Certain Grüss type inequalities involving the generalized fractional integral operator
© Wang et al.; licensee Springer. 2014
Received: 31 January 2014
Accepted: 25 March 2014
Published: 9 April 2014
A remarkably large number of Grüss type fractional integral inequalities involving the special function have been investigated by many authors. Very recently, Kalla and Rao (Matematiche LXVI(1):57-64, 2011) gave two Grüss type inequalities involving the Saigo fractional integral operator. Using the same technique, in this paper, we establish certain new Grüss type fractional integral inequalities involving the Gauss hypergeometric function. Moreover, we also consider their relevances for other related known results.
MSC: 26D10, 26A33.
1 Introduction and preliminaries
In recent years, a number of inequalities involving the fractional operators (like Erdélyi-Kober, Riemann-Liouville, Saigo fractional integral operators etc.) have been considered by many authors (see, e.g., [1–10]; for very recent work, see also  and ). The above-mentioned works largely have motivated us to perform the present study.
We begin by recalling some known functions and earlier works.
where is a best possible constant.
Very recently, Kalla and Rao  gave two Grüss type inequalities involving the Saigo fractional integral operator. Using the same technique, in this paper, we establish certain new Grüss type fractional integral inequalities involving the generalized fractional integral operator due to Curiel and Galué (see ). Moreover, we also consider their relevant connections with other known results.
Throughout the present paper, we shall investigate a fractional integral over the space introduced in  and defined as follows.
Definition 1.1 The space of functions , , the set of real numbers, consists of all functions , , that can be represented in the form with and , where is the set of continuous functions in the interval .
We define a fractional integral operator associated with the Gauss hypergeometric function as follows.
where is the Gauss hypergeometric fractional integral of order α and is defined in the following.
where ℕ denotes the set of positive integers.
where Ξ and denotes the sets of complex numbers and nonpositive integers, respectively.
Next, we discuss some results regarding the fractional integral operator which have been used in the present work.
where C is constant.
Applying the result (1.8) to (1.13), after a little simplification, we easily arrive at the required result (1.10).
Using the result (1.8) in (1.15), after a little simplification, we easily arrive at the required result (1.11).
This completes the proof of the Lemma 1.1. □
for all ; , , and with and .
then integrating with respect to v from 0 to x, we obtain the required result (1.16). This completes the proof of Lemma 1.2. □
2 Main results
In this section, we establish two inequalities involving the composition formula of the fractional integral (1.3) with a power function.
for all ; , , and with and .
Applying the well-known inequality ; and using in the right-hand side of the inequality (2.8) and simplifying it, we obtain the required result (2.1). This completes the proof of Theorem 2.1. □
for all ; , , and with and .
Proof For the synchronous function f and g, the inequality (1.9) holds for all .
Following the procedure of the Lemma 1.2 for applying the fractional integral , after a little simplification, we arrive at the required result (2.9). This completes the proof of Theorem 2.2. □
3 Concluding remarks
where , and are given by (3.1), (3.2), and (3.3), respectively.
We conclude our present investigation by remarking further that the results obtained here are useful in deriving various fractional integral inequalities involving such relatively more familiar fractional integral operators. For example, if we consider and make use of (3.1), Theorems 2.1 and 2.2 provide, respectively, the known fractional integral inequalities due to Kalla and Rao [, pp.60-62, Eqs. (14) and (22)].
Again, for and in Theorems 2.1 and 2.2 and making use of the relation (3.2), Theorems 2.1 and 2.2 provide, respectively, the known fractional integral inequalities due to Kalla and Rao [, p.62, Eqs. (24) and (25)].
Finally, taking and in Theorems 2.1 and 2.2 yields the known result due to Dahmani et al. [, Theorem 3.1].
It is noted that the results derived in this paper are general in character and give some contributions to the theory of integral inequalities and fractional calculus. Moreover, they are expected to lead to some applications for establishing uniqueness of solutions in fractional boundary value problems, and in fractional partial differential equations.
The authors are thankful to the worthy referees for their useful suggestions. This work is supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (No. 2012021002-3).
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