Certain recent fractional integral inequalities associated with the hypergeometric operators

Shilpi Jain; Praveen Agarwal; Bashir Ahmad; S.K.Q. Al-Omari

doi:10.1016/j.jksus.2015.04.002

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Original article

28 (

); 82-86

doi:

10.1016/j.jksus.2015.04.002

Certain recent fractional integral inequalities associated with the hypergeometric operators

Shilpi Jain^{^a}, Praveen Agarwal^{^b}, Bashir Ahmad^{^c}, S.K.Q. Al-Omari^{^d,⁎}

a Department of Mathematics, Poornima College of Engineering, Sita Pura, Jaipur 302022, India

b Department of Mathematics, Anand International College of Engineering, Jaipur 303012, Rajasthan, India

c Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

d Faculty of Science, Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa’ Applied University, Amman 11134, Jordan

⁎Corresponding author. s.k.q.alomari@fet.edu.jo (S.K.Q. Al-Omari)

Received: 2015-4-1, Accepted: 2015-4-10, Published: 2015-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

The principle aim of this paper is to establish some new (presumably) fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann–Liouville type fractional integral operators by using hypergeometric fractional integral operator. Some relevant connections of the results presented here with those earlier ones are also pointed out.

Keywords

Integral inequalities

Chebyshev functional

Riemann–Liouville fractional integral operator

Pólya and Szegö type inequalities

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1 Introduction and preliminaries

In recent years the study of fractional integral inequalities involving functions of independent variables is an important research subject in mathematical analysis because the inequality technique is also one of the very useful tools in the study of special functions and theory of approximations. During the last two decades or so, several interesting and useful extensions of many of the fractional integral inequalities have been considered by several authors (see, for example,Cerone and Dragomir, 2007; Choi and Agarwal, 2014a,b,c,d ; see also the very recent work Anber and Dahmani, 2013). The above-mentioned works have largely motivated our present study.

For our purpose, we begin by recalling the well-known celebrated functional introduced by Chebyshev (1882) and defined by

(1.1)

T (f, g) = \frac{1}{b - a} \int_{a}^{b} f (x) g (x) dx - (\frac{1}{b - a} \int_{a}^{b} f (x) dx) (\frac{1}{b - a} \int_{a}^{b} g (x) dx),

where

f (x)

and

g (x)

are two integrable functions which are synchronous on

[a, b]

, i.e.,

(1.2)

(f (x) - f (y)) (g (x) - g (y)) ⩾ 0,

for any

x, y \in [a, b]

The functional (1.1) has attracted many researchers’ attention due to diverse applications in numerical quadrature, transform theory, probability and statistical problems. Among those applications, the functional (1.1) has also been employed to yield a number of integral inequalities (see, e.g., Anastassiou, 2011; Dragomir, 2000; Sulaiman, 2011 ; for a very recent work, see also Wang et al., 2014).

In 1935, Grüss (1935) proved the inequality

(1.3)

|T (f, g)| ⩽ \frac{(M - m) (N - n)}{4},

where

f (x)

and

g (x)

are two integrable functions which are synchronous on

[a, b]

, i.e.,

(1.4)

m ⩽ f (x) ⩽ M, n ⩽ g (x) ⩽ N,

for any

m, M, n, N \in R

and

x, y \in [a, b]

In the sequel, Pólya and Szegö (1925) introduced the following inequality

(1.5)

\frac{\int_{a}^{b} f^{2} (x) dx \int_{a}^{b} g^{2} (x) dx}{{(\int_{a}^{b} f (x) dx \int_{a}^{b} g (x) dx)}^{2}} ⩽ \frac{1}{4} {(\sqrt{\frac{MN}{mn}} + \sqrt{\frac{mn}{MN}})}^{2},

Similarly, Dragomir and Diamond (2003) proved that

(1.6)

|T (f, g)| ⩽ \frac{(M - m) (N - n)}{4 {(b - a)}^{2} \sqrt{mMnN}} \int_{a}^{b} f (x) dx \int_{a}^{b} g (x) dx,

where

f (x)

and

g (x)

are two positive integrable functions which are synchronous on

[a, b]

, i.e.,

(1.7)

0 < m ⩽ f (x) ⩽ M < \infty, 0 < n ⩽ g (x) ⩽ N < \infty .

Here, motivated essentially by above works, the main objective of this paper is to establish certain new (presumably) Pólya–Szegö type inequalities associated with Gaussian hypergeometric fractional integral operators. Relevant connections of the results presented here with those involving Riemann–Liouville fractional integrals are also indicated. Nowadays, the fractional calculus (fractional integral and derivative operators) has become one of the most rapidly growing research subjects of all branches of science due to its numerous applications. Recently many authors have showed the far-reaching development of the fractional calculus by their remarkably large number of contributions (see, e.g., Bhrawy and Zaky, 2015a,b; Bhrawy and Abdelkawy, in press; Bhrawy et al., 2015; Cattani, 2010; Jumarie, 2009; Komatsu, 1966, 1967; Li et al., 2011, 2013; Liu et al., 2014; Saxena, 1967; Yang et al., 2013a,b; Yang and Baleanu, 2013 , and the related references therein).

Here, we start by recalling the following definition.

Definition 1

Let $α > 0, μ > - 1, β, η \in R$ , then a generalized fractional integral $I_{t}^{α, β, η, μ}$ (in terms of the Gauss hypergeometric function) of order α for a real-valued continuous function $f (t)$ is defined by Choi and Agarwal (2014b, p. 285, Eq. (1.8)):

(1.8)

I_{t}^{α, β, η, μ} \{f (t)\} = \frac{t^{- α - β - 2 μ}}{Γ (α)} \int_{0}^{t} τ^{μ} {(t - τ)}^{α - 1}_{2} F_{1} (α + β + μ, - η; α; 1 - \frac{τ}{t}) f (τ) d τ,

where, the function

_{2} F_{1} (-)

appearing as a kernel for the operator (1.8) is the Gaussian hypergeometric function defined by

(1.9)

_{2} F_{1} (a, b; c; t) = \sum_{n = 0}^{\infty} \frac{{(a)}_{n} {(b)}_{n}}{{(c)}_{n}} \frac{t^{n}}{n!},

and

{(a)}_{n}

is the Pochhammer symbol:

{(a)}_{n} = a (a + 1) \dots (a + n - 1), {(a)}_{0} = 1 .

For

f (t) = t^{λ - 1}

in (1.8), we get (see Baleanu et al., 2014)

(1.10)

I_{t}^{α, β, η, μ} \{t^{λ - 1}\} = \frac{Γ (μ + λ) Γ (λ - β + η)}{Γ (λ - β) Γ (λ + μ + α + η)} t^{λ - β - μ - 1} .

where

α, β, η, λ \in R, μ > - 1, μ + λ > 0

and

λ - β + η > 0

2 Certain fractional integral inequalities associate with hypergeometric operator

In this section, we establish certain Pólya–Szegö type integral inequalities for the synchronous functions involving the hypergeometric fractional integral operator (1.8), some of which are (new) presumably ones.

Theorem 1

Let f and g be two positive integrable functions on $[0, \infty)$ . Assume that there exist four positive integrable functions $u_{1}, u_{2}$ , $v_{1}$ and $v_{2}$ on $[0, \infty)$ such that: $(A_{1}) 0 < u_{1} (τ) ⩽ f (τ) ⩽ u_{2} (τ), 0 < v_{1} (τ) ⩽ g (τ) ⩽ v_{2} (τ), (τ \in [0, t], t > 0) .$ Then for $t > 0$ and $α > 0$ , the following inequality holds:

(2.1)

\frac{I_{t}^{α, β, η, μ} {v_{1} v_{2} f^{2}} (t) I_{t}^{α, β, η, μ} {u_{1} u_{2} g^{2}} (t)}{{(I_{t}^{α, β, η, μ} {(v_{1} u_{1} + v_{2} u_{2}) fg} (t))}^{2}} ⩽ \frac{1}{4} .

Proof

To prove (2.1), we start from $(A_{1})$ , for $τ \in [0, t], t > 0$ , we have

(2.2)

\frac{f (τ)}{g (τ)} ⩽ \frac{u_{2} (τ)}{v_{1} (τ)},

which yields

(2.3)

(\frac{u_{2} (τ)}{v_{1} (τ)} - \frac{f (τ)}{g (τ)}) ⩾ 0 .

Analogously, we have

(2.4)

\frac{u_{1} (τ)}{v_{2} (τ)} ⩽ \frac{f (τ)}{g (τ)},

from which one has

(2.5)

(\frac{f (τ)}{g (τ)} - \frac{u_{1} (τ)}{v_{2} (τ)}) ⩾ 0 .

Multiplying (2.3) and (2.5), we obtain

(\frac{u_{2} (τ)}{v_{1} (τ)} - \frac{f (τ)}{g (τ)}) (\frac{f (τ)}{g (τ)} - \frac{u_{1} (τ)}{v_{2} (τ)}) ⩾ 0,

(2.6)

(\frac{u_{2} (τ)}{v_{1} (τ)} + \frac{u_{1} (τ)}{v_{2} (τ)}) \frac{f (τ)}{g (τ)} ⩾ \frac{f^{2} (τ)}{g^{2} (τ)} + \frac{u_{1} (τ) u_{2} (τ)}{v_{1} (τ) v_{2} (τ)} .

After some manipulation (2.6) can be written as

(2.7)

(u_{1} (τ) v_{1} (τ) + u_{2} (τ) v_{2} (τ)) f (τ) g (τ) ⩾ v_{1} (τ) v_{2} (τ) f^{2} (τ) + u_{1} (τ) u_{2} (τ) g^{2} (τ) .

Now, multiplying both sides of (2.7) by

\frac{t^{- α - β - 2 μ}}{Γ (α)} τ^{μ} {(t - τ)}^{α - 1}_{2} F_{1} (α + β + μ, - η; α; 1 - \frac{τ}{t})

and integrating with respect to τ from 0 to t, we get

I_{t}^{α, β, η, μ} {(u_{1} v_{1} + u_{2} v_{2}) fg} (t) ⩾ I_{t}^{α, β, η, μ} {v_{1} v_{2} f^{2}} (t) + I_{t}^{α, β, η, μ} {u_{1} u_{2} g^{2}} (t) .

Applying the AM–GM inequality, i.e.,

a + b ⩾ 2 \sqrt{ab}, a, b \in R^{+}

, we have

I_{t}^{α, β, η, μ} {(u_{1} v_{1} + u_{2} v_{2}) fg} (t) ⩾ 2 \sqrt{I_{t}^{α, β, η, μ} {v_{1} v_{2} f^{2}} (t) I_{t}^{α, β, η, μ} {u_{1} u_{2} g^{2}} (t)},

which leads to

I_{t}^{α, β, η, μ} {v_{1} v_{2} f^{2}} (t) I_{t}^{α, β, η, μ} {u_{1} u_{2} g^{2}} (t) ⩽ \frac{1}{4} {(I_{t}^{α, β, η, μ} {(u_{1} v_{1} + u_{2} v_{2}) fg} (t))}^{2} .

Therefore, we obtain the inequality (2.1) as requested.

□

Theorem 2

Let f and g be two positive integrable functions on $[0, \infty)$ . Assume that there exist four positive integrable functions $u_{1}, u_{2}$ , $v_{1}$ and $v_{2}$ satisfying $(A_{1})$ on $[0, \infty)$ . Then for $t > 0$ and $α, β > 0$ , the following inequality holds:

(2.8)

\frac{I_{t}^{α, β, η, μ} {u_{1} u_{2}} (t) I_{t}^{γ, δ, ζ, ν} {v_{1} v_{2}} (t) I_{t}^{α, β, η, μ} {f^{2}} (t) I_{t}^{γ, δ, ζ, ν} {g^{2}} (t)}{{(I_{t}^{α, β, η, μ} {u_{1} f} (t) I_{t}^{γ, δ, ζ, ν} {v_{1} g} (t) + I_{t}^{α, β, η, μ} {u_{2} f} (t) I_{t}^{γ, δ, ζ, ν} {v_{2} g} (t))}^{2}} ⩽ \frac{1}{4} .

Proof

To prove (2.8), using the condition $(A_{1})$ , we obtain

(2.9)

(\frac{u_{2} (τ)}{v_{1} (ρ)} - \frac{f (τ)}{g (ρ)}) ⩾ 0,

and

(2.10)

(\frac{f (τ)}{g (ρ)} - \frac{u_{1} (τ)}{v_{2} (ρ)}) ⩾ 0,

which imply that

(2.11)

(\frac{u_{1} (τ)}{v_{2} (ρ)} + \frac{u_{2} (τ)}{v_{1} (ρ)}) \frac{f (τ)}{g (ρ)} ⩾ \frac{f^{2} (τ)}{g^{2} (ρ)} + \frac{u_{1} (τ) u_{2} (τ)}{v_{1} (ρ) v_{2} (ρ)} .

Multiplying both sides of (2.11) by

v_{1} (ρ) v_{2} (ρ) g^{2} (ρ)

, we have

(2.12)

u_{1} (τ) f (τ) v_{1} (ρ) g (ρ) + u_{2} (τ) f (τ) v_{2} (ρ) g (ρ) ⩾ v_{1} (ρ) v_{2} (ρ) f^{2} (τ) + u_{1} (τ) u_{2} (τ) g^{2} (ρ) .

Multiplying both sides of (2.12) by

\begin{matrix} \frac{t^{- α - β - γ - δ - 2 (μ + ν)}}{Γ (α) Γ (γ)} τ^{μ} {(t - τ)}^{α - 1} ρ^{ν} {(t - ρ)}^{γ - 1} \\ \times_{2} F_{1} (α + β + μ, - η; α; 1 - \frac{τ}{t})_{2} F_{1} (γ + δ + ν, - ζ; γ; 1 - \frac{ρ}{t}) \end{matrix}

and double integrating with respect to τ and ρ from 0 to t, we have

\begin{matrix} I_{t}^{α, β, η, μ} {u_{1} f} (t) I_{t}^{γ, δ, ζ, ν} {v_{1} g} (t) + I_{t}^{α, β, η, μ} {u_{2} f} (t) I_{t}^{γ, δ, ζ, ν} {v_{2} g} (t) \\ ⩾ I_{t}^{α, β, η, μ} {f^{2}} (t) I_{t}^{γ, δ, ζ, ν} {v_{1} v_{2}} (t) + I_{t}^{α, β, η, μ} {u_{1} u_{2}} (t) I_{t}^{γ, δ, ζ, ν} {g^{2}} (t) . \end{matrix}

Applying the AM–GM inequality, we get

\begin{matrix} I_{t}^{α, β, η, μ} {u_{1} f} (t) I_{t}^{γ, δ, ζ, ν} {v_{1} g} (t) + I_{t}^{α, β, η, μ} {u_{2} f} (t) I_{t}^{γ, δ, ζ, ν} {v_{2} g} (t) \\ ⩾ 2 \sqrt{I_{t}^{α, β, η, μ} {f^{2}} (t) I_{t}^{γ, δ, ζ, ν} {v_{1} v_{2}} (t) I_{t}^{α, β, η, μ} {u_{1} u_{2}} (t) I_{t}^{γ, δ, ζ, ν} {g^{2}} (t)}, \end{matrix}

which leads to the desired inequality in (2.8). The proof is completed.

□

Theorem 3

(2.13)

I_{t}^{α, β, η, μ} {f^{2}} (t) I_{t}^{γ, δ, ζ, ν} {g^{2}} (t) ⩽ I_{t}^{α, β, η, μ} {(u_{2} fg) / v_{1}} (t) I_{t}^{γ, δ, ζ, ν} {(v_{2} fg) / u_{1}} (t)

Proof

From (2.2), we have

(2.14)

\frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f^{2} (τ) d τ ⩽ \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} \frac{u_{2} (τ)}{v_{1} (τ)} f (τ) g (τ) d τ,

which implies

(2.15)

I_{t}^{α, β, η, μ} {f^{2}} (t) ⩽ I_{t}^{α, β, η, μ} {(u_{2} fg) / v_{1}} (t) .

By (2.4), we get

\frac{1}{Γ (β)} \int_{0}^{t} {(t - ρ)}^{β - 1} g^{2} (ρ) d ρ ⩽ \frac{1}{Γ (β)} \int_{0}^{t} {(t - ρ)}^{β - 1} \frac{v_{2} (ρ)}{u_{1} (ρ)} f (ρ) g (ρ) d ρ,

from which one has

(2.16)

I_{t}^{γ, δ, ζ, ν} {g^{2}} (t) ⩽ I_{t}^{γ, δ, ζ, ν} {(v_{2} fg) / u_{1}} (t) .

Multiplying (2.15) and (2.16), we get the desired inequality in (2.13).

□

3 Special cases and concluding remarks

We now, briefly consider some consequences of the results derived in the previous sections. Following Curiel and Galué (1996), the operator (1.2) would reduce immediately to the extensively investigated Saigo, Erdélyi–Kober and Riemann–Liouville type fractional integral operators, respectively, given by the following relationships (see also Curiel and Galué, 1996 and Kiryakova, 1994):

(3.1)

I_{0, t}^{α, β, η} \{f (t)\} = I_{t}^{α, β, η, 0} \{f (t)\} = \frac{t^{- α - β}}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1}_{2} F_{1} (α + β, - η; α; 1 - \frac{τ}{t}) f (τ) d τ (α > 0, β, η \in R)

(3.2)

I^{α, η} \{f (t)\} = I_{t}^{α, 0, η, 0} \{f (t)\} = \{f (t)\} = \frac{t^{- α - η}}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} τ^{η} f (τ) d τ (α > 0, η \in R),

and

(3.3)

R^{α} \{f (t)\} = I_{t}^{α, - α, η, 0} \{f (t)\} = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f (τ) d τ (α > 0) .

For example, if we set $μ = 0$ in Theorem 1 and $μ = ν = 0$ in Theorem 2 and 3, using (3.1), the inequality (2.1), (2.8) and (2.13) gives the following results involving Saigos fractional integral operators, which are believed to be new:

Corollary 1

(3.4)

\frac{I_{0, t}^{α, β, η} {v_{1} v_{2} f^{2}} (t) I_{0, t}^{α, β, η} {u_{1} u_{2} g^{2}} (t)}{{(I_{0, t}^{α, β, η} {(v_{1} u_{1} + v_{2} u_{2}) fg} (t))}^{2}} ⩽ \frac{1}{4} .

Corollary 2

(3.5)

\frac{I_{0, t}^{α, β, η} {u_{1} u_{2}} (t) I_{0, t}^{γ, δ, ζ} {v_{1} v_{2}} (t) I_{0, t}^{α, β, η} {f^{2}} (t) I_{0, t}^{γ, δ, ζ} {g^{2}} (t)}{{(I_{0, t}^{α, β, η} {u_{1} f} (t) I_{0, t}^{γ, δ, ζ} {v_{1} g} (t) + I_{0, t}^{α, β, η} {u_{2} f} (t) I_{0, t}^{γ, δ, ζ} {v_{2} g} (t))}^{2}} ⩽ \frac{1}{4} .

Corollary 3

(3.6)

I_{0, t}^{α, β, η} {f^{2}} (t) I_{0, t}^{γ, δ, ζ} {g^{2}} (t) ⩽ I_{0, t}^{α, β, η} {(u_{2} fg) / v_{1}} (t) I_{0, t}^{γ, δ, ζ} {(v_{2} fg) / u_{1}} (t)

Similarly, if we set $μ = β = 0$ in Theorem 1 and $μ = ν = β = δ = 0$ in Theorem 2 and 3, using (3.2), the inequality (2.1), (2.8) and (2.13) gives the following results involving Erd $\overset{́}{e}$ lyi–Kober fractional integral operators, which are also believed to be new:

Corollary 4

(3.7)

\frac{I_{t}^{α, η} {v_{1} v_{2} f^{2}} (t) I_{t}^{α, η} {u_{1} u_{2} g^{2}} (t)}{{(I_{t}^{α, η} {(v_{1} u_{1} + v_{2} u_{2}) fg} (t))}^{2}} ⩽ \frac{1}{4} .

Corollary 5

(3.8)

\frac{I_{t}^{α, η} {u_{1} u_{2}} (t) I_{t}^{γ, ζ} {v_{1} v_{2}} (t) I_{t}^{α, η} {f^{2}} (t) I_{t}^{γ, ζ} {g^{2}} (t)}{{(I_{t}^{α, η} {u_{1} f} (t) I_{t}^{γ, ζ} {v_{1} g} (t) + I_{t}^{α, η} {u_{2} f} (t) I_{t}^{γ, ζ} {v_{2} g} (t))}^{2}} ⩽ \frac{1}{4} .

Corollary 6

Let f and g be two positive integrable functions on $[0, \infty)$ . Assume that there exist four positive integrable functions $u_{1}, u_{2}$ , $ψ_{1}$ and $ψ_{2}$ satisfying $(A_{1})$ on $[0, \infty)$ . Then for $t > 0$ and $α, β > 0$ , the following inequality holds:

(3.9)

I_{t}^{α, η} {f^{2}} (t) I_{t}^{γ, ζ} {g^{2}} (t) ⩽ I_{t}^{α, η} {(u_{2} fg) / v_{1}} (t) I_{t}^{γ, ζ} {(v_{2} fg) / u_{1}} (t) .

For another example, if we put $μ = 0$ in Theorem 1 and $μ, ν = 0$ in Theorem 2 and 3, replace β by $- α$ and $β, δ$ by $- α, - γ$ in Theorem 1 and 2, respectively, and use (3.3), the inequalities (2.1), (2.8) and (2.13) gives known results involving Riemann–Liouville fractional integral operators (see Ntouyas et al., submitted).

Furthermore, we also get some more special cases of Theorem 1–3 , as follows:

Corollary 7

Let f and g be two positive integrable functions on $[0, \infty)$ satisfying $(A_{2}) 0 < m ⩽ f (τ) ⩽ M < \infty, 0 < n ⩽ g (τ) ⩽ N < \infty, (τ \in [0, t], t > 0) .$ Then for $t > 0$ and $α > 0$ , we have

(3.10)

\frac{(I_{t}^{α, β, η, μ} \{f^{2}\} (t)) (I_{t}^{α, β, η, μ} \{g^{2}\} (t))}{{(I_{t}^{α, β, η, μ} \{fg\} (t))}^{2}} ⩽ \frac{1}{4} {(\sqrt{\frac{mn}{MN}} + \sqrt{\frac{MN}{mn}})}^{2} .

Corollary 8

Let f and g be two positive integrable functions on $[0, \infty)$ satisfying $(A_{2})$ . Then for $t > 0$ and $α, β > 0$ , we have

(3.11)

\frac{t^{α + β}}{Γ (α + 1) Γ (β + 1)} \frac{(I_{t}^{α, β, η, μ} \{f^{2}\} (t)) (I_{t}^{γ, δ, ζ, ν} \{g^{2}\} (t))}{{(I_{t}^{α, β, η, μ} \{f\} (t) I_{t}^{γ, δ, ζ, ν} \{g\} (t))}^{2}} ⩽ \frac{1}{4} {(\sqrt{\frac{mn}{MN}} + \sqrt{\frac{MN}{mn}})}^{2} .

Corollary 9

Let f and g be two positive integrable functions on $[0, \infty)$ satisfying $(A_{2})$ . Then for $t > 0$ and $α, β > 0$ , we have

(3.12)

\frac{(I_{t}^{α, β, η, μ} \{f^{2}\} (t)) (I_{t}^{γ, δ, ζ, ν} \{g^{2}\} (t))}{I_{t}^{α, β, η, μ} \{fg\} (t) I_{t}^{γ, δ, ζ, ν} \{fg\} (t)} ⩽ \frac{MN}{mn} .

4 Concluding remark

We conclude our present study with the remark that our main result here, being of a very general nature, can be specialized to yield numerous interesting fractional integral inequalities including some known results. Furthermore, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems in the fractional partial differential equations.

Acknowledgements

The authors should express their deep gratitude for the reviewers’s critical, kind, and enduring guidance to clarify and improve this paper.

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Yang A.M., Yang X.-J., Li Z.B., . Local fractional series expansion method for solving wave and diffusion equations on Cantor sets. Abstr. Appl. Anal. 2013 Article ID 351057
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Wang G., Agarwal P., Chand M., . Certain Grüss type inequalities involving the generalized fractional integral operator. J. Inequal. Appl. 2014:147.
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Introduction and preliminaries

Certain fractional integral inequalities associate with hypergeometric operator

Special cases and concluding remarks

Concluding remark

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[14] Choi J., Agarwal P., . Certain new pathway type fractional integral inequalities. Honam Math. J.. 2014;36:437-447.
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[15] Choi J., Agarwal P., . Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions. Abstr. Appl. Anal. 2014 Article ID 735946, 7p
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[16] Dragomir S.S., . Some integral inequalities of Grüss type. Indian J. Pure Appl. Math.. 2000;31(4):397-415.
[Google Scholar]

[17] Dragomir S.S., Diamond N.T., . Integral inequalities of Grüss type via Pólya–Szegö and Shisha–Mond results. East Asian Math. J.. 2003;19(1):27-39.
[Google Scholar]

[18] Grüss G., . Über das Maximum des absoluten Betrages von $\frac{1}{b - a} \int_{a}^{b} f (x) g (x) dx - \frac{1}{{(b - a)}^{2}} \int_{a}^{b} f (x) dx \int_{a}^{b} g (x) dx$ . Math. Z.. 1935;39:215-226.
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[19] Jumarie G., . Table of some basic fractional calculus formulae derived from modified Riemann–Liouville derivative for non-differentiable functions. Appl. Math. Lett.. 2009;22(3):378-385.
[Google Scholar]

[20] Kiryakova V.S., . Generalized Fractional Calculus and Applications (Pitman Res. Notes Math. Ser. 301). Harlow: Longman Scientific & Technical; 1994.

[21] Komatsu H., . Fractional powers of operators. Pacific J. Math.. 1966;19(2):285-346.
[Google Scholar]

[22] Komatsu H., . Fractional powers of operators. II, Interpolation spaces. Pacific J. Math.. 1967;21(1):89-111.
[Google Scholar]

[23] Li M., Lim S.C., Chen S.Y., . Exact solution of impulse response to a class of fractional oscillators and its stability. Math. Probl. Eng.. 2011;2011 Article ID 657839, 9p
[Google Scholar]

[24] Li M., Lim S.C., Cattani C., Scalia M., . Characteristic roots of a class of fractional oscillators. Adv. High Energy Phys.. 2013;2013 Article ID 853925, 7p
[Google Scholar]

[25] Liu K., Hu R.-J., Cattani C., Xie G.N., Yang X.-J., Zhao Y., . Local fractional Z transforms with applications to signals on Cantor sets. Abstr. Appl. Anal.. 2014;2014 Article ID 638648
[Google Scholar]

[26] Ntouyas, S.K., Agarwal, P., Tariboon, J., On Pólya-Szegö and Chebyshev types inequalities involving the Riemann-Liouville fractional integral operators, submitted.

[27] Pólya G., Szegö G., . Aufgaben und Lehrsatze aus der Analysis. Band 1, Die Grundlehren der mathmatischen Wissenschaften 19. Berlin: Springer; 1925.

[28] Saxena R.K., . On fractional integration operators. Math. Z.. 1967;96(4):288-291.
[Google Scholar]

[29] Sulaiman W.T., . Some new fractional integral inequalities. J. Math. Anal.. 2011;2(2):23-28.
[Google Scholar]

[30] Yang X.-J., Srivastava H.M., He J.-H., Baleanu D., . Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A. 2013;377(28–30):1696-1700.
[Google Scholar]

[31] Yang X.-J., Baleanu D., . Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci.. 2013;17(2):625-628.
[Google Scholar]

[32] Yang A.M., Yang X.-J., Li Z.B., . Local fractional series expansion method for solving wave and diffusion equations on Cantor sets. Abstr. Appl. Anal. 2013 Article ID 351057
[Google Scholar]

[33] Wang G., Agarwal P., Chand M., . Certain Grüss type inequalities involving the generalized fractional integral operator. J. Inequal. Appl. 2014:147.
[Google Scholar]