Some (p, q)-Hardy type inequalities for (p, q)-integrable functions

1.   Introduction

In mathematics, quantum calculus, also known as $q$-calculus, is the study of calculus without limit. In $q$-calculus, we obtain $q$-analogues of mathematical formulas that can be recaptured as $q$ tends to one. The history of $q$-calculus can be traced back to Euler, who first introduced $q$-calculus in the track of Newton's work on infinite series. Then, in 1910, F. H. Jackson ^[1] presented a systematic study of $q$-calculus and defined the $q$-definite integral, which is known as the $q$-Jackson integral. In recent years, the interest in $q$-calculus been arising due to high demand of mathematics in this field. The $q$-calculus numerous applications in various fields of mathematics and other areas such as combinatorics, dynamical systems, fractals, number theory, orthogonal polynomials, special functions, mechanics and also for scientific problems in some applied areas, see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14]} for more details.

Along with the development of the theory and application of $q$-calculus, the theory of $q$-calculus based on two parameters $(p, q)$-integers has also presented and recieved more attention during the last few dacades. In 1991, R. Chakrabati and R. Jagannathan ^[15] introduced the $(p, q)$-calculus. Next, P. N. Sadjang ^[16] studied the fundamental theorem of $(p, q)$-calculus and some $(p, q)$-Taylor formulas. Recently, M. Tunç and E. Göv ^[17] defined the $(p, q)$-derivative and $(p, q)$-integral on finite intervals. Moreover, they studied some properties of $(p, q)$-calculus and $(p, q)$-analogue of some important integral inequalities. The $(p, q)$-integral inequalities have been studied and rapidly developed during this period by many authors, see ^{[18,19,20,21,22,23,24,25,26]} and the references therein.

Mathematical inequalities was applied in various branches of mathematics as analysis, differential equations, geometry, etc. One typical such example is Hardy inequality. Let us just mention that in 1920, G. H. Hardy ^[27] presented the following famous inequality for $f$ is a non-negative integrable function and $s > 1$, then

which is now known as Hardy inequality.

Hardy inequality has been studied by a large number of authors during the twentieth century. Over the last twenty years a large number of papers have appeared in the literature which deal with the simple proofs, various generalizations and discrete analogue of Hardy inequality, see ^{[28,29,30,31,32,33,34,35]} for more details.

In 2014, L. Maligranda et al. ^[36] studied a $q$-analogue of Hardy inequality (1.1) and some related inequalities. It seems to be a huge new research area to study these so called $q$-Hardy type inequalities. They obtained more general results on $q$-Hardy type inequalities. By taking $q \to 1$, we obtain classical results on Hardy inequality (1.1). Next, L. E. Persson and S. Shaimardan ^[37] studied some $q$-analogue of Hardy type inequalities for the Riemann-Liouville fractional integral operator, see ^[38,39] for more details.

The purpose of this paper is to study some $(p, q)$-Hardy type inequalities for $(p, q)$-integrable functions by using $(p, q)$-derivative and $(p, q)$-integral. Moreover, we also study $(p, q)$-Hölder integral inequality and $(p, q)$-Minkowski integral inequality for two variables. By taking $q\to 1$ and $p = 1$, our results reduce to classical results on Hardy type inequalities, Hölder integral inequality and Minkowski integral inequality for two variables.

2.   Preliminaries

In this section, we recall some known concepts and basic results of $(p, q)$-calculus. Throughout this paper, we let $p, q$ be constants with $0 < q < p\leq 1$ and $\left[a, b\right] \subseteq \mathbb{R}$. We give some definitions and theorems for $(p, q)$-calculus, which will be used in these papers ^{[16,17,18,19,20,21,22,23]}.

First, we give some $(p, q)$-notation, which would appear in this study quite frequently. For any real number $n$, the $(p, q)$-analogue of $n$ is defined by

and

If $p = 1$, then (2.1) reduces to

which is $q$-analogue of $n$.

Definition 2.1. ^[18] If $f: [a, b]\to \mathbb{R}$ is a continuous function, then $(p, q)$-derivative of the function $f$ on $[a, b]$ at $x$ is defined by

The function $f$ is said to be a $(p, q)$-differentiable function on $[a, b]$ if $_aD_{p, q}f(x)$ exists for all $x\in[a, b]$.

Since $f:[a, b]\to \mathbb{R}$ is a continuous function, we have $_aD_{p, q}f(a) = \lim_{x\to a}D_{p, q}f(x)$. In Definition 2.1, if $a = 0$, then $_0D_{p, q}f = D_{p, q}f$ is defined by

And, if $p = 1$, then $D_{p, q}f(x) = D_q f(x)$, which is the $q$-derivative of the function $f$, and also if $q\to1$ in (2.5), then it reduces to a classical derivative.

Example 1. Define function $f:[a, b] \to \mathbb{R}$ by $f(x) = x^2+x+c$, where $c$ is a constant. Then for $x \neq a$, we have

Theorem 2.2. If $f, g: [a, b]\to \mathbb{R}$ are continuous functions and $c, d$ are constants, then the following formulas hold:

(i) $_aD_{p, q}[cf(x)\pm dg(x)] = c_aD_{p, q}f(x)\pm d_aD_{p, q}g(x)$;

(ii) $_aD_{p, q}[f(x)g(x)] = f(px+(1-p)a)_aD_{p, q}g(x)+g(qx+(1-q)a)_aD_{p, q}f(x)$;

(iii) $_aD_{p, q}\left[\frac{f(x)}{g(x)}\right] = \frac{g(px+(1-p)a)_aD_{p, q}f(x)-f(px+(1-p)a)_aD_{p, q}g(x)}{g(px+(1-p)a)g(qx+(1-q)a)}$.

The proof of this theorem is given in ^[17].

Definition 2.3. ^[18] If $f:[a, b]\to \mathbb{R}$ is a continuous function and $0 < a < b$, then the $(p, q)$-integral is defined by

And, $f$ is said to be a $(p, q)$-integrable function on $[a, b]$ if $\int_a^b{f(x)\ _ad_{p, q}x}$ exists for all $x\in[a, b].$ If $a = 0$ in (2.7), then one can get the classical $(p, q)$-integral defined by

and

If $p = 1$ in (2.8), then we have the classical $q$-integral ^[11].

Example 2. Define function $f:[a, b] \to \mathbb{R}$ by $f(x) = 2x+c$, where $c$ is a constant. Then we have

If $p = 1$, we have (2.10) reduces to the $q$-integral of $f(x) = 2x+c$. Furthermore, if $a = 0$, $p = 1$ and $q \to 1$, (2.10) reduces to the classical integral.

The proofs of the following theorems are given in ^[17].

Theorem 2.4. If $f, g: [a, b]\to \mathbb{R}$ are continuous functions, $t\in [a, b]$ and $\alpha$ is a constant, then the following formulas hold:

(i) $_aD_{p, q}\int_a^t f(x)\ _ad_{p, q}x = f(t)$;

(ii) $\int_c^t\ _aD_{p, q}f(x)\ _ad_{p, q}x = f(t)-f(c)$ for $c\in(a, t)$;

(iii) $\int_a^t{[f(x)+g(x)]\ _ad_{p, q}x} = \int_a^b{f(x)\ _ad_{p, q}x}+\int_a^b{g(x)\ _ad_{p, q}x}$;

(iv) $\int_a^t\alpha f(x)\ _ad_{p, q}x = \alpha\int_a^bf(x)\ _ad_{p, q}x$;

(v) $\int_0^t{x^\alpha}d_{p, q}x = \frac{b^{\alpha+1}}{[\alpha+1]_{p, q}}$;

(vi) $\int_c^t{f(px+(1-p)a)\ _aD_{p, q}g(x)}\ _ad_{p, q}x = (fg)(x)\big |_c^t-\int_c^t{g(qx+(1-q)a)\ _aD_{p, q}f(x)\ _ad_{p, q}x}$.

Theorem 2.5. If $f, g: [a, b]\to \mathbb{R}$ are continuous functions and $r > 1$ with $1/r+1/s = 1$, then

Theorem 2.6. If $f, g: [a, b]\to \mathbb{R}$ are continuous functions and $0 < r < 1$ with $1/r+1/s = 1$, then

The proof is similar to the proof of Lemma 2.2 in ^[32].

Definition 2.7. ^[16] The improper $(p, q)$-integral of $f(x)$ on $[0, \infty]$ is defined by

And, $f$ is said to be a $(p, q)$-integrable function on $[0, \infty]$ if $\int_0^\infty f(x)d_{p, q}x$ exists for all $x\in[0, \infty].$

And, if $p = 1$ then (2.13) reduces to

which is an improper $q$-integral of $f(x)$ on $[0, \infty]$ and appeared in ^[13].

Definition 2.8. ^[23] Let $h:[0, b] \times [0, d] \subset \mathbb{R}^2\to \mathbb{R}$ be a continuous function in each variable; let $0 < q_i < p_i\leq 1$, where $i = 1, 2$. The definite $(p_1p_2, q_1q_2)$-integral on $[0, b] \times [0, d]$ is defined by

for $(s, t)\subset [0, b] \times [0, d]$. And, $h$ is said to be a $(p_1p_2, q_1q_2)$-integrable function on $[0, b] \times [0, d]$ if $\int_0^t\int_0^s{h(z, w)}d_{p_1, q_1}z\, d_{p_2, q_2}w$ exists for all $(s, t)\subset [0, b] \times [0, d]$.

Example 3. Define function $h:[0, b] \times [0, d] \subset \mathbb{R}^2\to \mathbb{R}$ by $h(z, w) = z^nw^m$, where $m, n$ are constants. Then we have

Theorem 2.9. If $f:[0, b]\to \mathbb{R}$ is a non-negative function, $r > 1$ and $f^r$ is a $q$-integrable function on $[0, \infty]$, then

The proof of this theorem is given in ^[36].

3.   Main results

In this section, we first present the $(p, q)$-Hölder inequality for two variables.

Theorem 3.1. If $h$ and $g$ are functions defined on $[0, b]\times[0, d]$ and $m_1, m_2 > 1$ with $1/m_1+1/m_2 = 1$, then

Proof. From Definition 2.8 and the discrete Hölder inequality, we get

The proof is completed.

Next, we present the $(p, q)$-Minkowski integral inequality for two variables.

Theorem 3.2. If $h$ is $(p_1p_2, q_1q_2)$-integrable function on $[0, b] \times [0, d]$, $\alpha\in(0, 1]$ and $1\leq r < \infty$, then

Proof. For $r = 1$, (3.2) holds by Fubini's Theorem. Next, for $1 < r < \infty$ and $1/r+1/z = 1$, by using Fubini's Theorem and Theorem 2.5, we have

Consequently, we obtain

Therefore,

The proof is completed.

Remark 3.3. If $p = 1$, then (3.2) reduces to

One easily see that when $q\to1$ in (3.4), the inequality turns into a Minkowski integral inequality for two variables.

Theorem 3.4. If $f:[0, b]\to \mathbb{R}$ is a non-negative function, $r > 1$ and $f^r$ is $(p, q)$-integrable function on $[0, \infty)$, then

Proof. By Definition 2.3 and Theorem 3.2, it is easy to see that

Thus, by Theorem 2.4(vi), we get

This completes the proof.

Remark 3.5. If $p = 1$, then (3.5) reduces to

which is $q$-Hardy inequality in ^[36] and when $q$ $\to$ $1$ in (3.7), the inequality reduces to (1.1).

Theorem 3.6. If $f:[0, b]\to \mathbb{R}$ is a non-negative function, $r\geq 1, z < r-1$ and $t^zf^r$ is $(p, q)$-integrable function on $[0, \infty)$, then

Proof. The proof is similar to the proof of Theorem 3.4.

Remark 3.7. If $p = 1$, then (3.8) reduces to

when $z = 0$, we get the inequality (3.7).

Theorem 3.8. If $f_1, f_2, \ldots, f_n$ are non-negative $(p, q)$-integrable functions on $[0, b]$, then

where

for $i = 1, 2, \ldots, n.$

Proof. By Jensen's inequality, we have

Thus, we obtain

Applying Theorem 3.4 to (3.11), we have

This completes the proof.

Remark 3.9. If $p = 1$, then (3.10) reduces to

which is $q$-Hardy inequality with many functions, and when $q\to1$ in (3.13), the inequality above appeared in ^[33].

Theorem 3.10. If $f:[0, b]\to \mathbb{R}$ is a non-negative function, $r > 1, 0 < a < b < \infty$ and $f^r$ is $(p, q)$-integrable function on $[a, b]$, then

Proof. By Definition 2.3, it is easy to see that

From Theorem 3.2, we have

Thus, by Theorem 2.4(vi), we get

This completes the proof.

Remark 3.11. If $p = 1$, then (3.14) reduces to

which is $q$-Hardy inequality about integration from $a$ to $b$, and when $q$ $\to$ $1$ in (3.15), the inequality reduces to ^[29].

Theorem 3.12. Let $f\geq0, g > 0, \frac{x}{g(x)}$ be non-increasing, $r > 1$ and let $f$ be $(p, q)$-integrable function on $[0, x]$. Then

Proof. From Theorem 2.5, we get

And by Theorem 2.4(vi), we obtain

Since the assumption of the function $\frac{x}{g(x)}$, we have

This proof is completed.

Remark 3.13. If $p = 1$, then (3.16) reduces to a generalization of $q$-Hardy inequality as

Also if $q\to1$, then (3.17) reduces to the well known generalization of Hardy inequality as

which appeared in ^[34].

The following results concern the converse inequalities.

Theorem 3.14. Let $f\geq0, g > 0, \frac{x}{g(x)}$ be non-decreasing, $0 < r < 1$ and let $f$ be $(p, q)$-integrable function on $[0, x]$. Then

Proof. From Theorem 2.6, we get

And by Theorem 2.4(vi), we obtain

Since the assumption of the function $\frac{x}{g(x)}$, we have

This proof is completed.

Remark 3.15. If $p = 1$, then (3.19) reduces to a converse generalization of $q$-Hardy inequality as

Also if $q\to1$, then (3.20) reduces to the well known converse generalization of Hardy inequality as

which appeared in ^[34].

4.   Conclusion

In this paper, we establish $(p, q)$-Hardy type inequalities for $(p, q)$-integral functions. We also obtain $(p, q)$-Hölder integral inequality and $(p, q)$-Minkowski integral inequality for two variables. Our work improves the results of Hardy type inequalities and the generalization of Hardy type inequalities. By taking $q\to 1$ and $p = 1$, our results gives classical inequality formulas.

Acknowledgments

This research was supported by Science Achievement Scholarship of Thailand and the Program Management Unit for Human Resources & Institutional Development, Research and Innovation [grant number B05F630104].

Conflict of interest

The authors declare that there are no conflict of interest in this paper.