1. Introduction and Preliminaries

Under various assumptions (Chebyshev inequality, Grüss inequality, Minkowski inequality, Hermite- Hadamard inequality, Ostrowski inequality etc.), inequalities are playing a very significant role in all fields of mathematics, particularly in the theory of approximations (see ^{[2, 6, 7, 13, 14, 17, 23]}). Therefore, in the literature we found several extensions and generalizations of these integral inequalities for the functions of bounded variation, synchronous, Lipschitzian, monotonic, absolutely continuous and n-times differentiable mappings etc. ([10,11,12,15,16,19,20,21,22,26,27,28]). In the past recent years, one more dimension have been added to this study, by introducing number of integral inequalities involving various fractional calculus and q-calculus operators. For detailed account, one may refer [1,3,4,5,8,9,18,24,25,29-35] and the references cited therein.

Recently, Tariboon et al. ^[33] investigated certain new integral inequalities for the integrable functions, whose bounds are also integrable functions, involving the Riemann-Liouville fractional integral operators. Our aim in this paper, is to obtain a general extensions of the results due to Tariboon et al. ^[33]. Main results investigated here provide certain new integral inequalities associated with the integrable functions, whose bounds are also integrable functions, involving the Saigo’s fractional integral operators. We also give some consequent results and special cases of the main results.

Firstly, we mention below the basic definitions and notations of some well-known operators of fractional calculus, which shall be used in the sequel.

Let then the Saigo fractional integral of order for a real-valued continuous function is defined by (^[36], see also [^[37], p. 19]):

(1.1)

where, the function appearing as a kernel for the operator (1.1) is the Gaussian hypergeometric function defined by

(1.2)

and is the Pochhammer symbol

The operator includes both the Riemann-Liouville and the Erdélyi-Kober fractional integral operators given by the following relationships:

(1.3)

and

(1.4)

Following ^[36], for in (1.1), we get

(1.5)

2. Main Results

In this section, we obtain certain integral inequalities, related to the integrable functions, whose bounds are also integrable functions, involving Saigo’s fractional hypergeometric operators. The results are given in the form of the following theorems:

Theorem 1. Let and are integrable functions defined on such that

(2.1)

Then, for we have

(2.2)

where

Proof. By the hypothesis of inequality (2.1), for any we have

which follows that

(2.3)

Consider

(2.4)

which remains positive, for all under the conditions stated with Theorem 1. Multiplying both sides of (2.3) by (where is given by (2.4)) and integrating the resulting identity with respect to from 0 to , and using (1.1), we get

(2.5)

Next, on multiplying both sides of (2.5) by

(2.6)

which also remains positive, for all Upon integrating the resulting inequality so obtained with respect to from 0 to , and using the operator (1.1), we easily arrive at the desired result (2.1).

It may be noted that, for the Theorem 1 immediately reduces to the following result:

Corollary 1. Let and are integrable functions defined on and satisfying inequality (2.1). Then, for we have

(2.7)

where

Theorem 2. Let and be two integrable functions on and and are four integrable functions on , such that

(2.8)

Then, for the following inequalities holds true:

(2.9)

(2.10)

(2.11)

(2.12)

Proof. Let f and g are two integrable functions and satisfying inequality (2.8), then to prove (2.9), we can write

which follows that

(2.13)

On multiplying both sides of (2.13) by (where is given by (2.4)) and integrating with respect to from 0 to , then by making use of (1.1), we get

(2.14)

Next, multiplying both sides of (2.13) by where is given by (2.6), and integrating with respect to from 0 to , we easily arrive at the desired result (2.9).

Following the similar procedure, one can easily establish the remaining inequalities (2.10) to (2.12) by using the following inequalities, respectively

and

Therefore, we omit the further details of the proof of these results.

3. Consequent Results and Special Cases

The Saigo’s fractional integral operator defined by (1.1), possess the advantage that the Erdélyi-Kober and the Riemann-Liouville type fractional integral operators happen to be the particular cases of this operator. Therefore, by suitably specializing the parameters, we now briefly consider some special cases of the result derived in the preceding section. To this end, let us set and and make use of the relation (1.4), then Theorems 1 & 2 yields the following inequalities involving the Erdélyi-Kober type fractional integral operators:

Corollary 2. Let and are integrable functions defined on and satisfying inequality (2.1), then for we have

(3.1)

where

Corollary 3. Let and be two integrable functions on and and are four integrable functions on and satisfying inequality (2.8). Then, for and the following inequalities holds true:

(3.2)

(3.3)

(3.4)

(3.5)

Next, if we replace by by and make use of the relation (1.3), then Theorems 1 & 2 corresponds to the known integral inequalities involving Riemann-Liouville type fractional integral operators, due to Tariboon et al ^[33].

Further, if we put and where and make use of formula (1.5), then the Theorems 1 & 2 leads to the following particluar results:

Corollary 4. Let f be an integrable function defined on , such that

(3.6)

Then, for we have

(3.7)

where and

Corollary 5. Let and be two integrable functions on such that

(3.8)

Then, for and the following inequalities holds true:

(3.9)

(3.10)

(3.11)

(3.12)

Again, if we set and and make use of formula (1.5), then the Theorem 1 and Corollary 1, further leads to the following integral inequalities:

Corollary 6. Let f be an integrable function defined on , such that

Then, for and we have

(3.13)

Corollary 7. Let be an integrable function defined on such that

Then, for we have

(3.14)

In this paper, we have introduced certain general integral inequalities, related to the integrable and bounded functions and involving Saigo’s fractional integral operators. Therefore, we conclude with the remark that, by suitably specializing the arbitrary function and one can further easily obtain additional integral inequalities involving the Riemann-Liouville, Erd´elyi-Kober and Saigo type fractional integral operators from our main results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

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