, and related cauchy means,” Journal of Inequalities and Applications, vol. 2010, Article ID 728251, 2010. 5 G. A. Anastassiou, Fractional Differentiation Inequalities, Springer Science-Businness Media, Dordrecht, The Netherlands, 2009. 6 G. D. Handley, J. J. Koliha, and J. Peˇcarić, “Hilbert-Pachpatte type integral inequalities for fractional derivatives,” Fractional Calculus & Applied Analysis, vol. 4, no. 1, pp. 37–46, 2001. 7 J. J. Trujillo, M. Rivero, and B. Bonilla, “On a Riemann-Liouville generalized Taylor’s formula,” Journal of Mathematical Analysis and Applications, vol. 231, no. 1, pp. 255–265, 1999. 8 J. E. Peˇcarić, I. Perić, and H. M. Srivastava, “A family of the Cauchy type mean-value theorems,” Journal of Mathematical Analysis and Applications, vol. 306, no. 2, pp. 730–739, 2005.