Some new Chebyshev type inequalities utilizing generalized fractional integral operators

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http://www.aimspress.com/journal/Math DOI:10.3934/math.2020079 Received: 03 October 2019 Accepted: 28 November 2019 Published: 14 January 2020 Research article

Some new Chebyshev type inequalities utilizing generalized fractional

integral operators

Fuat Usta∗_{, H ¨}_{useyin Budak and Mehmet Zeki Sarıkaya}

Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-Turkey * Correspondence: Email: fuatusta@duzce.edu.tr; Tel: +905554168681.

Abstract: Chebyshev type inequalities for the generalized fractional integral operators are studied based on two synchronous functions in a rather general form. The main results of this paper generalize some previous results obtained by the authors. We also present the special cases of related inequalities for this type of fractional integral is obtained.

Keywords: Chebyshev inequality; fractional integral operators Mathematics Subject Classification: 26D15, 26A33, 26D10

1. Introduction

In recent years, there has been an increasing interest in the fractional calculus. One of the significant motivations for such deep interest in the subject is its capability to model a number of natural phenomena, see, for example, the papers [9, 12]. On the other hand Chebyshev inequality has broad practicability in statistical problems, numerical quadrature, probability and transform theory, and the bounding of special functions. Its basic appeal develops out of a desire to approximate, for instance, information in the form of a particular measure of the product of functions in terms of the products of the individual function measures. It is, also, of great interest in differential and difference equations [7, 13].

The essential destination of the present study is to prove a Chebyshev type inequality for the generalized fractional integral operators. After some preliminaries and summarization of some previous known results in Section 2, Section 3 deals with general Chebyshev type inequalities for generalized fractional integral operators. Finally, some concluding remarks are given in Section 4.

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2. Preliminaries

In this section we recall some basic definitions and previous results which will be used in what follows.

In 1882, Chebyshev [3] proved the following inequality:

Theorem 2.1. Let f and g be two integrable functions in [0, 1]. If both functions are simultaneously increasing or decreasing for the same values of x in[0, 1], then

1 Z 0 f(x)g(x)dx ≥ 1 Z 0 f(x)dx 1 Z 0 g(x)dx. (2.1)

If one function is increasing and the other is decreasing for the same values of x in [0, 1], then (2.1) reverses. In the last years, many papers were devoted to the generalization of the inequalities (2.1), we can mention the works [1, 2, 4–6, 8, 11, 15–19].

2.1. Generalized fractional integral operators

In [14], Raina defined the following results connected with the general class of fractional integral operators. F_ρ,λσ (x)= F_ρ,λσ(0),σ(1),...(x)= ∞ X k=0 σ (k) Γ (ρk + λ)x k _{(ρ, λ > 0; |x| < R) ,} (2.2) where the coefficients σ (k) (k ∈ N0= N∪ {0}) is a bounded sequence of positive real numbers and R is

the set of real numbers. With the help of (2.2), in [14], Raina defined the following fractional integral operators, as follows: J_ρ,λ,a+;ωσ f(x)= Z x a (x − t)λ−1F_ρ,λσ ω (x − t)ρ_{f (t)dt, x > a,} (2.3) The importance of these operators stems indeed from their generality. Many useful fractional integral operators can be obtained by specializing the coefficient σ (k). Here, we just point out that the classical Riemann-Liouville fractional integrals I_aα+ of order α defined by (see, [10])

I_aα+f (x) := 1 Γ (α) Z x a (x − t)α−1 f(t)dt (x > a; α > 0) , (2.4) follow easily by setting

λ = α, σ (0) = 1, w = 0, (2.5)

in (2.3).

In [19], Usta et. al gave the following Chebychev type inequalities for the generalized fractional integral operators:

Theorem 2.2. Let f and g be two synchronous functions on [0, ∞), that is they are having the same sense of variation on[0, ∞). Then for all t, ρ, λ > 0 and w ∈ R, we have

J_ρ,λ,0+;ωσ (1)(t)J_ρ,λ,0+;ωσ ( f g) (t) ≥ J_ρ,λ,0+;ωσ f(t)J_ρ,λ,0+;ωσ g(t), (2.6) where the coefficients σ (k) (k ∈ N0 = N∪ {0}) is a bounded sequence of positive real numbers.

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Theorem 2.3. Let f and g be two synchronous functions on [0, ∞), that is they are having the same sense of variation on[0, ∞). Then for all t > 0 and ρ1, ρ2, λ1, λ2 > 0 and w1, w2 ∈ R, we have

tλ2F_ρ2,λ2+1σ2 [ω2tρ2] J_{ρ1,λ1,0+;ω1}σ1 ( f g) (t)+ tλ1Fρ1σ1,λ1+1[ω2tρ1] J_{ρ2,λ2,0+;ω2}σ2 ( f g) (t) (2.7)

≥ J_ρ1,λ1σ1 _,0+;ω1( f ) (t)J_{ρ2,λ2,0+;ω2}σ2 (g) (t)+ J_{ρ1,λ1,0+;ω1}σ1 (g) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f ) (t)

where the coefficients σ1(k), σ2(k) (k ∈ N0 = N∪ {0}) are bounded sequences of positive real numbers.

For more details, one may consult [19]. 3. Main findings & Cumulative results

In this section, we will present some fractional integral inequalities for functions defined on the positive real line with the help of fractional integral operators given above. The inequalities to be given in this section are a generalization of the former inequalities.

Theorem 3.1. Let f and g be two synchronous functions on [0, ∞), that is they are having the same sense of variation on[0, ∞) and h ≥ 0. Then for all t > 0 and ρ1, ρ2, λ1, λ2 > 0 and w1, w2 ∈ R, we

have tλ2F_ρ2,λ2+1σ2 [ω2tρ2] J_{ρ1,λ1,0+;ω1}σ1 ( f gh) (t)+ tλ1F_ρ1σ1_,λ1+1[ω2tρ1] J_{ρ2,λ2,0+;ω2}σ2 ( f gh) (t) ≥ J_ρ1,λ1σ1 _,0+;ω1(h f ) (t)J_{ρ2,λ2,0+;ω2}σ2 (g) (t)+ J_{ρ1,λ1,0+;ω1}σ1 (hg) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f ) (t) +Jσ1 ρ1,λ1,0+;ω1( f ) (t)Jρ2,λ2,0+;ω2σ2 (hg) (t)+ Jρ1,λ1,0+;ω1σ1 (g) (t)Jρ2,λ2,0+;ω2σ2 (h f ) (t) −J_ρ1σ1_,λ1,0+;ω1( f g) (t)J_{ρ2,λ2,0+;ω2}σ2 (h) (t) − J_{ρ1,λ1,0+;ω1}σ1 (h) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f g) (t)

Proof. Let f , g and h be three functions satisfying the conditions of Theorem 3.1. Then, we have ( f (η) − f (ξ)) (g(η) − g(ξ)) (h(η)+ h(ξ)) ≥ 0.

Therefore

f(η)g(η)h(η)+ f (ξ)g(ξ)h(ξ) ≥ h(η) f (η)g(ξ) + h(η) f (ξ)g(η) + h(ξ) f (η)g(ξ) (3.1) + h(ξ) f (ξ)g(η) − f (η)g(η)h(ξ) − f (ξ)g(ξ)h(η).

Now, by multiplying both sides of (3.1) by (t − η)λ1−1(t −ξ)λ2−1F_ρσ1

1,λ1 ω 1(t −η)ρ1 Fρσ22,λ2 ω 2(t −ξ)ρ2, we obtain: (t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2σ2_,λ2ω2(t −ξ)ρ2 f (η)g(η)h(η) (3.2)

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+ (t − η)λ1−1_{(t −}_ξ)λ2−1_Fσ1 ρ1,λ1ω1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 f (ξ)g(ξ)h(ξ) ≥ (t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2σ2_,λ2ω2(t −ξ)ρ2 h(η) f (η)g(ξ) + (t − η)λ1−1 (t −ξ)λ2−1F_ρσ1 1,λ1 ω 1(t −η)ρ1 Fρσ22,λ2 ω 2(t −ξ)ρ2 h(η) f (ξ)g(η) + (t − η)λ1−1 (t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 Fρ2,λ2σ2 ω2(t −ξ)ρ2 h(ξ) f (η)g(ξ) + (t − η)λ1−1 (t −ξ)λ2−1Fσ1 ρ1,λ1ω1(t −η)ρ1 F σ2 ρ2,λ2ω2(t −ξ)ρ2 h(ξ) f (ξ)g(η) −(t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 f (η)g(η)h(ξ) −(t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 f (ξ)g(ξ)h(η).

Finally, if we double integrate (3.2) with respect to η and ξ over (0, t) × (0, t), we get the desired

result.

Corollary 3.2. Choosingλ1 = λ2= λ, σ1 = σ2 = σ, ρ1 = ρ2= ρ and w1 = w2 = w in Theorem 3.1, we

obtain the following inequality

tλF_ρ,λ+1σ [ωtρ] J_ρ,λ,0+;ωσ ( f gh) (t)+ J_ρ,λ,0+;ωσ (h) (t)J_ρ,λ,0+;ωσ ( f g) (t)

≥ J_ρ,λ,0+;ωσ (h f ) (t)J_ρ,λ,0+;ωσ (g) (t)+ J_ρ,λ,0+;ωσ (hg) (t)J_ρ,λ,0+;ωσ ( f ) (t)

where the coefficients σ (k) (k ∈ N0 = N∪ {0}) is a bounded sequence of positive real numbers.

Remark 3.3. If we choose h= 1 in Theorem 3.1, Theorem 3.1 reduce to Theorem 2.3 which was proved by Usta et al. in [19].

Theorem 3.4. Let f, g and h be three monotonic functions defined on [0, ∞), satisfying the following ( f (η) − f (ξ)) (g(η) − g(ξ)) (h(η) − h(ξ)) ≥ 0

for allη, ξ ∈ [0, t] , then for all t > 0 and ρ1, ρ2, λ1, λ2 > 0 and w1, w2 ∈ R, we have

tλ2Fσ2 ρ2,λ2+1[ω2tρ2] J σ1 ρ1,λ1,0+;ω1( f gh) (t) − tλ1Fρ1σ1,λ1+1[ω2tρ1] J σ2 ρ2,λ2,0+;ω2( f gh) (t) ≥ J_ρ1,λ1σ1 _,0+;ω1(h f ) (t)J_{ρ2,λ2,0+;ω2}σ2 (g) (t)+ J_{ρ1,λ1,0+;ω1}σ1 (hg) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f ) (t) −J_ρ1σ1_,λ1,0+;ω1( f ) (t)J_{ρ2,λ2,0+;ω2}σ2 (hg) (t) − J_{ρ1,λ1,0+;ω1}σ1 (g) (t)J_{ρ2,λ2,0+;ω2}σ2 (h f ) (t) +Jσ1 ρ1,λ1,0+;ω1( f g) (t)Jρ2,λ2,0+;ω2σ2 (h) (t) − Jρ1,λ1,0+;ω1σ1 (h) (t)Jρ2,λ2,0+;ω2σ2 ( f g) (t)

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Proof. The proof is similar to previous theorem. Theorem 3.5. Let f, g be two functions on [0, ∞). Then for all t > 0 and ρ1, ρ2, λ1, λ2 > 0 and

w1, w2 ∈ R, we have tλ2_Fσ2 ρ2,λ2+1[ω2tρ2] J_{ρ1,λ1,0+;ω1}σ1 f2(t)+ tλ1_Fσ1 ρ1,λ1+1[ω2tρ1] J_{ρ2,λ2,0+;ω2}σ2 g2(t) ≥ 2J_ρ1,λ1σ1 _,0+;ω1( f ) (t)J_{ρ2,λ2,0+;ω2}σ2 (g) (t),

Proof. As

(( f (η) − g(ξ))2≥ 0 we have

f2(η)+ g2(ξ) ≥ 2 f (η)g(ξ). (3.3)

Now, by multiplying both sides of (3.3) by

(t −η)λ1−1(t −ξ)λ2−1_Fσ1 ρ1,λ1ω1(t −η)ρ1 F_ρ2σ2_,λ2ω2(t −ξ)ρ2, we get (t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 Fρ2,λ2σ2 ω2(t −ξ)ρ2 f2(η) (3.4) + (t − η)λ1−1 (t −ξ)λ2−1Fσ1 ρ1,λ1ω1(t −η)ρ1 F σ2 ρ2,λ2ω2(t −ξ)ρ2 g2(ξ) ≥ 2 (t − η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 f (η)g(ξ).

Finally by double integration (3.4) over (0, t) × (0, t), we get the desired result. Corollary 3.6. Choosingλ1 = λ2= λ, σ1 = σ2 = σ, ρ1 = ρ2= ρ and w1 = w2 = w in Theorem 3.5, we

tλF_ρ,λ+1σ [ωtρ] J_ρ,λ,0+;ωσ f2(t)+ tλF_ρ,λ+1σ [ωtρ] J_ρ,λ,0+;ωσ g2(t)

≥ 2J_ρ,λ,0+;ωσ ( f ) (t)J_ρ,λ,0+;ωσ (g) (t). In particular

tλF_ρ,λ+1σ [ωtρ] J_ρ,λ,0+;ωσ f2(t) ≥hJ_ρ,λ,0+;ωσ ( f ) (t)i2

Theorem 3.7. Let f, g be two functions on [0, ∞). Then for all t > 0 and ρ1, ρ2, λ1, λ2 > 0 and

w1, w2 ∈ R, we have J_ρσ1 1,λ1,0+;ω1 f2(t)J_ρσ2 2,λ2,0+;ω2 g2(t)+ J_ρσ2 2,λ2,0+;ω2 f2(t)J_ρσ1 1,λ1,0+;ω1 g2(t) ≥ 2Jσ1 ρ1,λ1,0+;ω1( f g) (t)Jρ2,λ2,0+;ω2σ2 ( f g) (t).

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Proof. As

(( f (η)g(ξ) − f (ξ)g(η))2 ≥ 0 we have

f2(η)g2(ξ)+ f2(ξ)g2(η) ≥ 2 f (η)g(η)g(ξ) f (ξ). (3.5)

Then by multiplying both sides of (3.5) by

(t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω

1(t −η)ρ1 F_ρ2σ2_,λ2ω2(t −ξ)ρ2 and following the similar steps in Theorem

3.5, we get the desired result.

Corollary 3.8. Choosingλ1 = λ2= λ, σ1 = σ2 = σ, ρ1 = ρ2= ρ and w1 = w2 = w in Theorem 3.7, we

J_ρ,λ,0+;ωσ f2(t)J_ρ,λ,0+;ωσ g2(t) ≥hJ_ρ,λ,0+;ωσ ( f g) (t)i2

Lemma 3.9. Let f : R → R, and define

f(x)=

x

Z

0

f(t)dt,

then for all t, ρ > 0, λ > 1 and w ∈ R, we have

J_{ρ,λ−1,0+;ω}σ f(t)= J_ρ,λ,0+;ωσ ( f ) (t)

Proof. J_ρ,λ,0+;ωσ f(t) = t Z 0 (t −τ)λ−1F_ρ,λσ ω (x − τ)ρ           τ Z 0 f(u)du           dτ = t Z 0 f(u) t Z u (t − u)λ−1F_ρ,λσ ω (t − u)ρ dτdu = t Z 0 (t −τ)λF_ρ,λ+1σ ω (x − t)ρ_{f (u)du} = Jσ ρ,λ+1,0+;ω( f ) (t). Theorem 3.10. Let f and g be two functions on [0, ∞), Then for all t, ρ > 0, λ1, λ2 > 1 and w1, w2 ∈ R,

we have tλ2F_ρ2σ2_,λ2+1[ω2tρ2] J_{ρ1,λ1−1,0+;ω1}σ1 f g(t)+ tλ1F_ρ1,λ1σ1 ₊₁[ω2tρ1] Jρ2σ2,λ2−1,0+;ω2 f g(t)

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≥ J_ρ1,λ1σ1 _−1,0_+;ω1f(t)J_{ρ2,λ2−1,0}σ2 _+;ω2(g) (t)+ J_{ρ1,λ1−1,0}σ1 _+;ω1(g) (t)J_{ρ2,λ2−1,0}σ2 _+;ω2f(t),

Proof. From Lemma 3.9 and inequality (2.7), we have tλ2F_ρ2,λ2+1σ2 [ω2tρ2] J_{ρ1,λ1−1,0}σ1 _+;ω1 f g(t)+ tλ1F_ρ1σ1_,λ1+1[ω2tρ1] J_{ρ2,λ2−1,0}σ2 _+;ω2 f g(t) = tλ2_Fσ2 ρ2,λ2+1[ω2tρ2] J_{ρ1,λ1,0+;ω1}σ1 ( f g) (t)+ tλ1F_ρ1,λ1+1σ1 [ω2tρ1] J_{ρ2,λ2,0+;ω2}σ2 ( f g) (t) ≥ J_ρ1σ1_,λ1,0+;ω1( f ) (t)J_ρ2,λ2σ2 _,0+;ω2(g) (t)+ J_{ρ1,λ1,0+;ω1}σ1 (g) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f ) (t) = Jσ1 ρ1,λ1−1,0+;ω1 f(t)J_{ρ2,λ2−1,0+;ω2}σ2 (g) (t)+ J_{ρ1,λ1−1,0+;ω1}σ1 (g) (t)J_ρ2,λ2σ2 _−1,0+;ω2f(t)

which completes the proof.

Corollary 3.11. Choosingλ1 = λ2 = λ, σ1 = σ2 = σ, ρ1= ρ2 = ρ and w1 = w2 = w in Theorem 3.10,

we obtain the following inequality

tλF_ρ,λ+1σ [ωtρ] J_{ρ,λ−1,0+;ω}σ f g(t) ≥ J_{ρ,λ−1,0+;ω}σ f(t)J_{ρ,λ−1,0+;ω}σ (g) (t)

Theorem 3.12. Let f and g be two differentiable function on [0, ∞).Then for all t, ρ > 0, λ1, λ2 > 0

and w1, w2 ∈ R, we have t λ2_Fσ2 ρ2,λ2+1[ω2tρ2] J_ρ1,λ1σ1 _,0+;ω1( f g) (t)+ tλ1F_ρ1,λ1+1σ1 [ω2tρ1] J_{ρ2,λ2,0+;ω2}σ2 ( f g) (t) −J_ρσ1 1,λ1,0+;ω1( f ) (t)J σ2 ρ2,λ2,0+;ω2(g) (t) − J σ1 ρ1,λ1,0+;ω1(g) (t)J σ2 ρ2,λ2,0+;ω2( f ) (t) ≤ k f0k_∞kg0k_∞tλ1+λ2+2Fσ1 ρ1,λ1+1[ω1tρ1] F σ2 ρ2,λ2+1[ω2tρ2] , where k f0k_∞ = sup x∈[0,∞) | f0(x)| < ∞,

and the coefficients σ1(k), σ2(k) (k ∈ N0 = N∪ {0}) are bounded sequences of positive real numbers.

Proof. With basic calculation, we have

tλ2F_ρ2,λ2σ2 ₊₁[ω2tρ2] J_{ρ1,λ1,0+;ω1}σ1 ( f g) (t)+ tλ1F_ρ1,λ1+1σ1 [ω2tρ1] J_{ρ2,λ2,0+;ω2}σ2 ( f g) (t) −J_ρ1σ1_,λ1,0+;ω1( f ) (t)J_ρ2,λ2σ2 _,0+;ω2(g) (t) − J_{ρ1,λ1,0+;ω1}σ1 (g) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f ) (t) = t Z 0 t Z 0 (t −η)λ1−1(t −ξ)λ2−1F_ρσ1 1,λ1 ω 1(t −η)ρ1 Fρσ22,λ2 ω 2(t −ξ)ρ2 ( f (η) − f (ξ)) (g(η) − g(ξ)) dηdξ.

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Then taking modulus of the above equality, we find that t λ2_Fσ2 ρ2,λ2+1[ω2tρ2] J_ρ1σ1_,λ1,0+;ω1( f g) (t)+ tλ1F_ρ1,λ1σ1 ₊₁[ω2tρ1] J_ρ2σ2_,λ2,0+;ω2( f g) (t) −J_{ρ1,λ1,0+;ω1}σ1 ( f ) (t)J_{ρ2,λ2,0+;ω2}σ2 (g) (t) − J_{ρ1,λ1,0+;ω1}σ1 (g) (t)J_{ρ2,λ2,0+;ω2}σ2 ( f ) (t) = t Z 0 t Z 0 (t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 ( f (η) − f (ξ)) (g(η) − g(ξ)) dηdξ = t Z 0 t Z 0 (t −η)λ1−1(t −ξ)λ2−1F_ρσ1 1,λ1 ω 1(t −η)ρ1 Fρσ22,λ2 ω 2(t −ξ)ρ2            η Z ξ f0(u)du                       η Z ξ g0(v)dv            dηdξ ≤ k f0k_∞kg0k_∞ t Z 0 t Z 0 (t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2σ2_,λ2ω2(t −ξ)ρ2 (η − ξ)2dηdξ ≤ k f0k_∞kg0k_∞t2F_ρ1,λ1σ1 ₊₁[ω1tρ1] F_ρ2σ2_,λ2+1[ω2tρ2]

which completes the proof.

we obtain the following inequality t λ_Fσ ρ,λ+1[ωtρ] Jρ,λ,0+;ωσ ( f g) (t) − Jρ,λ,0+;ωσ ( f ) (t)Jρ,λ,0+;ωσ (g) (t) ≤ 1 2k f 0_k ∞kg 0_k ∞t2λ+2 h F_ρ,λ+1σ [ωtρ]i2 where the coefficients σ (k) (k ∈ N0 = N∪ {0}) is a bounded sequence of positive real numbers.

Remark 3.14. If we choose λ1 = α, λ2 = β, σ1(0) = σ2(0) = 1, and w1 = w2 = 0, in Theorem

3.1, Theorem 3.4, Theorem 3.5, Theorem 3.7, Theorem 3.10 and Theorem 3.12, then the inequalities reduces to Theorem 2.1, Theorem 2.2, Theorem 2.3, Theorem 2.4, Theorem 2.6 and Theorem 2.7 proved by Sulaiman in [18], respectively.

Theorem 3.15. Let f and g be two synchronous functions on [0, ∞), that is they are having the same sense of variation on[0, ∞) , and let v1, v2 : [0, ∞) → [0, ∞) . Then for all t, ρ > 0, λ1, λ2 > 0 and

w1, w2 ∈ R, we have

J_ρ2σ2_,λ2,0+;ω2(v2) (t)J_{ρ1,λ1,0+;ω1}σ1 (v1f g) (t)+ J_{ρ2,λ2,0+;ω2}σ2 (v2f g) (t)J_{ρ1,λ1,0+;ω1}σ1 (v1) (t) (3.6)

≥ J_ρ2σ2_,λ2,0+;ω2(v2g) (t)J_{ρ1,λ1,0+;ω1}σ1 (v1f) (t)+ J_{ρ2,λ2,0+;ω2}σ2 (v2f) (t)J_{ρ1,λ1,0+;ω1}σ1 (v1g) (t),

Proof. As f and g be two synchronous functions on [0, ∞) , then for all η, ξ ≥ 0 we have ( f (η) − f (ξ)) (g(η) − g(ξ)) ≥ 0.

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Therefore

f(η)g(η)+ f (ξ)g(ξ) ≥ f (η)g(ξ) + f (ξ)g(η). (3.7)

Mutlipying both sides of (3.7) by (t − η)λ1−1Fσ1

ρ1,λ1ω1(t −η)ρ1 v1(η), η ∈ (0, t) , we find that

(t −η)λ1−1F_ρ1,λ1σ1 ω

1(t −η)ρ1 v1(η) f (η)g(η)+ (t − η)λ1−1F_ρ1,λ1σ1 ω1(t −η)ρ1 v1(η) f (ξ)g(ξ)(3.8)

≥ (t −η)λ1−1F_ρ1,λ1σ1 ω

1(t −η)ρ1 v1(η) f (η)g(ξ)+ (t − η)λ1−1F_ρ1,λ1σ1 ω1(t −η)ρ1 v1(η) f (ξ)g(η).

Integrating (3.7) with repect to η over (0, t) , we get

J_{ρ1,λ1,0+;ω1}σ1 (v1f g) (t)+ f (ξ)g(ξ)Jρ1,λ1σ1 ,0+;ω1(v1) (t) ≥ g(ξ)J_{ρ1,λ1,0+;ω1}σ1 (v1f) (t)+ f (ξ)J_{ρ1,λ1,0+;ω1}σ1 (v1g) (t).

(3.9) Now, similarly, by multiplying both sides of (3.9) by (t − ξ)λ2−1F_ρ2,λ2σ2 ω

2(t −ξ)ρ2 v2(ξ), ξ ∈ (0, t) and

integrating with respect to ξ over (0, t) , we get the desired result.

J_ρ,λ,0+;ωσ (v2) (t)J_ρ,λ,0+;ωσ (v1f g) (t)+ J_ρ,λ,0+;ωσ (v2f g) (t)J_ρ,λ,0+;ωσ (v1) (t)

≥ J_ρ,λ,0+;ωσ (v2g) (t)J_ρ,λ,0+;ωσ (v1f) (t)+ J_ρ,λ,0+;ωσ (v2f) (t)J_ρ,λ,0+;ωσ (v1g) (t),

Remark 3.17. If we choose v1 = v2 = 1 in Theorem 3.15, the inequality (3.14) reduce to inequality

(2.7).

Theorem 3.18. Let f and g be two synchronous functions on [0, ∞), that is they are having the same sense of variation on [0, ∞) , and let p, q, r : [0, ∞) → [0, ∞) . Then for all t, ρ > 0, λ1, λ2 > 0 and

w1, w2 ∈ R, we have J_ρσ1 1,λ1,0+;ω1(r) (t)J σ2 ρ2,λ2,0+;ω2(q) (t)J σ1 ρ1,λ1,0+;ω1(p f g) (t) +2Jσ1 ρ1,λ1,0+;ω1(r) (t)Jρ2,λ2,0+;ω2σ2 (q f g) (t)Jρ1σ1,λ1,0+;ω1(p) (t) +Jρ1,λ1σ1 ,0+;ω1(p) (t)Jρ2,λ2,0+;ω2σ2 (q) (t)Jρ1,λ1,0+;ω1σ1 (r f g) (t) +Jσ1 ρ1,λ1,0+;ω1(q) (t)Jρ2,λ2σ2 ,0+;ω2(r) (t)Jρ1,λ1,0+;ω1σ1 (p f g) (t) +Jσ1 ρ1,λ1,0+;ω1(q) (t)Jρ2,λ2σ2 ,0+;ω2(r f g) (t)Jρ1σ1,λ1,0+;ω1(p) (t) ≥ J_ρσ1 1,λ1,0+;ω1(r) (t)J σ2 ρ2,λ2,0+;ω2(qg) (t)J σ1 ρ1,λ1,0+;ω1(p f ) (t)

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+Jρ1,λ1σ1 ,0+;ω1(r) (t)Jρ2,λ2,0+;ω2σ2 (q f ) (t)Jρ1,λ1,0+;ω1σ1 (pg) (t) +Jρ1,λ1σ1 ,0+;ω1(q) (t)Jρ2,λ2σ2 ,0+;ω2(rg) (t)Jρ1,λ1,0+;ω1σ1 (p f ) (t) +Jσ1 ρ1,λ1,0+;ω1(q) (t)Jρ2,λ2σ2 ,0+;ω2(r f ) (t)Jρ1,λ1,0+;ω1σ1 (pg) (t) +Jσ1 ρ1,λ1,0+;ω1(p) (t)Jρ2,λ2,0+;ω2σ2 (qg) (t)Jρ1,λ1,0+;ω1σ1 (r f ) (t) +Jσ1 ρ1,λ1,0+;ω1(p) (t)Jρ2,λ2,0+;ω2σ2 (q f ) (t)Jρ1σ1,λ1,0+;ω1(rg) (t),

Proof. If we choose v1 = p and v2 = q in Theorem 3.15, we can write:

J_ρ2σ2_,λ2,0+;ω2(q) (t)J_{ρ1,λ1,0+;ω1}σ1 (p f g) (t)+ J_{ρ2,λ2,0+;ω2}σ2 (q f g) (t)J_ρ1,λ1σ1 _,0+;ω1(p) (t) (3.10)

≥ J_ρ2σ2_,λ2,0+;ω2(qg) (t)J_ρ1σ1_,λ1,0+;ω1(p f ) (t)+ J_{ρ2,λ2,0+;ω2}σ2 (q f ) (t)J_ρ1,λ1σ1 _,0+;ω1(pg) (t). Multiplying both sides of (3.10) by J_ρ1σ1_,λ1,0+;ω1(r) (t), we get

J_{ρ1,λ1,0+;ω1}σ1 (r) (t)J_{ρ2,λ2,0+;ω2}σ2 (q) (t)J_{ρ1,λ1,0+;ω1}σ1 (p f g) (t) (3.11)

+J_{ρ1,λ1,0+;ω1}σ1 (r) (t)J_ρ2σ2_,λ2,0+;ω2(q f g) (t)J_ρ1,λ1σ1 _,0+;ω1(p) (t) ≥ J_{ρ1,λ1,0+;ω1}σ1 (r) (t)J_{ρ2,λ2,0+;ω2}σ2 (qg) (t)J_{ρ1,λ1,0+;ω1}σ1 (p f ) (t)

+Jσ1

ρ1,λ1,0+;ω1(r) (t)Jρ2σ2,λ2,0+;ω2(q f ) (t)Jρ1,λ1σ1 ,0+;ω1(pg) (t).

If we choose v1 = r and v2 = q in Theorem 3.15 and multiplying by J_{ρ1,λ1,0+;ω1}σ1 (p) (t), then we find that

J_{ρ1,λ1,0+;ω1}σ1 (p) (t)J_{ρ2,λ2,0+;ω2}σ2 (q) (t)J_{ρ1,λ1,0+;ω1}σ1 (r f g) (t) (3.12)

+J_{ρ1,λ1,0+;ω1}σ1 (p) (t)J_ρ2σ2_,λ2,0+;ω2(q f g) (t)J_{ρ1,λ1,0+;ω1}σ1 (r) (t) ≥ J_{ρ1,λ1,0+;ω1}σ1 (p) (t)J_{ρ2,λ2,0+;ω2}σ2 (qg) (t)J_ρ1,λ1σ1 _,0+;ω1(r f ) (t)

+Jσ1

ρ1,λ1,0+;ω1(p) (t)Jρ2σ2,λ2,0+;ω2(q f ) (t)Jρ1,λ1,0+;ω1σ1 (rg) (t).

Similarly, if we choose v1 = p and v2 = r in Theorem 3.15 and multiplying by J_ρ1,λ1σ1 _,0+;ω1(q) (t), then

we find that

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+J_{ρ1,λ1,0+;ω1}σ1 (q) (t)J_{ρ2,λ2,0+;ω2}σ2 (r f g) (t)J_ρ1,λ1σ1 _,0+;ω1(p) (t) ≥ J_{ρ1,λ1,0+;ω1}σ1 (q) (t)J_{ρ2,λ2,0+;ω2}σ2 (rg) (t)J_{ρ1,λ1,0+;ω1}σ1 (p f ) (t)

+Jσ1

ρ1,λ1,0+;ω1(q) (t)Jρ2,λ2,0+;ω2σ2 (r f ) (t)Jρ1,λ1σ1 ,0+;ω1(pg) (t).

Then by adding the inequalities of (3.11)-(3.13), the desired inequality has been obtained. Corollary 3.19. Choosingλ1 = λ2 = λ, σ1 = σ2 = σ, ρ1= ρ2 = ρ and w1 = w2 = w in Theorem 3.18,

2J_ρ,λ,0+;ωσ (r) (t)J_ρ,λ,0+;ωσ (q) (t)J_ρ,λ,0+;ωσ (p f g) (t) +2Jσ ρ,λ,0+;ω(r) (t)Jρ,λ,0+;ωσ (p) (t)Jρ,λ,0+;ωσ (q f g) (t) +2Jσ ρ,λ,0+;ω(p) (t)Jρ,λ,0+;ωσ (q) (t)Jρ,λ,0+;ωσ (r f g) (t) ≥ J_ρ,λ,0+;ωσ (r) (t)hJ_ρ,λ,0+;ωσ (qg) (t)J_ρ,λ,0+;ωσ (p f ) (t)+ J_ρ,λ,0+;ωσ (q f ) (t)J_ρ,λ,0+;ωσ (pg) (t)i +Jσ ρ,λ,0+;ω(q) (t) h J_ρ,λ,0+;ωσ (rg) (t)J_ρ,λ,0+;ωσ (p f ) (t)+ J_ρ,λ,0+;ωσ (r f ) (t)J_ρ,λ,0+;ωσ (pg) (t)i +Jσ ρ,λ,0+;ω(p) (t) h J_ρ,λ,0+;ωσ (qg) (t)J_ρ,λ,0+;ωσ (r f ) (t)+ J_ρ,λ,0+;ωσ (q f ) (t)J_ρ,λ,0+;ωσ (rg) (t)i where the coefficients σ (k) (k ∈ N0 = N∪ {0}) is a bounded sequence of positive real numbers.

Remark 3.20. If we chooseλ1 = α, λ2 = β, σ1(0) = σ2(0) = 1, and w1 = w2 = 0, in Theorem 3.15

and Theorem 3.18 then the inequalities reduces to Lemma 3 and Theorem 4 proved by Dahmani in [5], respectively.

Theorem 3.21. Let v1and v2be two positive functions on[0, ∞) and let f and g be two differentiable

functions on [0, ∞).If f0 ∈ Lp([0, ∞)) , g0 ∈ Lq([0, ∞)) , p > 1, 1_p + 1_q = 1, then for all t > 0 and

ρ1, ρ2, λ1, λ2> 0 and w1, w2∈ R, we have T( f , g; v1, v2) ≤ t k f0kpkg 0_k qJ σ1 ρ1,λ1,0+;ω1(v1) (t)J_ρ2σ2_,λ2,0+;ω2(v2) (t), where T( f , g; v1, v2) := J σ2 ρ2,λ2,0+;ω2(v2) (t)J σ1 ρ1,λ1,0+;ω1(v1f g) (t)+ J σ2 ρ2,λ2,0+;ω2(v2f g) (t)J σ1 ρ1,λ1,0+;ω1(v1) (t) − J_ρ2σ2_,λ2,0+;ω2(v2g) (t)J_{ρ1,λ1,0+;ω1}σ1 (v1f) (t) − J_{ρ2,λ2,0+;ω2}σ2 (v2f) (t)J_{ρ1,λ1,0+;ω1}σ1 (v1g) (t)

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Proof. Following the similar steps of proof of Theorem 8, we can write T( f , g; v1, v2) := t Z 0 t Z 0 H(η, ξ) (t − η)λ1−1_{(t −}ξ)λ2−1_Fσ1 ρ1,λ1ω1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 v1(η)v2(ξ)dηdξ (3.14) where H(η, ξ)= ( f (η) − f (ξ)) (g(η) − g(ξ)) = η Z ξ η Z ξ f0(x)g0(y)dxdy.

Using the well-known H¨older inequality for double integral, we find that

|H(η, ξ)| ≤ η Z ξ η Z ξ | f0(x)|pdxdy 1 p η Z ξ η Z ξ |g0(y)|qdxdy 1 q = |η − ξ| η Z ξ | f0(x)|pdx 1 p η Z ξ |g0(y)|qdy 1 q . (3.15) Substituting (3.15) into (3.14), we have

|T ( f , g; v1, v2)| ≤ t Z 0 t Z 0 |η − ξ| (t − η)λ1−1_{(t −}_ξ)λ2−1_Fσ1 ρ1,λ1ω1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 v1(η)v2(ξ) (3.16) × η Z ξ | f0(x)|pdx 1 p η Z ξ |g0(y)|qdy 1 q dηdξ.

Appying again H¨older inequality to the right hand side of (3.16), we find that |T ( f , g; v1, v2)| ≤            t Z 0 t Z 0 |η − ξ| (t − η)λ1−1_{(t −}_ξ)λ2−1_Fσ1 ρ1,λ1ω1(t −η)ρ1 F_ρ2σ2_,λ2ω2(t −ξ)ρ2 v1(η)v2(ξ) η Z ξ | f0(x)|pdx dηdξ            1 p ×            t Z 0 t Z 0 |η − ξ| (t − η)λ1−1_{(t −}_ξ)λ2−1_Fσ1 ρ1,λ1ω1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 v1(η)v2(ξ) η Z ξ |g0(x)|qdx dηdξ            1 q ≤ k f0k_pkg0k_q ×           t Z 0 t Z 0 |η − ξ| (t − η)λ1−1 (t −ξ)λ2−1Fσ1 ρ1,λ1ω1(t −η)ρ1 F σ2 ρ2,λ2ω2(t −ξ)ρ2 v1(η)v2(ξ)dηdξ           1 p

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×           t Z 0 t Z 0 |η − ξ| (t − η)λ1−1 (t −ξ)λ2−1F_ρσ1 1,λ1 ω 1(t −η)ρ1 Fρσ22,λ2 ω 2(t −ξ)ρ2 v1(η)v2(ξ)           1 q . Now using the fact that |η − ξ| ≤ t and 1_p + 1_q = 1, we have

|T ( f , g; v1, v2)| ≤ t k f0k_pkg0k_q           t Z 0 t Z 0 (t −η)λ1−1(t −ξ)λ2−1F_ρ1σ1_,λ1ω 1(t −η)ρ1 Fρ2,λ2σ2 ω2(t −ξ)ρ2 v1(η)v2(ξ)dηdξ           1 p ×           t Z 0 t Z 0 (t −η)λ1−1(t −ξ)λ2−1F_ρ1,λ1σ1 ω 1(t −η)ρ1 F_ρ2,λ2σ2 ω2(t −ξ)ρ2 v1(η)v2(ξ)           1 q = t k f0_k pkg 0_k q h J_{ρ1,λ1,0+;ω1}σ1 (v1) (t) i1p h J_{ρ2,λ2,0+;ω2}σ2 (v2) (t) i1p h J_{ρ1,λ1,0+;ω1}σ1 (v1) (t) i1q h J_{ρ2,λ2,0+;ω2}σ2 (v2) (t) i1q = t k f0_k pkg 0_k qJ σ1 ρ1,λ1,0+;ω1(v1) (t)J σ2 ρ2,λ2,0+;ω2(v2) (t).

Thus the proof is completed.

Corollary 3.22. If we chooseλ1= λ2 = λ, σ1 = σ2= σ, ρ1 = ρ2= ρ, w1 = w2 = w and v1 = v2 = v in

Theorem 3.21, we have the following inequality

T( f , g; v, v) ≤ t k f0k_pkg0k_qhJ_ρ,λ,0+;ωσ (v) (t)i2

Remark 3.23. In particular, puttingλ1 = λ2 = α, σ1(0)= σ2(0)= 1, and w1 = w2= 0, then Corollary

3.22 reduce to Theorem 3.1 proved by Dahmani et. al in [6].

Corollary 3.24. If we choose v1 = v2 = v in Theorem 3.21, we have the following inequality

T( f , g; v, v) ≤ t k f0k_pkg0k_qJ_ρ1,λ1σ1 _,0+;ω1(v) (t)J_ρ2,λ2σ2 _,0+;ω2(v) (t)

Remark 3.25. In particular, putting λ1 = α, λ2 = β, σ1(0) = σ2(0) = 1, and w1 = w2 = 0, then

Corollary 3.24 reduce to Theorem 3.2 proved by Dahmani et. al in [6].

Corollary 3.26. If we choose v1 = v2 = 1 in Theorem 3.21, we have the following inequality

T( f , g) ≤ tλ1+λ2+1F_ρ1σ1_,λ1+1[ω1tρ1] F_ρ2,λ2+1σ2 [ω2tρ2] k f0kpkg 0_k

q.

Remark 3.27. In particular, putting λ1 = α, λ2 = β, σ1(0) = σ2(0) = 1, and w1 = w2 = 0, then

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4. Concluding remarks

We have introduced a general version of Chebyshev type integral inequality for the generalised fractional integral operators based on two synchronous functions. The established results are generalization of some existing Chebychev type integral inequalities in the previous published studies. For further investigations we propose to consider the Chebyshev type inequalities for other fractional integral operators.

Conflict of interest

All authors declare no conflicts of interest in this paper. References

1. S. Belarbi, Z. Dahmani, On Some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10 (2009), 1–12.

2. K. Boukerrioua, A. G. Lakoud, On generalization of ˇCebyˇsev type inequalities, J. Inequal. Pure Appl. Math., 8 (2007), 1–9.

3. P. L. ˇCebyˇsev, Sur less expressions approximatives des integrales definies par les autres prises entre les memes limites,Proc. Math. Soc. Charkov, 2 (1882), 93–98.

4. J. Choi, P. Agarwal, Certain fractional integral inequalities involving hypergeometric operators, East Asian Mathematical Journal, 30 (2014), 283–291.

5. Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9 (2010), 493–497. 6. Z. Dahmani, O. Mechouar, S. Brahami, Certain inequalities related to the Chebyshev functional

involving a type Riemann-Liouville operator, Bull. Math. Anal. Appl., 3 (2011), 38–44.

7. S. S. Dragomir, A generalization of Gr¨uss inequality in inner product spaces and applications, J. Math. Annal. Appl., 237 (1999), 74–82.

8. A. G. Lakoud, F. Aissaouinew, New ˇCebyˇsev type inequalities for double integrals, J. Math. Inequal., 5 (2011), 453–462.

9. T. Hayat, S. Nadeem, S. Asghar, Periodic unidirectional fows of a viscoelastic fluid with the fractional Maxwell model, Appl. Math. Comput., 151 (2004), 153–161.

10. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.

11. S. M. Malamud, Some complements to the Jensen and Chebyshev inequalities and a problem of W. Walter,Proc. Amer. Math. Soc., 129 (2001), 2671–2678.

12. S. Momani, Z. Odiba, Numerical approach to differential equations of fractional order, J. Comput. Appl. Math., 207(2007), 96–110.

13. D. S. Mitrinovic, J. E. Pecaric, A. M. Fink, Classical and new inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

(15)

14. R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian mathematical journal, 21 (2005), 191–203.

15. B. G. Pachpatte, On ˇCebyˇsev-Gr¨uss type inequalities via Pecaric’s extention of the Montgomery identity,J. Inequal. Pure Appl. Math., 7 (2006), 1–10.

16. J. E. Pecaric, On the Cebyˇsev inequality,Bul. Sti. Tehn. Inst. Politehn ”Tralan Vuia”ˇ Timis¸ora(Romania), 25 (1980), 5–9.

17. M. Z. Sarıkaya, N. Aktan, H. Yıldırım, Weighted ˇCebyˇsev-Gr¨uss type inequalities on time scales, J. Math. Inequal., 2 (2008), 185–195.

18. W. T. Sulaiman, Some new fractional integral inequalities, J. Math. Anal., 2 (2011), 23–28. 19. F. Usta, H. Budak and M. Z. Sarıkaya, On Chebyshev type inequalities for fractional integral

operators, AIP Conf. Proc., 1833 (2017), 1–4. c

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