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

1.   Introduction

In 1960, Opial ^[1] presented an inequality involving integral of a function and its derivative as follows

Theorem 1.1. Let $x \in C^1[0, h]$ be such that $x(t) > 0$ in $(0, h)$. Then, the following inequalities hold:

i) If $x(0) = x(h) = 0$, then

ii) If $x(0) = 0$, then

In (1.1), the constant $\frac{h}{4}$ 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.

Recently, Mirkovi et.al ^[8] established a new integral inequality of the $q$-Opial type as follows:

Theorem 1.2. Let $f \in C^1[0, 1]$ be $q$-decreasing function with $f(b q^0) = 0$. Then, for any $p \ge 0$,

In ^[9], Alp et al. gave the following $q$-Opial type inequality for quantum integral :

Theorem 1.3. Let $x(t) \in C^1[0, h]$ be such that $x(0) = x(h) = 0$, and $x(t) > 0$ in $(0, h)$. Then, the following inequality holds :

The $(p, q)$-calculus is known as two parameter quantum calculus, a generalization of $q$-calculus, whose applications play important roles in physics, chemistry, orthogonal polynomials and number theory ^[10,11]. In ^[12], Mursaleen et al. applied $(p, q)$-calculus in approximation theory and investigated first $(p, q)$-analogue of Bernstein operators. Recently, Sadjang in ^[13] studied the $(p, q)$-derivative, the $(p, q)$-integration, and obtained some of their properties and the fundamental theorem of $(p, q)$-calculus. About the recent results of $(p, q)$-calculus, please see ^{[14,15,16,17,18]}.

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 $q \to 1^-$, then all the results we have obtained in this paper reduce to the classical cases.

2.   Preliminaries

In what follows, $p, q$ are two real numbers satisfying $0 < q < p \le 1$, and $(p, q)$-bracket is defined as

Definition 2.1. (^[13]) The $(p, q)$-derivative of the function $f$ is defined as

and $(D_{p, q} f)(0) = \lim_{x \to 0} D_{p, q} f(x) = f^{\prime}(0)$, privided that $f$ is differentiable at $0$.

Definition 2.2. (^[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 $a < b$, we have

Lemma 2.1. (^[13]) The $(p, q)$-derivative fulfills the following product rules

Theorem 2.2. (^[13]) (Fundamental theorem of $(p, q)$-calculus). If $F(x)$ is a $(p, q)$-antiderivative of $f(x)$ and $F(x)$ is continuous at $x = 0$, we have

where $0 \le a < b \le \infty$.

3.   Main results

Theorem 3.1. Let $f \in C^1[0, h]$ be such that $f(t) > 0$ in $(0, h)$. If $f(0) = 0$ and $m \in \mathbb{N}$. Then, the following inequality holds:

Proof. Choosing $g(x)$ as

such that

and for $x \in [0, h]$, we have

and

By Lemma 2.1, we can obtain that

Thus, by (3.2)–(3.7), it follows that

By using the H${\rm \ddot{o}}$lder's inequality and (3.8) with (3.2) for $(p, q)$-integral with indices $m+1$ and $\frac{m+1}{m}$, we obtain

which proves the theorem.

Remark 3.1. If $p = 1$ and $m = 1$, then Theorem 3.1 reduces to Theorem 3.3 in ^[9]. If $p = 1$ and $m = \alpha$, then Theorem 3.1 reduces to Theorem 3.9 in ^[9]. Moreover, if $p = 1$, $m = 1$, and $q \to 1^-$, then (3.1) reduces to (1.2) in Theorem 1.1.

Theorem 3.2. Let $f \in C^1[0, h]$ be such that $f(t) > 0$ in $(0, h)$. If $f(0) = f(h) = 0$ and $m \in \mathbb{N}$. 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 $x \in [0, h]$, we have

and

From (3.8), one has

Similarly, by (3.10)–(3.14), we can get that

Adding (3.15) to (3.16), we obtain

Using the H${\rm \ddot{o}}$lder's inequality with indices $m+1$ and $\frac{m+1}{m}$, we can write that

Similarly, we obtain

Therefore, (3.17)–(3.19) imply that (3.9) holds.

Remark 3.2. If $p = 1$ and $m = 1$, then Theorem 3.2 reduces to Theorem 3.1 in ^[9]. In Theorem 3.2 if we take $p = 1$, $m = 1$, and $q \to 1^-$, we recapture the inequality (1.1).

Theorem 3.3. Let $m > 0$. Assume that $\mu(t)$ is a nonnegative and continuous function on $[0, h]$, $m \in \mathbb{N}$, and $f \in C^1[0, 1]$ with $f(0) = 0$. Then, the following inequality holds:

Proof. Using the H${\rm \ddot{o}}$lder's inequality for $(p, q)$-integral with indices $\frac{m+1}{m}$ and $m+1$, we obtain

By using (3.4) and from the H${\rm \ddot{o}}$lder's inequality for $(p, q)$-integral with indices $\frac{m+1}{m}$ and $m+1$, we get

Substituting (3.22) into (3.21), we have

This completes the proof.

Theorem 3.4. If $f$ and $g$ are absolutely continuous functions on $[0, h]$, and $f(0) = g(0) = 0$, then

Proof. As in (3.4), we have

By using Lemma 2.1, Theorem 2.2, (3.24) and Cauchy inequality, we get

Theorem 3.5. Let $m, r > 0$. Assume that $f$ and $g$ are absolutely continuous functions on $[0, h]$, $f(0) = g(0) = 0$, $f(h) = g(h) = 0$, and $f(t), g(t) > 0$ in $(0, h)$. Then

Proof. Let

Then, we have

and

By (3.26)–(3.28), we obtain

Similarly, we have

On the other hand, the following elementary inequality in ^[8] holds :

From (3.29)–(3.31), we get

By using H${\rm \ddot{o}}$lder's inequality on the right side of (3.32) with indices $\frac{m+r}{m+r-1}$, $m+r$, we obtain

Integrating (3.33) on $[0, h]$, we have

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:

4.   Example

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

Example 4.1. Let $p = \frac{2}{3}$, $q = \frac{1}{2}$ and $m = 5$. Set $\mu(t) = t,$ it is clear that $\mu(t)$ is a nonnegative and continuous function on $[0, \frac{\pi}{2}].$ Set $h = \frac{\pi}{2},$ and $f(x) = \sin x$, then $f \in C^1[0, 1]$ with $f(0) = 0$. Thus, by Theorem 3.3, we have

5.   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.

Acknowledgments

The authors thanks anonymous referees for their remarkable comments, suggestions which help to improve this paper. This work is supported by Natural Science Foundation of China (11571136).

Conflict of interest

The authors declare that there are no conflicts of interest.