On some integral inequalities using Hadamard fractional integral

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On some integral inequalities using Hadamard fractional integral

Vaijanath L. Chinchane,

a,∗

_{and Deepak B. Pachpatte}

b

a_{Department of Mathematics, Deogiri Institute of Engineering and Management Studies, Aurangabad-431005, India.}

b_{Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, India.}

Abstract

In this paper, using Hadamard fractional integral, we establish two main new result on fractional integral inequalities by considering the extended Chebyshev functional in case of synchronous function. The first result concerns with some inequalities using one fractional parameter and other with two parameter.

Keywords:Chebyshev functional, Hadamard fractional integral, Hadamard fractional derivative and fractional integral inequality.

1

Introduction

In recent years, many authors have worked on fractional integral inequalities and its application which plays important role in classical differential and integral equations, see [3, 5, 6, 7, 8, 9, 10]. Dahmani gave the following fractional integral inequalities, using the Riemann-Liouville fractional integral for extended Chebyshev functional, see for instance [6].

Theorem 1.1. Let f and g be two synchronous function on [0,∞[ and let r, p, q : [0,∞[→ [0,∞[ for all

t >0, α >0 and then

2Jαr(t) [Jαp(t)Jα(qf g)(t) +Jαq(t)Jα(pf g)(t)] + 2Jαp(t)Jαq(t)Jα(rf g)(t)≥

Jαr(t) [Jα(pf)(t)Jα(qg)(t) +Jα(qf)(t)Jα(pg)(t)]Jαp(t)[Jα(rf)(t)Jα(qg)(t) +Jα(qf)(t)Jα(rg)(t)] +Jαq(t) [Jα(rf)(t)Jα(pg)(t) +Jα(pf)(t)Jα(rg)(t)].

(1.1)

Theorem 1.2. Let f and g be two synchronous function on [0,∞[ and let r, p, q : [0,∞[→ [0,∞[ for all

t >0, α >0 β >0 then we have,

Jαr(t)

Jαq(t)Jβ(pf g)(t) + 2Jαp(t)Jβ(qf g)(t) +Jβq(t)Jα(pf g)(t)

+

Jαp(t)Jβq(t) +Jβp(t)Jαq(t)

Jα(rf g)(t)≥ Jαr(t)

Jα(pf)(t)Jβ(qg)(t) +Jβ(qf)(t)Jα(pg)(t)

Jαp(t)[Jα(rf)(t)Jβ(qg)(t) +Jβ(qf)(t)Jα(rg)(t)] +Jαq(t)Jα(rf)(t)Jβ(pg)(t) +Jβ(pf)(t)Jα(rg)(t).

(1.2)

The main objective of this paper is to establish some inequalities for the extended Chebyshev functional given in [6], using Hadamard fractional integrals. The paper has been organized as follows. In Section 2, we define basic definitions and proposition related to Hadamard fractional derivatives and integrals. In Section 3, we give the main results.

∗_{Corresponding author.}

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2

Preliminaries

Recently many authors have studied integral inequalities on fractional calculus using Riemann-Liouville, Caputo derivative, see [3, 5, 6, 7, 8, 9, 10]. The necessary background details are given in the book A.A. Kilbas [1], and in book of S.G. Samko et al. [4], here we present some definitions of Hadamard derivative and integral as given in [2, p.159-171].

Definition 2.1. The Hadamard fractional integral of order α∈R+ _{of function} _f₍_x₎_{, for all} _{x >}₁ _{is defined} as,

HD−1,xαf(x) =

1 Γ(α)

Z x

1

ln(x

t)

α−1_f₍_t₎dt

t , (2.1)

where Γ(α) =R₀∞e−uuα−1du.

Definition 2.2. The Hadmard fractional derivative of orderα∈[n−1, n),n∈Z+_{, of function}_f₍_x₎_{is given} as follows

HD1α,xf(x) =

1 Γ(n−α)(x

d dx)

nZ x

1

ln(x

t)

n−α−1_f₍_t₎dt

t . (2.2)

From the above definitions, we can see obviously the difference between Hadamard fractional and Riemann-Liouville fractional derivative and integrals, which include two aspects. The kernel in the Hadamard integral has the form of ln(x

t) instead of the form of (x−t), which is involves both in the Riemann-Liouville and Caputo

integral. The Hadamard derivative has the operator(x_dxd)n_{, whose construction is well suited to the case of the}

half-axis and is invariant relation to dilation [4, p.330], while the Riemann-Liouville derivative has the operator (d

dx) n_.

We give some image formulas under the operator (2.1) and (2.2), which would be used in the derivation of our main result.

Proposition 2.1. [2] If0< α <1, the following relation hold:

HD−1,xα(lnx)

β−1₌ Γ(β)

Γ(β+α)(lnx)

β+α−1_, _(2.3)

HD1α,x(lnx)

β−1₌ Γ(β)

Γ(β−α)(lnx)

β−α−1_, _(2.4)

respectively.

For the convenience of establishing the result, we give the semigroup property,

(HD−1,xα)(HD− β

1,x)f(x) =HD

−(α+β)

1,x f(x). (2.5)

3

Main Results

In this section, we present and prove the main results.

Lemma 3.1. Let f and g be two synchronous function on [0,∞[. and x, y : [0,∞) → [0,∞). Then for all

t >0,α >0, we have,

HD−1,tαx(t)HD−1,tα(yf g)(t) +HD−1,tαy(t)HD−1,tα(xf g)(t)≥ HD−1,tα(xf)(t)HD1−,tα(yg)(t) +HD−1,tα(yf)(t)HD−1,tα(xg)(t).

(3.1)

Proof. Sincef andg are synchronous on [0,∞[ for allτ≥0,ρ≥0, we have

(f(τ)−f(ρ))(g(τ)−g(ρ))≥0. (3.2)

From (3.2),

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Now, multiplying both side of (3.3) by (ln(τt)) α−1_x₍_τ₎

τΓ(α) , τ∈(0, t),t >0. Then the integrating resulting identity

with respect toτ from 1 tot we obtain

1 Γ(α) Z t 1 ln(t τ)

α−1_x₍_τ₎_f₍_τ₎_.g₍_τ₎dτ

τ + 1 Γ(α) Z t 1 ln(t τ)

α−1_x₍_τ₎_f₍_ρ₎_.g₍_ρ₎dτ

τ ≥ 1 Γ(α) Z t 1 ln(t τ)

α−1_x₍_τ₎_f₍_τ₎_.g₍_ρ₎dτ

τ + 1 Γ(α) Z t 1 ln(t τ)

α−1_x₍_τ₎_f₍_ρ₎_.g₍_τ₎dτ

τ .

(3.4)

Consequently,

HD1−,tα(xf g)(t) +f(ρ).g(ρ)HD−1,tα(x)(t)≥g(ρ)HD−1,tα(xf)(t) +f(ρ)HD1−,tα(xg)(t). (3.5)

Multiplying both side of (3.5) by (ln(

t ρ))

α−1_y₍_ρ₎

ρΓ(α) , ρ ∈ (0, t), t > 0. Then integrating resulting identity with

respect toρfrom 1 totwe obtain

HD1−,tα(xf g)(t)

1 Γ(α) Z t 1 ln(t ρ)

α−1_y₍_ρ₎dρ

ρ +HD

−α

1,t(x)(t)

α−1_y₍_ρ₎_f₍_ρ₎_g₍_ρ₎dρ

ρ

≥H D1−,tα(xf)(t)

α−1_y₍_ρ₎_g₍_ρ₎dρ

ρ +HD

−α

1,t(xg)(t)

1 Γ(α), Z t 1 ln(t ρ)

α−1_y₍_ρ₎_f₍_ρ₎dρ

ρ ,

(3.6)

and this ends the proof of inequality 3.1.

Now, we gave our main result here.

Theorem 3.2. Let f and g be two synchronous function on[0,∞[, andr, p, q: [0,∞)→[0,∞). Then for all

t >0,α >0, we have

2HD−1,tαr(t)

HD−1,tαp(t)HD1−,tα(qf g)(t) +HD−1,tαq(t)HD1−,tα(pf g)(t)

+ 2HD−1,tαp(t)HD1−,tαq(t)HD−1,tα(rf g)(t)≥

HD1−,tαr(t)

HD1−,tα(pf)(t)HD1−,tα(qg)(t) +HD1−,tα(qf)(t)HD−1,tα(pg)(t)

+

HD1−,tαp(t)

HD−1,tα(rf)(t)HD1−,tα(qg)(t) +HD1−,tα(qf)(t)HD−1,tα(rg)(t)

+

HD1−,tαq(t)

HD1−,tα(rf)(t)HD−1,tα(pg)(t) +HD1−,tα(pf)(t)HD1−,tα(rg)(t)

(3.7)

Proof. To prove above theorem, puttingx=p, y=q, and using lemma 3.1, we get

HD−1,tαp(t)HD1−,tα(qf g)(t) +HD1−,tαq(t)HD1−,tα(pf g)(t)≥ HD−1,tα(pf)(t)HD−1,tα(qg)(t) +HD−1,tα(qf)(t)HD−1,tα(pg)(t).

(3.8)

Now, multiplying both side of (3.8) byHD−1,tαr(t), we have

HD1−,tαr(t)

HD−1,tαp(t)HD−1,tα(qf g)(t) +HD1−,tαq(t)HD1−,tα(pf g)(t)

≥

HD1−,tαr(t)

HD−1,tα(pf)(t)HD1−,tα(qg)(t) +HD−1,tα(qf)(t)HD−1,tα(pg)(t)

, (3.9)

puttingx=r, y=q, and using lemma 3.1, we get

HD1−,tαr(t)HD1−,tα(qf g)(t) +

−α

1,t q(t)HD1−,tα(rf g)(t)≥ HD1−,tα(rf)(t)HD−1,tα(qg)(t) +HD−1,tα(qf)(t)HD−1,tα(rg)(t),

(3.10)

multiplying both side of (3.10) byHD1−,tαp(t), we have

HD1−,tαp(t)

HD−1,tαr(t)HD−1,tα(qf g)(t) +

−α

1,t q(t)HD1−,tα(rf g)(t)

≥

HD1−,tαp(t)

HD−1,tα(rf)(t)HD1−,tα(qg)(t) +HD−1,tα(qf)(t)HD−1,tα(rg)(t)

. (3.11)

With the same arguments as before, we can write

HD1−,tαq(t)

HD1−,tαr(t)

−α

1,t(pf g)(t) +HD1−,tαp(t)HD−1,tα(rf g)(t)

≥

HD1−,tαq(t)

HD1−,tα(rf)(t)HD−1,tα(pg)(t) +HD−1,tα(pf)(t)HD−1,tα(rg)(t)

. (3.12)

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Lemma 3.3. Letf andg be two synchronous function on[0,∞[. andx, y: [0,∞[→[0,∞[. Then for allt >0,

α >0, we have

HD−1,tαx(t)HD− β

1,t(yf g)(t) +HD− β

1,ty(t)HD1−,tα(xf g)(t)≥ HD−1,tα(xf)(t)HD−1,tβ(yg)(t) +HD−1,tβ(yf)(t)HD−1,tα(xg)(t).

(3.13)

Proof. Now multiplying both side of (3.5) by (ln(

t ρ))

β−1_y₍_ρ₎

ρΓ(β) ,ρ∈(0, t),t >0 we obtain:

(ln(_ρt))β−1_y₍_ρ₎

ρΓ(β) .HD

−α

1,t(xf g)(t) +

(ln(_ρt))β−1_y₍_ρ₎

ρΓ(β) .f(ρ)g(ρ)HD

−α

1,tx(t)≥

(ln(_ρt))β−1_y₍_ρ₎

ρΓ(β) .g(ρ)HD

−α

1,t(xf)(t) +

(ln(t_ρ))β−1_y₍_ρ₎

ρΓ(β) .f(ρ)HD

−α

1,t(xg)(t),

(3.14)

then integrating (3.14) over (1,t), we obtain

HD−1,tα(xf g)(t)

1 Γ(β) Z t 1 ln(t ρ)

β−1_y₍_ρ₎dρ

ρ +HD

−α

1,t(x)(t)

β−1_y₍_ρ₎_f₍_ρ₎_g₍_ρ₎dρ

ρ

≥H D−1,tα(xf)(t)

β−1

y(ρ)g(ρ)dρ

ρ +HD

−α

1,t(xg)(t)

1 Γ(β), Z t 1 ln(t ρ)

β−1

y(ρ)f(ρ)dρ

ρ,

(3.15)

this ends the proof of inequality (3.13).

Theorem 3.4. Let f andg be two synchronous function on[0,∞[, andr, p, q: [0,∞)→[0,∞). Then for all

t >0,α >0, we have

HD1−,tαr(t)

h

HD−1,tαq(t)HD1−,tβ(pf g)(t) + 2HD1−,tαp(t)HD1−,tβ(qf g)(t) +HD1−,tβq(t)HD−1,tα(pf g)(t)

i

+hHD−1,tαp(t)HD− β

1,tq(t) +HD− β

1,tp(t)HD1−,tαq(t)

i

H

D−₁_,tα(rf g)(t)≥

HD1−,tαr(t)

h

HD−1,tα(pf)(t)HD−1,tβ(qg)(t) +HD1−,tβ(qf)(t)HD−1,tα(pg)(t)

i

+

HD1−,tαp(t)

h

HD1−,tα(rf)(t)HD−1,tβ(qg)(t) +HD1−,tβ(qf)(t)HD−1,tα(rg)(t)

i

+

HD1−,tαq(t)

h

HD−1,tα(rf)(t)HD−1,tβ(pg)(t) +HD1−,tβ(pf)(t)HD−1,tα(rg)(t)

i

.

(3.16)

Proof. To prove above theorem, puttingx=p, y=q, and using lemma 3.3 we get

HD1−,tαp(t)HD−1,tβ(qf g)(t) +HD−1,tβq(t)HD−1,tα(pf g)(t)≥

HD1−,tα(pf)(t)HD− β

1,t(qg)(t) +HD− β

1,t(qf)(t)HD1−,tα(pg)(t).

(3.17)

Now, multiplying both side of (3.16) by HD1−,tαr(t), we have

HD−1,tαr(t)

h

HD1−,tαp(t)HD−1,tβ(qf g)(t) +HD−1,tβq(t)HD−1,tα(pf g)(t)

i

≥

HD−1,tαr(t)

h

HD1−,tα(pf)(t)HD1−,tβ(qg)(t) +HD−1,tβ(qf)(t)HD−1,tα(pg)(t)

i

,

(3.18)

putting x=r, y=q, and using lemma 3.3, we get

HD−1,tαr(t)HD−1,tβ(qf g)(t) +

−β

1,t q(t)HD−1,tα(rf g)(t)≥ HD−1,tα(rf)(t)HD−

β

1,t(qg)(t) +HD− β

1,t(qf)(t)HD1−,tα(rg)(t),

(3.19)

multiplying both side of (3.19) byHD1−,tαp(t), we have

HD−1,tαp(t)

h

HD−1,tαr(t)HD−1,tβ(qf g)(t) +HD1−,tβq(t)HD−1,tα(rf g)(t)≥

i

HD−1,tαp(t)

h

HD−1,tα(rf)(t)HD−1,tβ(qg)(t) +HD−1,tβ(qf)(t)HD1−,tα(rg)(t)

i

.

(3.20)

With the same argument as before, we obtain

HD−1,tαq(t)

h

HD1−,tαr(t)

−β

1,t(pf g)(t) +HD−1,tβp(t)HD−1,tα(rf g)(t)

i

≥

HD−1,tαq(t)

h

HD1−,tα(rf)(t)HD−1,tβ(pg)(t) +HD−1,tβ(pf)(t)HD1−,tα(rg)(t)

i

.

(3.21)

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Remark 3.1. Applying theorem 3.4 forα=β, we obtain Theorem 3.2.

Remark 3.2. If f, g, r, p and qsatisfies the following condition,

1. The function f and g is asynchronous on [0,∞).

2. The function r,p,q are negative on [0,∞).

3. Two of the function r,p,q are positive and the third is negative on [0,∞).

then the inequality 3.7 and 3.16 are reversed.

References

[1] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Application of Fractional Differential Equations,Elsevier, Amersterdam, 2006.

[2] D. Baleanu, J.A.T. Machado and C.J. Luo,Fractional Dynamic and Control,Springer, 2012, pp.159-171.

[3] G. A. Anastassiou, Fractional Differentiation Inequalities, Springer Publishing Company, Incorpo-rated, New York, NY, 2009.

[4] S.G. Somko, A.A. Kilbas and O.I. Marichev,Fractional Integral and Derivative Theory and Applica-tion,Gordon and Breach, Yverdon, Switzerland, 1993.

[5] S. Belarbi and Z. Dahmani, On some new fractional integral inequality, J. Inequal. Pure and Appl. Math., 10(3)(2009), Art. 86, 5 pp.

[6] Z. Dahmani, New inequalities in fractional integrals,Int. J. Nonlinear Sci., 9(4)(2010), 493-497.

[7] Z. Dahmani, On Minkowski and Hermit-Hadamard integral inequalities via fractional integration,

Ann. Funct. Anal., 1(1)(2010), 51-58.

[8] Z. Dahmani, The Riemann-Liouville operator to genarate some new inequalities, Int. J. Nonlinear Sci., 12(4)(2011), 452-455.

[9] Z. Dahmani, Some results associate with fractional integrals involving the extended Chebyshev,Acta Univ. Apulensis Math. Inform., 27(2011), 217-224.

[10] W. Yang, Some new fractional quantum integral inequalities,Appl. Math. Latt., 25(6)(2012), 963-969.

Received: October 7, 2012;Accepted: October 25, 2012