#### Abstract

The main objective of this paper is to establish some, presumably, new Saigo type fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville and Erdélyi-Kober type fractional integral operators, respectively. We also investigate and present -extensions of the above results and some other presumably new ones. Relevant connections of the results presented here with those earlier ones are also indicated.

#### 1. Introduction and Preliminaries

Throughout this paper, , , , and denote the sets of positive integers, real numbers, complex numbers, and nonpositive integers, respectively, and . Consider the following functional:
where are two integrable functions on and and are positive integrable functions on . If and are* synchronous* on , that is,
for any , then we have (see, e.g., [1, 2])
The inequality in (2) is reversed, if and are asynchronous on , that is,
for any . If , for any , we get the Chebyshev inequality (see [3]). Ostrowski [4] established the following generalization of the Chebyshev inequality.

If and are two differentiable and synchronous functions on and is a positive integrable function on with and , for , then we have If and are asynchronous on , then we have

If and are two differentiable functions on with and , for , and is a positive integrable function on , then we have

The functional (1) has attracted many researchers’ attention due mainly to diverse applications in numerical quadrature, transform theory, and probability and statistical problems. Among those applications, the functional (1) has also been employed to yield a number of integral inequalities (see, e.g., [5–12]; for a very recent work, see also [13]).

Here, in this paper, we aim at establishing certain, presumably, new inequalities involving Saigo fractional integral operator (8) whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville fractional integral operator (11) and Erdélyi-Kober fractional integral operator (12). We also investigate and present -extensions of our results and some other presumably new ones. Relevant connections of some of the results presented here with those earlier ones are also pointed out.

For our purpose, we also need to recall the following definitions and some earlier works.

*Definition 1. *A real-valued function is said to be in the space , if there exists a real number such that , where .

*Definition 2. *A function is said to be in the space , if .

*Definition 3. *Let and . Then the Saigo fractional integral (in terms of the Gauss hypergeometric function) of order for a real-valued continuous function is defined by (see [14]; see also [15])
where the function is the Gaussian hypergeometric function defined by (see, e.g., [16, Section 1.5])
and is the familiar Gamma function. Here, is the Pochhammer symbol defined, for , by (see, e.g., [16, p. 2 and pp. 4–6])

It is noted that Saigo fractional integral operator includes both Riemann-Liouville and Erdélyi-Kober fractional integral operators, respectively, given by the following relationships (see, e.g., [17]), for :

For our purpose, we also recall the following definitions (see, e.g., [16, Section 6]) and some earlier works.

The -*shifted factorial * is defined by
where and it is assumed that .

The -*shifted factorial* for negative subscript is defined by
We also write
It follows from (13), (14), and (15) that
which can be extended to as follows:
where the principal value of is taken.

We begin by noting that F. J. Jackson was the first to develop -calculus in a systematic way. The -*derivative* of a function is defined by
It is noted that
if is differentiable.

The function is a -*antiderivative* of , if . It is denoted by
The* Jackson integral* of is thus defined, formally, by
which can be easily generalized as follows:

Suppose that . The definite -integral is defined as follows: A more general version of (23) is given by

The classical Gamma function (see, e.g., [16, Section 1.1]) was found by Euler, while he was trying to extend the factorial to real numbers. The -factorial function of , defined by can be rewritten as follows: By replacing by in (27), Jackson [18] defined the -Gamma function by

The -analogue of is defined by the polynomial

*Definition 4. *Let ; let and be real or complex numbers. Then a -analogue of Saigo's fractional integral is given for by (see [19, p. 172, Equation ])

The integral operator includes both the -analogues of the Riemann-Liouville and Erdélyi-Kober fractional integral operators given by the following definitions.

*Definition 5. *A -analogue of Riemann-Liouville fractional integral operator of a function of an order is given by (see [20])
where is given by (17).

*Definition 6. *A -analogue of the Erdélyi-Kober fractional integral operator for , , and is given by (see [20])

#### 2. Certain Inequalities Involving Saigo Fractional Integral Operator

Here, we start with presenting an inequality involving Saigo fractional integral (8) asserted by the following lemma.

Lemma 7. *Let and be two continuous and synchronous functions on and let be continuous functions. Then the following inequality holds true:
**
for all , , and , with and .*

*Proof. *Let and be two continuous and synchronous functions on . Then, for all , with , we have
or, equivalently,
Consider the following function defined by
We observe that each term of the above series is nonnegative under the conditions in Lemma 7, and, hence, the function remains nonnegative for all .

Now, by multiplying both sides of (35) by defined by (36), integrating the resulting inequality with respect to from to , and using (8), we get

Next, by multiplying both sides of (37) by , where is given when is replaced by in (36), integrating the resulting inequality with respect to from to , and using (8), we are led to the desired result (33).

Theorem 8. *Let and be two continuous and synchronous functions on and let be continuous functions. Then the following inequality holds true:
**
for all , , and , with and .*

*Proof. *By setting and in Lemma 7, we get
Since under the given conditions, by multiplying both sides of (39) by , we have
Similarly, by replacing , by , and , by , , respectively, in (33) and then multiplying both sides of the resulting inequalities by and both of which are nonnegative under the given assumptions, respectively, we get the following inequalities:
Finally, by adding (40) and (41), side by side, we arrive at the desired result (38).

We present another inequality involving the Saigo fractional integral operator in (8) asserted by the following lemma.

Lemma 9. *Let and be two continuous and synchronous functions on and let be continuous functions. Then the following inequality holds true:
**
for all , , , and with , , , and .*

*Proof. *By multiplying both sides of (37) by
which remains nonnegative under the conditions in (42), integrating the resulting inequality with respect to from to , and using (8), we get the desired result (42).

Theorem 10. *
for all , , , and with , , , and .*

*Proof. *By setting and in (42), we have
By multiplying both sides of (45) by , after a little simplification, we get
Now, by replacing , by , and , by , in (42), respectively, and then multiplying both sides of the resulting inequalities by and , respectively, we get the following two inequalities:
Finally, we find that the inequality (44) follows by adding the inequalities (46) and (47), side by side.

*Remark 11. *It may be noted that the inequalities (38) and (44) in Theorems 8 and 10, respectively, are reversed, if the functions are asynchronous on . The special case of (44) in Theorem 10, when , , and , is easily seen to yield the inequality (38) in Theorem 8.

Here, we derive certain, presumably, new integral inequalities by setting in (38) and in (44), respectively, and applying the integral operator (12) to the resulting inequalities, we obtain two integral inequalities involving Erdélyi-Kober fractional integral operators stated in Corollaries 12 and 13 below.

Corollary 12. *
for all , , and .*

Corollary 13. *
for all , , , and .*

*Remark 14. *The special cases in Theorem 8 and and in Theorem 10 are seen to yield the known fractional integral inequalities due to Dahmani [21].

#### 3. Saigo Fractional -Integral Inequalities

We establish certain -integral inequalities which are the -analogues (or extensions) of the results derived in the preceding section, some of which are presumably new ones. For our purpose, we begin with providing comparison properties for the fractional -integral operators asserted by the following lemma.

Lemma 15. *Let and be continuous functions with , for all . Then we have the following inequalities:*(i)*the Saigo fractional -integral operator of the function in (30):
for all and , with and ;*(ii)*the -analogue of Riemann-Liouville fractional integral operator of the function of an order in (31):
for all ;*(iii)*the -analogue of Erdélyi-Kober fractional integral operator of the function in (32):
for all and .*

*Proof. *By applying (23) to the -integral in (30), we have
It is easy to see that
for all and . Next, for simplicity,
Then we claim that , for all . We find from (29) that
It is easy to see that, if , then . On the other hand, if , then we have
since , for all with . Finally, we find that, under the given conditions, each term in the double series of (53) is nonnegative. This completes the proof of (50). The other two inequalities in (51) and (52) may be easily proved.

For convenience and simplicity, we define the following function by where , ; , with and ; ; is a continuous function. As in the process of Lemma 15, it is seen that under the conditions given in (58).

Here, we present two -integral inequalities involving the Saigo fractional -integral operator (30) stated in Lemmas 16 and 17 below.

Lemma 16. *Let ; let and be two continuous and synchronous functions on ; let be a continuous function. Then the following inequality holds true, for ,
**
for all and with and .*

*Proof. *Since and are two synchronous functions on , for all , the inequality (35) is satisfied. By multiplying both sides of (35) by in (58) together with (59) and taking -integration of the resulting inequality with respect to from to with the aid of Definition 4, we get

Next, by multiplying both sides of (61) by in (58) together with (59), taking -integration of the last resulting inequality with respect to from to , and using Definition 4, we are led to the desired result (60).

Lemma 17. *Let ; let and be two continuous and synchronous functions on ; let be continuous functions. Then the following inequality holds true, for ,
**
for all , , and , , , with , , , and .*

*Proof. *By multiplying both sides of (61) by in (58) together with (59) and taking the -integration of the resulting inequality with respect to from to with the aid of Definition 4, we get the desired result (62).

Theorem 18. *Let ; let , be two continuous synchronous functions; let be continuous functions. Then the following inequality holds true, for ,
**
for all and with and .*

*Proof. *We start with (60) in Lemma 16; by putting and , we get
By multiplying both sides of (64) by , we have
Similarly, by replacing , by , and , by , in (60), respectively and then multiplying both sides of the resulting inequalities by and , respectively, we get
Finally, by adding (65) and (66), side by side, we arrive at the desired result in Theorem 18.

Theorem 19. *Let ; let , *