Integral inequalities involving generalized Erdélyi-Kober fractional integral operators

Dumitru Baleanu; Sunil Dutt Purohit; Jyotindra C. Prajapati

doi:10.1515/math-2016-0007

40% Rabatt

auf Fachbücher bei De Gruyter Brill *

Artikel Open Access

Integral inequalities involving generalized Erdélyi-Kober fractional integral operators

Dumitru Baleanu , Sunil Dutt Purohit und Jyotindra C. Prajapati

Veröffentlicht/Copyright: 13. Februar 2016

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 14 Heft 1

Abstract

Using the generalized Erdélyi-Kober fractional integrals, an attempt is made to establish certain new fractional integral inequalities, related to the weighted version of the Chebyshev functional. The results given earlier by Purohit and Raina (2013) and Dahmani et al. (2011) are special cases of results obtained in present paper.

Keywords: Fractional integral inequalities; Chebyshev functional; Erdélyi-Kober operators

MSC 2010: 26A33; 26D10; 26D15

1 Introduction

Fractional integral inequalities have many applications in numerical quadrature, transform theory, probability, and statistical problems, but the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems. Moreover, they also provide upper and lower bounds to the solutions of the above equations. Therefore, a significant development in the classical and fractional integral inequalities, particularly in analysis, has been witnessed; see, for instance, the papers [1]-[5] and the references cited therein. Moreover, these inequalities are also playing an important role to interpret the stability of a class of fractional oscillators (see [6]).

The following inequality is well known in the literature as Chebyshev inequality [7]:

If f, g : [a, b] → ℝ⁺ are absolutely continuous functions, whose first derivatives are bounded and

(1)T(f,g)=1b−a∫abf(t)g(t)dt−(1b−a∫abf(t)dt)(1b−a∫abg(t)dt),

then

|T(f,g)|≤112(b−a)2‖f'‖∞‖g'‖∞,

where ||·||_∞ denotes the norm in L_∞[a, b].

Under suitable assumptions (Chebyshev inequality, Grüss inequality, Minkowski inequality, Hermite-Hadamard inequality, Ostrowski inequality etc.), inequalities are playing a significant role in the field of mathematical sciences, particularly, in the theory of approximations. Therefore, in the literature we found several extensions and generalizations of these classical integral inequalities, including fractional calculus and q-calculus operators also (see [8]-[26]).

Further, a weighted version of the Chebyshev functional (see [7]) is defined as:

(2)T(f,g,p)=∫abp(t)dt∫abf(t)g(t)p(t)dt−∫abf(t)p(t)dt∫abg(t)p(t)dt,

where f and g are integrable functions on [a, b] and p(t) is a positive and integrable function on [a, b]. In 2000, Dragomir [27] derived the following inequality, related to the weighted Chebyshev functional (2):

(3)2|T(f,g,p)|≤‖f'‖r‖g'‖s[∫ab∫ab|x−y|p(x)p(y)dxdy],

where f, g are differentiable functions and f′ Î L_r(a, b), g′ Î L_s(a, b), r > 1, r⁻¹ + s⁻¹ = 1. The several extensions of inequality (3) are studied by many authors. Recently, Dahmani et al. [28], Purohit and Raina [29], Baleanu et al. [30] and Ntouyas et al. [31] obtained certain generalized Chebyshev type integral inequalities involving various type of fractional integral operators.

In present paper, we add one more dimension to this study by establishing certain integral inequalities for the functional (2) associated with the differentiable functions whose derivatives belong to the space L_p([0, ∞)), involving the generalized Erdélyi-Kober fractional integrals. Further, we also derive certain known and new integral inequalities for the fractional integrals by suitably choosing the special cases of our main results.

2 Fractional calculus

Authors mention the preliminaries and notations of some well-known operators of fractional calculus, which shall be used in the sequel.

The generalized Erdélyi-Kober fractional integral operator I_β^η,α of order α for a real-valued continuous function f(t) is defined as (see [32, p. 14, eqn. (1.1.17)]):

(4)Iβη,α﹛f(t)﹜=t−B(η+α)Γ(α)∫0tτβη(tβ−τβ)α−1f(τ)d(τβ)=βt−β(η+α)Γ(α)∫0tτβ(η+1)−1(tβ−τβ)α−1f(τ)dτ,

where α > 0, β > 0 and η Î ℝ.

Following Kiryakova [32], the generalized Erdélyi-Kober fractional integral operators (4), possess the advantage that a number of generalized integration and differentiation operators happen to be the particular cases of this operator. Some important special cases of the integral operator I_β^η,α are mentioned below:

(i) For η = 0, α = n (integer > 0) and β = 1, the operator (4) yields the following ordinary n-fold integrations:

(5)ln﹛f(t)﹜=tnI10,nf(t)=1(n−1)!∫0t(t−τ)n−1f(τ)dτ.

(ii) If we set η = 0 and β = 1, the operator (4) reduces to the Riemann-Liouville fractional integral operators with the following relationship:

(6)Rα﹛f(t)﹜=tαI10,α﹛f(t)﹜=1Γ(α)∫0t(t−τ)α−1f(τ)dτ.

(iii) Again, for η = 0, α = 1 and β = 1, the operator (4) leads to the Hardy-Littlewood (Cesaro) integration operator:

(7)L1,0﹛f(t)﹜=I10,1﹛f(t)﹜=1t∫0tf(τ)dτ,

and its generalization for integers n > m − 1 (when η = n, α = 1 and β = 1), we have

(8)Lm,n﹛f(t)﹜=tn−m+1I1n,1﹛f(t)﹜=t−m∫0tτnf(τ)dτ.

(iv) When β = 1, operator (4) reduces to the fractional integral operator, which was originally considered by Kober [33] and Erdélyi [34]:

(9)Iα,η﹛f(t)﹜=I1η,α﹛f(t)﹜=t−α−ηΓ(α)∫0t(t−τ)α−1τηf(τ)dτ (α>0,η∈ℝ).

(v) Also for β = 2, the operator (4) yields the Erdélyi-Kober fractional integral operator I_n,α (Sneddon [35]):

(10)Iη,α=I2η,α﹛f(t)﹜=2t−2(η+α)Γ(α)∫0tτ2η+1(t2−τ2)α−1f(τ)dτ.

(vi) Further, if we set η=−12,β=2 and α is replaced by α+12, the Uspensky integral transform ([36]) can easily be obtained as under:

(11)Pα﹛f(t)﹜=12I2−12,α+12﹛f(t)﹜=1Γ(α+12)∫01(1−τ2)α−12f(tτ)dτ.

For a detailed information about fractional integral operator (4) and its more special cases one may refer the book [32, pp. 15-17].

Next, we recall a composition formula of fractional integral (4) with a power function (see also as special case of image formula [32, p. 29, eqn. (1.2.26)]).

(12)Iβη,α﹛tλ﹜=Γ(1+η+λβ)Γ(1+α+η+λβ)tλ(λ>−β(η+1)).

3 Fractional integral inequalities

In this section, we obtain certain integral inequality which gives an estimation for the fractional integral of a product in terms of the product of the individual function fractional integrals, involving generalized Erdélyi-Kober fractioal integral operators. We give our results related to the Chebyshev’s functional (2) in the case of differentiable mappings whose derivatives belong to the space L_p([0, ∞)) and satisfy the Holder’s inequality.

Theorem 3.1.

Suppose that p be a positive function, f and g be differentiable functions on [0, ∞), f′ Î L_r([0, ∞)), g′ Î L_s([0, ∞)) such that r⁻¹ + s⁻¹ = 1 with r > 1. Then for all t > 0, α > 0, β > 0, η Î ℝ and η > −1:

(13)2|Iβη,α﹛p(t)﹜Iβη,α﹛p(t)f(t)g(t)﹜−Iβη,α﹛p(t)f(t)﹜Iβη,α﹛p(t)g(t)﹜|≤β2t−2β(η+α)‖f'‖r‖g'‖sΓ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1× ×p(τ)p(ρ)|τ−ρ|dτdρ≤‖f'‖r‖g'‖st(Iβη,α﹛t﹜)2.

Proof. Let us define

(14)ℋ(τ,ρ)=(f(τ)−f(ρ))(g(τ)−g(ρ)),

and

(15)F(t,τ)=βt−β(η+a)τβ(η+1)−1Γ(α)(tβ−τβ)α−1,τ∈(0,t),t>0.

Under the conditions stated in the theorem, we observed that the function F(t, τ) > 0, for all τ Î (0, t) (t > 0). Upon multiplying (14) by F(t, τ)p(τ) and integrating with respect to τ from 0 to t, we get

(16)βt−β(η+α)Γ(α)∫0tτβ(η+1)−1(tβ−τβ)α−1p(τ)ℋ(τ,ρ)dτ=Iβη,α﹛p(t)f(t)g(t)﹜−f(ρ)Iβη,α﹛p(t)g(t)﹜−g(ρ)Iβη,α﹛p(t)f(t)﹜+f(ρ)g(ρ)Iβη,α﹛p(t)﹜.

Next, on multiplying above relation (16) by F(t, ρ)p(ρ), and then integrating with respect to ρ from 0 to t, we obtain

(17)β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1p(τ)p(ρ)ℋ(τ,ρ)dτdρ=2(Iβη,α﹛p(t)﹜Iβη,α﹛p(t)f(t)g(t)﹜−Iβη,α﹛p(t)f(t)﹜Iβη,α﹛p(t)g(t)﹜).

Now, in view of (14), we have

ℋ(τ,ρ)=∫τρ∫τρf'(y)g'(z)dydz.

On making use of the Hölder’s inequality, namely

we obtain

(18)|ℋ(τ,ρ)|≤|∫τρ∫τρ|f'(y)|rdydz|r−1|∫τρ∫τρ|g'(z)|sdydz|s−1.

Since

|∫τρ∫τρ|f'(y)|rdydz|r−1=|τ−ρ|r−1|∫τρ|f'(y)|rdy|r−1

and

|∫τρ∫τρ|g'(z)|sdydz|s−1=|τ−ρ|s−1|∫τρ|g'(z)|sdz|s−1,

therefore, inequality (18) reduces to

(19)|ℋ(τ,ρ)|≤|τ−ρ||∫τρ|f'(y)|rdy|r−1|∫τρ|g'(z)|sdz|s−1.

It follows from (17) that

(20)β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1p(τ)p(ρ)|τ−ρ|××|∫τρ|f'(y)|rdy|r−1|∫τρ|g'(z)|sdz|s−1dτdρ.

Again using Hölder’s inequality on the right-hand side of (20), we get

β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−τβ)α−1p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤[βrt−rβ(η+α)Γr(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1× ×p(τ)p(ρ)|τ−ρ||∫τρ|f'(y)|rdy|dτdρ]r−1××[βst−sβ(η+α)Γs(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1××p(τ)p(ρ)|τ−ρ||∫τρ|g'(z)|sdz|dτdρ]s−1.

Using the fact that

|∫τρ|f(y)|pdy|≤‖f‖pp,

we get

(21)β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤[βr−t−rβ(η+α)‖f'‖rrΓr(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−τβ)α−1××p(τ)p(ρ)|τ−ρ|dτdρ]r−1××[βst−sβ(η+α)‖g'‖ssΓs(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−τβ)α−1××p(τ)p(ρ)|τ−ρ|dτdρ]s−1.

From (21), we arrived at

(22)β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρ)α−1p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤β2t−2β(η+α)‖f'‖r‖g'‖sΓ2(α)[∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−τβ)α−1××p(τ)p(ρ)|τ−ρ|dτdρ]r−1××[∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1p(τ)p(ρ)|τ−ρ|dτdρ]s−1.

Using the relation r⁻¹ + s⁻¹ = 1, the above inequality yields to

(23)β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤β2t−2β(η+α)‖f'‖r‖g'‖sΓ2(α)[∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−τβ)α−1××p(τ)p(ρ)|τ−ρ|dτdρ.

On the other hand (17) gives

(24)2|Iβη,α﹛p(t)﹜Iβη,α﹛p(t)f(t)g(t)﹜−Iβη,α﹛p(t)f(t)﹜Iβη.α﹛p(t)g(t)﹜|≤β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1××p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ.

On making use of (23) and (24), one can easily arrive at the left-hand side of the inequality (13).

Now, to derive the right-hand side of the inequality (13), we have 0 ≤ τ ≤ t, 0 ≤ ρ ≤ t, and therefore,

0≤|τ−ρ|≤t.

Evidently, from (23), we obtain

β2t−2β(η+α)Γ2(α)∫0t∫0tτβ(η+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1×p(τ)p(ρ)|ℋ(τ,ρ)dτdρ|≤β2t1−2β(η+α)‖f'‖r‖g'‖sΓ2(α)∫0t∫0tτβ(η+1)−1ρβ(η+1)−1(tβ−τβ)α−1×p(τ)p(ρ)dτdρ=‖f'‖r‖g'‖st(Iβη,α﹛p(t)﹜)2.

This leads to the proof of Theorem 3.1. □

Next, we establish a further generalization of Theorem 3.1.

Theorem 3.2.

Suppose that p is a positive function, f and g are differentiable functions on [0, ∞) and f′ Î L_r([0, ∞)), g′ Î L_s([0, ∞)) such that r > 1 and r⁻¹ + s⁻¹ = 1. Then for all t > 0 the following inequality holds:

|Iβη,α﹛p(t)﹜Iδζ,γ﹛p(t)f(t)g(t)﹜+Iδζ,γ﹛p(t)﹜Iβη,α﹛p(t)f(t)g(t)﹜|−Iβη,α﹛p(t)f(t)﹜Iδζ,γ﹛p(t)g(t)﹜−Iδζ,γ﹛p(t)f(t)﹜Iβη,α﹛p(t)g(t)﹜|≤βδt−β(η+α)−δ(ζ+γ)‖f'‖r‖g'‖sΓ(α)Γ(γ)∫0t∫0tτβ(η+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tβ−τβ)α−1(tδ−ρδ)γ−1××p(τ)p(ρ)|τ−ρ|dτdρ≤‖f'‖r‖g'‖stIβη,α﹛p(t)﹜Iδζ,γ﹛p(t)﹜,

where α, β, γ, δ > 0, η, ζ Î ℝ such that η > −1 and ζ > −1.

Proof. To obtain the desired results, we multiply (16) by

δt−δ(ζ+γ)ρδ(ζ+1)−1p(ρ)Γ(γ)(tδ−τδ)γ−1,ρ∈(0,t),t>0,

and make integration from 0 to t (with respect to ρ), to obtain

(25)βδt−β(η+α)−δ(ζ+γ)Γ(α)Γ(γ)∫0t∫0tτβ(n+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1×p(τ)p(ρ)ℋ(τ,ρ)dτdρ=Iβη,α﹛p(t)﹜Iδζ,γ﹛p(t)f(t)g(t)﹜+Iδζ,γ﹛p(t)﹜Iβη,α﹛p(t)f(t)g(t)﹜−Iβη,α﹛p(t)f(t)﹜Iδζ,γ﹛p(t)g(t)﹜−Iδζ,γ﹛p(t)f(t)﹜Iβη,α﹛p(t)g(t)﹜.

On using (19), the (25) leads to

(26)βδt−β(η+α)−δ(ζ+γ)Γ(α)Γ(γ)∫0t∫0tτβ(η+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1× ×p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤βδt−β(η+α)−δ(ζ+γ)Γ(α)Γ(γ)∫0t∫0tτβ(η+1)−1ρδ(ζ+1)(tβ−τβ)α−1(tδ−ρδ)γ−1× ×p(τ)p(ρ)|τ−ρ||∫τρ|f'(y)|rdy|r−1|∫τρ|g'(z)|sdz|s−1dτdρ.

Making use of the Hölder’s inequality, we readily obtain

(27)βδt−β(η+α)−δ(ζ+γ)Γ(α)Γ(γ)∫0t∫0tτβ(η+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1× ×p(τ)p(ρ)|ℋ(τ,ρ)|dτdρ≤βδt−β(η+α)−δ(ζ+γ)Γ(α)Γ(γ)∫0t∫0tτβ(η+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1× ×p(τ)p(ρ)|τ−ρ|dτdρ.

Now, one can easily arrive at the left-sided inequality of Theorem 3.2, by taking relations (25) and (27) into account. Further, for 0 ≤ τ ≤ t, 0 ≤ ρ ≤ t, we have

0≤|τ−ρ|≤t.

Therefore, from (27), we get

β δt−β(n+α)−δ(ζ+γ)Γ(α)Γ(γ)∫0t∫0tτβ(n+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1p(τ)p(ρ)|ℋ(τ,ρ)| dτdρ≤β δt−β(n+α)−δ(ζ+γ)‖f′‖r‖g′‖sΓ(α)Γ(γ)∫0t∫0tτβ(n+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1× ×p(τ)p(ρ)|τ−ρ| dτdρ.=‖f′‖r ‖g′‖st Iβn,α ﹛p(t)﹜ Iδζ,γ ﹛p(t)﹜,

this leads to the proof of Theorem 3.2. □

Remark 3.3.

For β = α, Theorem 3.2 immediately reduces to Theorem 3.1.

4 Special cases

Now, we briefly consider some implications of main results. To this end, if we consider β = 1 (additionally δ = 1 for Theorem 3.2) and make use of the relation (9), the Theorems 3.1 and 3.2 provide the known fractional integral inequalities involving the Erdélyi-Kober operators, due to Purohit and Raina [29]. Again, if we set β = 1, η = 0 (additionally δ = 1 and ζ = 0 for Theorem 3.2), and make use of the relation (6), the main results recover the known results due to Dahmani et al. [28, pp. 39-42, Theorems 3.1 & 3.2].

Further, by setting η=−12, β = 2 and replacing α by α+12 (additionally ζ=12, δ = 2 and γ replaced by γ+12 for Theorem 3.2), and make use of the relation (11), the main results provide the following integral inequalities involving the Uspensky integral transform:

Corollary 4.1.

Suppose that p is a positive function, f and g are differentiable functions on [0, ∞), f′ Î L_r([0, ∞)), g′ Î L_s([0, ∞)) such that r⁻¹ + s⁻¹ = 1 with r > 1. Then for all t > 0 and α > 0:

|Pα﹛p(t)﹜Pα﹛p(t)f(t)﹜− Pα﹛p(t)f(t)﹜ Pα﹛p(t)g(t)﹜|≤t−4α‖f′‖r ‖g′‖sΓ2(α+12)∫0t∫0t(t2−τ2)α−12(t2−ρ2)α−12p(τ)p(ρ)|τ−ρ| dτdρ≤‖f′‖r ‖g′‖st(Pα﹛p(t)﹜)2.

Corollary 4.2.

|Pα﹛p(t)﹜ Pγ ﹛p(t)f(t)g(t)﹜+Pγ﹛p(t)﹜Pα﹛p(t)f(t)g(t)﹜ −Pα﹛p(t)f(t)﹜ Pγ ﹛p(t)g(t)﹜−Pγ﹛p(t)f(t)﹜Pα﹛p(t)g(t)﹜|≤t−2α−2γ‖f′‖r ‖g′‖sΓ(α+12)Γ(γ+12)∫0t∫0t(t2−τ2)α−12(t2−ρ2)γ−12p(τ)p(ρ)|τ−ρ| dτdρ≤‖f′‖r ‖g′‖st Pα﹛p(t)﹜ Pγ﹛p(t)﹜, where α, γ > 0.

Moreover, by using relations (5) to (11) with suitable values of parameters η, α and β, the results established in this paper can generate some interesting inequalities involving the various type of integral operator.

Now, by suitably choosing the function p(t), we consider some examples of our main results. For example, let us set p(t) = t^ƛ(ƛ Î [0, ∞), t Î (0, ∞)), then Theorems 3.1 and 3.2 yield the following results:

Example 4.3.

Suppose that f and g are two differentiable functions on [0, ∞) and if f′ Î L_r([0, ∞)), g′ Î L_s([0, ∞)), r > 1, r⁻¹ + s⁻¹ = 1 then for all t > 0, ƛ Î [0, ∞), α > 0, β > 0, η Î ℝ such that ƛ > −β(η + 1):

2|Γ(1+η+λβ)Γ(1+α+η+λβ)tλIβn,α﹛tλf(t)g(t)﹜−Iβn,α﹛tλf(t)﹜Iβn,α﹛tλg(t)﹜|≤β2t−2β(n+α)‖f′‖r‖g′‖sΓ2(α)∫0t∫0tτβ(n+1)+λ−1ρβ(n+1)+λ−1(tβ−τβ)α−1(tβ−ρβ)α−1|τ−ρ| dτdρ≤‖f′‖r‖g′‖sΓ2(1+η+λβ)Γ2(1+α+η+λβ)t1+2λ.

Example 4.4.

Let f and g be differentiable functions on [0, ∞) and if f′ Î L_r([0, ∞)), g′ Î L_s([0, ∞)), r > 1, r⁻¹ + s⁻¹ = 1 then

|Γ(1+η+λβ)Γ(1+α+η+λβ)tλIδζ,γ﹛tλf(t)g(t)﹜+Γ(1+ζ+λδ)Γ(1+γ+ζ+λδ)tλIβn,α﹛tλf(t)g(t)﹜−Iβn,α﹛tλf(t)﹜Iδζ,γ﹛tλg(t)﹜−Iδζ,γ﹛tλf(t)﹜Iβn,α﹛tλg(t)﹜|≤β δt−β(n+α)−δ(ζ+γ)‖f′‖r‖g′‖sΓ(α)Γ(γ)∫0t∫0tτβ(n+1)+λ−1ρδ(ζ+1)+λ−1(tβ−τβ)α−1(tδ−ρδ)γ−1× ×|τ−ρ| dτdρ≤‖f′‖r‖g′‖sΓ(1+η+λβ)Γ(1+ζ+λδ)Γ(1+α+η+λβ)Γ(1+γ+ζ+γδ)t1+2λ,

for all t > 0, α, β, γ, δ > 0, η, ζ Î ℝ, ƛ Î [0, ∞) such that ƛ > min ﹛–β(η + 1), −δ(ζ + 1)﹜.

Further, if we put ƛ = 0 in Examples 4.3 and 4.4 (or set p(t) = 1 in Theorems 3.1 and 3.2), we obtain the following results:

Example 4.5.

Suppose f and g are differentiable functions on [0, ∞) and if f′ Î L_r ([0, ∞)), g′ Î L_s([0, ∞)), r > 1, r⁻¹ + s⁻¹ = 1 then for all t > 0, α > 0, β > 0, η Î ℝ such than η > −1:

2|Γ(1+η)Γ(1+α+η)Iβn,α﹛f(t)g(t)﹜−Iβn,α﹛f(t)﹜Iβn,α﹛g(t)﹜|≤β2t−2β(n+α)‖f′‖r‖g′‖sΓ2(α)∫0t∫0tτβ(n+1)−1ρβ(n+1)−1(tβ−τβ)α−1(tβ−ρβ)α−1|τ−ρ|dτdρ≤‖f′‖r‖g′‖stΓ2(1+η)Γ2(1+α+η).

Example 4.6.

Let f and g be two differentiable functions on [0, ∞). If f′ Î L_r([0, ∞)), g′ Î L_s([0, ∞)), r > 1, r⁻¹ + s⁻¹ = 1, then

|Γ(1+η)Γ(1+α+η)Iδζ,γ﹛f(t)g(t)﹜+Γ(1+ζ)Γ(1+γ+ζ)Iβn,α﹛f(t)g(t)﹜ −Iβn,α﹛f(t)﹜Iδζ,γ﹛g(t)﹜−Iδζ,γ﹛f(t)﹜Iβn,α﹛g(t)﹜|≤β δt−β(n+α)−δ(ζ+γ)‖f′‖r‖g′‖sΓ(α)Γ(γ)∫0t∫0tτβ(n+1)−1ρδ(ζ+1)−1(tβ−τβ)α−1(tδ−ρδ)γ−1|τ−ρ| dτdρ≤‖f′‖r‖g′‖stΓ(1+η)Γ(1+ζ)Γ(1+α+η)Γ(1+γ+ζ),

for all t > 0, α, β, γ, δ > 0, η, ζ Î ℝ, such that η > −1 and ζ > −1.

The results established here are giving some contribution to the theory of integral inequalities and fractional calculus, and may find some applications in the theory of fractional differential equations. Moreover, by virtue of the unified nature of the generalized Erdélyi-Kober operator (4) and arbitray function p(t), one can further deduce number of new fractional integral inequalities involving various fractional calculus operators and special functions, from our main results.

Acknowledgement

The authors would like to express their appreciation to the referees for their valuable suggestions which helped to achieve better presentation of this paper.

References

[1] V. Lakshmikantham, A. S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 2007, 11(3-4), 395-402.Suche in Google Scholar

[2] Z. Denton, A. S. Vatsala, Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations, Opusc. Math., 2011, 31(3), 327-339.10.7494/OpMath.2011.31.3.327Suche in Google Scholar

[3] A. Debbouche, D. Baleanu, R. P. Agarwal, Nonlocal nonlinear integrodifferential equations of fractional orders, Boundary Value Problems, 2012, 2012, 78.10.1186/1687-2770-2012-78Suche in Google Scholar

[4] H.-R. Sun, Y.-N. Li, J. J. Nieto, Q. Tang, Existence of solutions for Sturm-Liouville boundary value problem of impulsive differential equations, Abstr. Appl. Anal., 2012, 2012, 707163.10.1155/2012/707163Suche in Google Scholar

[5] Z.-H. Zhao, Y.-K. Chang, and J. J. Nieto, Asymptotic behavior of solutions to abstract stochastic fractional partial integrodifferential equations, Abstr. Appl. Anal., 2013, 2013, 138068.10.1155/2013/138068Suche in Google Scholar

[6] M. Li, S. C. Lim, S.Y. Chen, Exact solution of impulse response to a class of fractional oscillators and its stability, Math. Probl. Eng., 2011, 2011, 657839.10.1155/2011/657839Suche in Google Scholar

[7] P.L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 1882, 2, 93-98.Suche in Google Scholar

[8] P. Cerone, S.S. Dragomir, New upper and lower bounds for the Chebyshev functional, J. Inequal. Pure App. Math., 2002, 3, 77.Suche in Google Scholar

[9] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, 1993.10.1007/978-94-017-1043-5Suche in Google Scholar

[10] G.A. Anastassiou, Fractional Differentiation Inequalities, Springer Publishing Company, New York, 2009.10.1007/978-0-387-98128-4Suche in Google Scholar

[11] G.A. Anastassiou, Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, 2011.10.1007/978-1-4614-0703-4Suche in Google Scholar

[12] J. Pečarić, I. Perić, Identities for the Chebyshev functional involving derivatives of arbitrary order and applications, J. Math. Anal. Appl., 2006, 313, 475-483.10.1016/j.jmaa.2005.06.025Suche in Google Scholar

[13] S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 2009, 10(3), 86.Suche in Google Scholar

[14] Z. Dahmani, L. Tabharit, Certain inequalities involving fractional integrals, Journal of Advanced Research in Scientific Computing, 2010, 2(1), 55-60.10.5373/jarpm.392.032110Suche in Google Scholar

[15] S.L. Kalla, Alka Rao, On Grüss type inequality for hypergeometric fractional integrals, Le Matematiche, 2011, 66 (1), 57-64.Suche in Google Scholar

[16] H. Öǧünmez, U.M. Özkan, Fractional quantum integral inequalities, J. Inequal. Appl. , 2011, 2011, 787939.10.1155/2011/787939Suche in Google Scholar

[17] W.T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal., 2011, 2(4), 23-28.10.5556/j.tkjm.42.2011.505-510Suche in Google Scholar

[18] M.Z. Sarikaya, H. Ogunmez, On new inequalities via Riemann-Liouville fractional integration, Abstr. Appl. Anal., 2012, 2012, 428983.10.1155/2012/428983Suche in Google Scholar

[19] M.Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamards inequalities for fractional integrals and related fractional inequalities, Math. Comput. Modeling, 2012, 57(9-10), 2403-2407.10.1016/j.mcm.2011.12.048Suche in Google Scholar

[20] Z. Dahmani, A. Benzidane, On a class of fractional q-integral inequalities, Malaya J. Matematik, 2013, 3(1), 1-6.10.26637/mjm103/001Suche in Google Scholar

[21] S.D. Purohit, R.K. Raina, Chebyshev type inequalities for the Saigo fractional integrals and their q-analogues, J. Math. Inequal., 2013, 7(2), 239-249.10.7153/jmi-07-22Suche in Google Scholar

[22] S.D. Purohit, F. Uçar, R.K. Yadav, On fractional integral inequalities and their q-analogues, Revista Tecnocientifica URU, 2013, 6, 53-66.Suche in Google Scholar

[23] S.D. Purohit, S.L. Kalla, Certain inequalities related to the Chebyshev’s functional involving Erdélyi-Kober operators, Sci., Ser. A, Math. Sci., 2014, 25, 55-63.Suche in Google Scholar

[24] D. Baleanu, S.D. Purohit, P. Agarwal, On fractional integral inequalities involving hypergeometric operators, Chinese J. Math., 2014, 2014, 609476.10.1155/2014/609476Suche in Google Scholar

[25] D. Baleanu, P. Agarwal, Certain inequalities involving the fractional q-integral operators, Abstr. Appl. Anal., 2014, 2014, 371274.10.1155/2014/371274Suche in Google Scholar

[26] J. Tariboon, S.K. Ntouyas, W. Sudsutad, Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Math. Sci., 2014, 2014, 869434.10.1155/2014/869434Suche in Google Scholar

[27] S.S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., 2000, 31(4), 397-415.10.1007/s13398-019-00712-6Suche in Google Scholar

[28] Z. Dahmani, O. Mechouar, S. Brahami, Certain inequalities related to the Chebyshev’s functional involving a Riemann-Liouville operator, Bull. Math. Anal. Appl., 2011, 3(4), 38-44.Suche in Google Scholar

[29] S.D. Purohit, R.K. Raina, Certain fractional integral inequalities involving the Gauss hypergeometric function, Rev. Téc. Ing. Univ. Zulia, 2014, 37(2), 167-175.Suche in Google Scholar

[30] D. Baleanu, S.D. Purohit, Chebyshev type integral inequalities involving the fractional hypergeometric operators, Abstr. Appl. Anal., 2014, 2014, 609160.10.1155/2014/609160Suche in Google Scholar

[31] S.K. Ntouyas, S.D. Purohit, J.Tariboon, Certain Chebyshev type integral inequalities involving the Hadamard’s fractional operators, Abstr. Appl. Anal., 2014, 2014, 249091.10.1155/2014/249091Suche in Google Scholar

[32] V. Kiryakova, Generalized Fractional Calculus and Applications, (Pitman Res. Notes Math. Ser. 301, Longman Scientific & Technical, Harlow, 1994.Suche in Google Scholar

[33] H. Kober, On a fractional integral and derivative, Quart. J. Math. Oxford, 1940, 11, 193-211.10.1093/qmath/os-11.1.193Suche in Google Scholar

[34] A. Erdélyi, On fractional integration and its applications to the theory of Hankel transforms, Quart. J. Math. Oxford, 1940, 11, 293-303.10.1093/qmath/os-11.1.293Suche in Google Scholar

[35] I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory, North-Holand Publ. Co., Amsterdam, 1966.Suche in Google Scholar

[36] J.V. Uspensky, On the development of arbitrary functions in series of Hermite’s and Laguerre’s polynomials, Ann. Math., 1927, 2(28), 593-619.10.2307/1968401Suche in Google Scholar

Received: 2015-7-15

Accepted: 2016-10-6

Published Online: 2016-2-13

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.