Abstract

We establish some inequalities involving Saigo fractional -integral operator in the theory of quantum calculus by using the two parameters of deformation, and , whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville and Kober fractional -integral operators, respectively. Furthermore, we also consider their relevance with other related known results.

1. Introduction and Preliminaries

The fractional -calculus is the -extension of the ordinary fractional calculus (see, e.g., [1] and the references therein). The theory of -calculus operators in the recent past has been applied in the areas like ordinary fractional calculus, optimal control problems, solutions of the -difference (differential) and -integral equations, -fractional integral inequalities, -transform analysis, and many more.

Fractional and -fractional integral inequalities have proved to be one of the most powerful and far-reaching tools for the development of many branches of pure and applied mathematics. These inequalities have gained considerable popularity and importance during the past few decades due to their distinguished 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 and in fractional partial differential equations. A detailed account of such fractional integral inequalities along with their applications can be found in the research contributions by many authors (see, e.g., [2–12] and [13]; for a very recent work, see also [14]).

In a recent paper, Brahim and Taf [15] investigated certain fractional integral inequalities in quantum calculus. Here we aim at establishing certain (presumable) new -integral inequalities associated with Saigo fractional -integral operator, introduced by Garg and Chanchlani [16]. Results due to Brahim and Taf [15] and Sroysang [17] follow as special cases of our results, respectively.

For our purpose, we also recall the following definitions (see, e.g., [12, 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 (1), (2), and (3) that which can be extended to as follows: where the principal value of is taken.

We begin by noting that 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 (11) is given by

The classical Gamma function (see, e.g., [12, 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: Replacing by in (15), Jackson [18] defined the -Gamma function by

The -analogue of is defined by the polynomial

Definition 1. Let , , and be real or complex numbers. Then a -analogue of Saigo’s fractional integral is given for by (see [16, page 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 2. A -analogue of Riemann-Liouville fractional integral operator of a function of an order is given by (see [1]) where is given by (5).

Definition 3. A -analogue of the Erdélyi-Kober fractional integral operator for , , and is given by (see [1])

2. Saigo Fractional -Integral Inequalities

In this section, we establish certain fractional -integral inequalities, 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 4 (Choi and Agarwal [13]). Let and be a continuous function with for all . Then one has the following inequalities: (i)the Saigo fractional -integral operator of the function in (18):  for all and , with and ;(ii)the -analogue of Riemann-Liouville fractional integral operator of the function of an order in (19):  for all ;(iii)the -analogue of Erdélyi-Kober fractional integral operator of the function in (20): for all and .

Proof. Applying (11) to the -integral in (18), we have It is easy to see that for all and . Next, for simplicity, Then we claim that for all . We find from (17) 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 (24) is nonnegative. This completes the proof of (21). The other two inequalities in (22) and (23) 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 4, it is seen that under the conditions given in (29).

Here we present six -integral inequalities involving the Saigo fractional -integral operator (18) stated in Theorems 5 to 12 below.

Theorem 5. Let , , , and be three continuous and synchronous functions on and be a continuous function. Then the following inequality holds true. For , for all , , and , , , with , , , and .

Proof. Let , , and be three continuous and synchronous functions on . Then, for all , we have which implies that Multiplying both sides of (33) by in (29) together with (30) and taking -integration of the resulting inequality with respect to from to with the aid of Definition 1, we get Next, multiplying both sides of (34) by in (29) together with (30), taking -integration of the resulting inequality with respect to from to , and using Definition 1, we are led to the desired result (31). This completes the proof of Theorem 5.

Theorem 6. Let , , , and be three continuous and synchronous functions on and be a continuous function. Then the following inequality holds true. For , for all , , and , , , with , , , and .

Proof. To prove the above result, multiplying both sides of (34) by in (29) together with (30) and taking the -integration of the resulting inequality with respect to from to with the aid of Definition 1, we get the desired result (35).

Remark 7. It may be noted that the inequalities in (31) and (35) are reversed if functions , , and are asynchronous. It is also easily seen that the special case of (35) in Theorem 6 reduces to that in Theorem 5.

Theorem 8. Let , be a continuous function, and , , and are be three continuous and synchronous functions on , satisfying the following condition: Then the following inequality holds true. For , for all , , and , , , with , , , and .

Proof. Let , , and be three continuous and synchronous functions on . Then, for all , we have from (36) which implies that Let us define a function Multiplying both sides of (40) by in (29) together with (30), taking -integration of the resulting inequality with respect to from to , and using (18), we get Next, multiplying both sides of (41) by in (29) together with (30), taking -integration of the resulting inequality with respect to from to , and using (18), we get Finally, by using (39) on to (42), we arrive at the desired result (37), involved in Theorem 8, after a little simplification.