Some Opial type inequalities in ( p , q ) -calculus

: In this paper, we establish 5 kinds of integral Opial-type inequalities in ( p , q )-calculus by means of H¨older’s inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in ( p , q )-calculus involving one function and its ( p , q ) derivative. Furthermore, Opial inequalities in ( p , q )-calculus involving two functions and two functions with their ( p , q ) derivatives are given. Our results are ( p , q )-generalizations of some known inequalities, such as Opial-type integral inequalities and ( p , q )-Wirtinger inequality.


Introduction
In 1960, Opial [1] presented an inequality involving integral of a function and its derivative as follows Theorem 1.1. Let x ∈ C 1 [0, h] be such that x(t) > 0 in (0, h). Then, the following inequalities hold: In (1.1), the constant h 4 is the best possible.
Since then, the study of generalizations, extensions and discretizations for inequalities (1.1) and (1.2) of Opial type inequalities has grown into a substantial field, with many important applications in theory of differential equations, approximations and probability, among others. For more details, we cite the readers to [2][3][4][5][6][7] and the references therein.
Inspired by the above mentioned works [8,9,13], in this paper, we will establish some (p, q)-Opial type inequalities by using (p, q)-calculus and analysis technique. If p = 1 and q → 1 − , then all the results we have obtained in this paper reduce to the classical cases.
Definition 2.1. ( [13]) The (p, q)-derivative of the function f is defined as 13]) Let f be an arbitrary function and a be a positive real number, the (p, q)-integral of f from 0 to a is defined by Also for two nonnegative numbers such that Lemma 2.1. ( [13]) The (p, q)-derivative fulfills the following product rules .
Then, the following inequality holds : Proof. Let g(x) be as in (3.2) and w(x) be as follows Then, we obtain by the condition f (h) = 0 that and for x ∈ [0, h], we have and (3.14) Similarly, by (3.10)-(3.14), we can get that Using the Hölder's inequality with indices m + 1 and m+1 m , we can write that Similarly, we obtain |D p,q f (t)| m+1 d p,q t.
Proof. Using the Hölder's inequality for (p, q)-integral with indices m+1 m and m + 1, we obtain (3.21) By using (3.4) and from the Hölder's inequality for (p, q)-integral with indices m+1 m and m + 1, we get |D p,q g(t)|d p,q t.
Proof. Let Then, we have and On the other hand, the following elementary inequality in [8] holds : By using Hölder's inequality on the right side of (3.32) with indices m+r m+r−1 , m + r, we obtain and the proof is completed. Remark 3.3. If m = r > 0 and f (x) = g(x), then the inequality (3.25) reduces to the following (p, q)-Wirtinger inequality:

Example
In the following, we will give an example to illustrate our main result.

Conclusion
It is known that (p, q)-calculus is a generalization of q-calculus. In this paper, we have established 5 new kinds of general Opial type integral inequalities in (p, q)-calculus. The methods we used to establish our results are quite simple and in virtue of some basic observations and applications of some fundamental inequalities and analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. We also discussed several particular cases. Our results are (p, q)-generalizations of Opial-type integral inequalities and (p, q)-Wirtinger inequality. An example is given to illustrate the effectiveness of our main result.