ON (p,q)-ANALOGUES OF SOME GENERALIZED OPIAL’S INTEGRAL INEQUALITIES

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this work, we obtain (p,q)-analogues of generalized Opial’s integral inequalities. We also present some further extensions of the new analogues. The fundamental theorem of (p,q)-calculus and the (p,q)-Hölder’s integral inequality were employed to establish the results.


INTRODUCTION
Opial established an inequality involving integral of a function and its derivative in [13] as (1)  This inequality, due to its significance, experienced a lot of extensions and generalizations over time in the classical field. See [3], [4], [5] and [16], among others.
In [16], generalizations of the classical Opial's inequality were established as where the coefficients (b − a)/2 and (b − a)/4 are their respective best constants possible.
(p, q)-Calculus is a generalization of q-calculus. There has been a lot of development in the study of (p, q)-calculus. Recently, Sadjang [15] investigated on fundamental concepts of (p, q)calculus. In [8], (p, q)-derivatives and (p, q)-integrals and their properties are also presented.
The Opial inequality plays essential role in establishing the existence and uniqueness of initial and boundary values problems for both ordinary and partial differential equations [2] and [7].
The objective of this paper is to establish (p, q)-analogues of the generalized Opial integral inequalities (2) and (3).

PRELIMINARIES
The basic concepts and terminologies of (p, q)-calculus which will be used to prove our results are presented in this section. The definitions provided can also be seen in [8], [9] [11], [14], [12] and [15].
Definition 2.1. [8] For any arbitrary function f in the real-line, the (p, q)-derivative is defined Definition 2.2. [8] For any positive real α, the twin basic number or the (p, q)-Number α is defined as The (p, q)-Derivative of sum or difference of f and g is defined as The (p, q)-Derivative of product of f and g is defined as The (p, q)-Derivative of a quotient of f and g is defined as , g(px)g(qx) = 0.
where k is real and index of g. Proof.
This completes the proof.
Definition 2.4. [15] Let f : [0, b] → R be a continuous function and 0 < q < p ≤ 1. The definite Remark 2.1. Taking p = 1, equation (12) reduces to the well known Jackson q-integral It is easily observed that if the function f is increasing (decreasing), then it is also (p, q)increasing ((p, q)-decreasing).

MAIN RESULTS
Lemma 3.1. Let h : [a, b] → R be an absolutely continuous and a differentiable function, such This completes the proof.

So that
By [15], it follows that p,q y(x)).
Remark 3.5. The inequality 55 is sharper than the inequality 39 .

It follows that
This completes the proof.

CONCLUSION
In this work, (p, q)-analogues of generalized Opial's integral inequalities and their further extensions were established. The basic definitions of (p, q)-calculus, the fundamental theorem of (p, q)-calculus and convexity properties of functions were employed to obtain the results. The (p, q)-Hölder's integral inequality was also applied in proving the theorems. It is hoped that these results will be very useful to the mathematics community.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.