Abstract
The aim of this work is to obtain quantum estimates for q-Hardy type integral inequalities on quantum calculus. For this, we establish new identities including quantum derivatives and quantum numbers. After that, we prove a generalized q-Minkowski integral inequality. Finally, with the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities. Additionally, by taking the limit \(q\rightarrow 1^{-}\), our results give classical results on the Hardy inequality.
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1 Introduction
Hardy’s integral inequality, proved by G.H. Hardy in 1920 [4] is
where \(p>1\), \(x>0\), f is a nonnegative measurable function on \(( 0,\infty ) \) and \(\int _{0}^{\infty }f^{p} ( t ) \,dt\) is convergent. Also the constant \(( \frac{p}{p-1} ) ^{p}\) is the best possible.
Hardy’s type inequalities have been studied by a large number of authors during the 20th century and has motivated some important lines of study which are currently active. Over the last 20 years a large number of papers have appeared in the literature which deal with the simple proofs, various generalizations and discrete analogues of Hardy’s inequality and its generalizations; see [5, 8, 11, 12, 15, 17–19].
The inequalities have become an important cornerstone in mathematical analysis and optimization and many uses of these inequalities have been discovered in a variety of settings. Recently, the Hermite–Hadamard type inequality has become the subject of intensive research. For recent results, refinements, counterparts, generalizations and new Hadamard’s-type inequalities, see [1, 7, 10, 14, 16, 20].
On the other hand, the study of calculus without limits is known as quantum calculus or q-calculus. The famous mathematician Euler initiated the study q-calculus in the 18th century by introducing the parameter q in Newton’s work of infinite series. In the early 20th century, Jackson [6] has started a symmetric study of q-calculus and introduced q-definite integrals. The subject of quantum calculus has numerous applications in various areas of mathematics and physics, such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions, quantum theory, mechanics and in theory of relativity. This subject has received outstanding attention by many researchers and hence it is considered as an in-corporative subject between mathematics and physics. The reader is referred to [2, 3, 9] for some current advances in the theory of quantum calculus and theory of inequalities in quantum calculus.
The purpose of this work is to establish quantum estimates for q-Hardy type integral inequalities on quantum calculus. For this, we establish new identities including quantum derivatives and quantum numbers. After that, we prove a generalized q-Minkowski integral inequality. Finally, with the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities. In addition, by taking the limit \(q\rightarrow 1^{-}\), our results give classical results on the Hardy inequality.
2 Preliminaries and definitions of q-calculus
Throughout this paper, let \(a< b\) and \(0< q<1\) be a constant. The following definitions, notations and theorems for q-derivative and q-integral of a function f on \([ a,b ]\) are given in [2, 3, 9].
The notation \([ z ] _{q}\) is defined by
A special case of (2.1) when \(z\in \mathbb{N} \) is
Also
Definition 1
Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a continuous function, then q-derivative of f at \(x\in [ a,b ] \) is characterized by the expression
Since \(f: [ a,b ] \rightarrow \mathbb{R} \) is a continuous function, thus we have \(D_{q}f ( a ) =\underset{x\rightarrow a}{\lim }D_{q}f ( x ) \) The function f is said to be q- differentiable on \([ a,b ] \) if \(D_{q}f ( t ) \) exists for all \(x\in [ a,b ] \). Also \(\underset{q\rightarrow 1^{-}}{\lim }D_{q}f ( x ) =f^{\prime } ( x ) \) is classic derivative.
Theorem 1
Assume that \(f,g:I\subset \mathbb{R} \rightarrow \mathbb{R} \) are continuous functions, then we have the properties of the q-derivative:
Definition 2
Suppose \(0< a< b\). The definite q-integral is defined as
and
where \(\sum_{n=0}^{\infty }q^{n}f ( q^{n}b ) \) and \(\sum_{n=0}^{\infty }q^{n}f ( q^{n}a ) \) are convergent.
Definition 3
([9])
The improper q-integral of \(f ( t ) \) on \([ 0,\infty ) \) is defined by
and
where \(\sum_{n=-\infty }^{\infty }q^{n}f ( q^{n} ) \) is convergent.
We have the following properties of the q-integral of (2.4):
Theorem 2
(q-Hölder inequality)
Let f, g be q-integrable on \([ a,b ] \) and \(0< q<1\) and \(\frac{1}{s}+\frac{1}{r}=1\) with \(s>1\). Then we have
3 Auxiliary results
The following results which will be used. There is no general change of variables property for the q-integral. However, the variable can be changed as follows.
Lemma 1
(q-Change of variables property)
Let \(f:I\rightarrow \mathbb{R} \) be a function and \(0< q<1\). Then we have
where \(b\neq 0\) and \(\int _{0}^{b}f ( t ) \,d_{q}t\) is convergent.
Proof
From the definition of the q-integral, we have
as desired. □
A general chain rule for q-derivative does not exist. However, a chain rule of \(( h ( t ) ) ^{p}\) and \(( h ( t ) ) ^{\frac{1}{p}}\) can be calculated as follows.
Lemma 2
Let \(h:I\subset \mathbb{R} \rightarrow \mathbb{R} \) be a function \(p\in \mathbb{Z} \) and \(0< q<1\). Then we have
In (3.2) if we choose \(q\rightarrow 1^{-}\) we have the classical derivative of \(( h ( t ) ) ^{p}\),
Proof
By the definition of the q-derivative we have
as desired. □
Lemma 3
Let \(h:I\subset \mathbb{R} \rightarrow \mathbb{R} \) be a function \(p\in \mathbb{Z} \) and \(0< q<1\). Then we have
In (3.3) if we choose \(q\rightarrow 1^{-}\) we have the classical derivative of \(( h ( t ) ) ^{\frac{1}{p}}\),
Proof
We consider
such that
and from (3.2) we know
Thus, we get
as desired. □
Similarly, we have more general result as follows.
Lemma 4
Let \(h:I\subset \mathbb{R} \rightarrow \mathbb{R} \) be a function \(\frac{n}{m}\in \mathbb{Q} \) and \(0< q<1\). Then we have
In (3.6) if we choose \(q\rightarrow 1^{-}\) we have the classical derivative of \(( h ( t ) ) ^{\frac{n}{m}}\),
Proof
We consider
such that
and from (3.2) we have
Thus, we get
as desired. □
4 Main results
Firstly, we will prove the generalized q-Minkokski type integral inequality which will be used in the next theorem.
Theorem 3
(Generalized q-Minkowski integral inequality)
Let \(\alpha \in ( 0,1 ] \), \(1\leq p\leq \infty \), \(f: [ a,b ] \times [ c,d ] \rightarrow \mathbb{R} \) be a q-integrable function. Then the following inequality holds:
where \(q\in ( 0,1 ) \).
Proof
The case \(p=1\) corresponds to Fubini’s theorem. For the case \(p=\infty \) we just notice that
Now assume that \(1< p<\infty \) and we can write
the last step coming from Fubini’s theorem. By applying the q-Hölder inequality to the inner integral with respect to x, we have
Finally dividing both sides by \(\int _{a}^{b} ( \vert \int _{c}^{d}f ( x,t ) \,d_{q}t \vert ^{p}\,d_{q}x ) ^{\frac{1}{r}}\) we have
i.e.
which gives the required inequality. □
Theorem 4
(q-Hardy inequality)
If f is a nonnegative function on \(( 0,\infty )\), \(p>1\) and \(\int _{0}^{\infty }f^{p} ( t ) \,d_{q}t\) is convergent, then the following inequality holds:
where \(q\in ( 0,1 ) \).
Proof
From (3.1) by the q-changing variables \(t=xs\) it follows that
Thus, we write
From the generalized q-Minkowski integral inequality and by using the q-changing variables \(xs=t\), we have
and the proof is completed. □
Remark 1
In (4.2) if we choose \(q\rightarrow 1^{-}\) we recapture the classical Hardy inequality.
The following theorem generalizes the q-Hardy type integral inequality by introducing power weights \(x^{r}\).
Theorem 5
If f is a nonnegative function on \(( 0,\infty )\), \(p\geq 1\), \(r< p-1\) and \(\int _{0}^{\infty }t^{r}f^{p} ( t ) \,d_{q}t\) is convergent, then the following inequality holds:
where \(q\in ( 0,1 ) \).
Proof
By the q-changing variables \(t=xs\) we get
So, from Minkowski q-integral inequality and by the changing variables \(xs=u\) the proof is completed as follows:
□
Remark 2
In Theorem 5 if we put \(r=0\) we obtain the inequality (4.2).
Definition 4
For a given weight r, we define the modified q-Hardy operator as
The following theorem will be proved using the q-Hardy operator.
Theorem 6
Assume f is a nonnegative function on \(( 0,\infty ) \), r being an absolutely continuous function on \(( 0,\infty )\), and \(p>1\). Also assume \(\int _{0}^{\infty }f^{p} ( x ) \,d_{q}x\) is convergent, and
for almost every \(x>0\) and for some \(\lambda >0\). Then we have the following inequality:
where
Proof
We assume \(0< a< b<\infty \) and
Then, defining \(H_{r,a}f ( x ) =\frac{1}{x}h_{r,a}f ( x ) \), and integrating by parts from (2.5) with \(w= ( h_{r,a}f ( x ) ) ^{p}\) and \(D_{q}g ( x ) =x^{-p}\) noting that \(g(x)=\frac{x^{1-p}}{ [ 1-p ] _{q}}\), we get
We notice that from (2.2)
is negative since \(p-1\in \mathbb{N} \), \(p-1>0\) and \(h_{q,r,a}f ( b ) >0\) with \(b>0\). Also, from the definition of \(h_{q,r,a}f ( x ) \) we have
Hence, by \([ 1-p ] _{q}=-\frac{1}{q^{ ( p-1 ) }} [ ( p-1 ) ] _{q}\)
or equivalently
Now, using (4.5) and the q-Hölder inequality, we have
where \(\frac{1}{p}+\frac{1}{p^{\prime }}=1\), that is,
If we take \(c>a\), then
Invoking the dominated convergence theorem, taking \(a\rightarrow \infty \), we get
for all \(c,b>0\). Finally, letting \(b\rightarrow \infty \) and \(c\rightarrow 0\),
□
In Theorem 6 if we take the limit \(q\rightarrow 1^{-}\) we obtain the following theorem, proved by N. Levinson in 1964 (cf. [13, Theorem 4]).
Remark 3
Let f be a nonnegative function on \(( 0,\infty ) \), r being absolutely continuous function on \(( 0,\infty ) \) and \(p>1\). Also assume \(\int _{0}^{\infty } ( f ( x ) ) ^{p}\,dx\) is convergent, and
for almost every \(x>0\) and for some \(\lambda >0\). Then we have the following inequality:
where
Theorem 7
Assume f is a nonnegative function on \(( 0,\infty ) \), u is absolutely continuous function on \(( 0,\infty ) \) and \(p>1\). Also assume \(\int _{a}^{b} ( f ( x ) ) ^{p}\,d_{q}x\) is convergent, and
for almost every \(x>0\) and for some \(\lambda >0\). Then we have the following inequality:
where
Proof
If we consider \(r ( x ) = ( \frac{1}{u ( x ) } ) ^{ \frac{1}{p}}\), then
and we apply Theorem 6 to g, we assume \(0< a< b<\infty \) and
Then, defining \(H_{q,r,a}g ( x ) =\frac{1}{x}h_{q,r,a}g ( x ) \), and integrating by parts from (2.5) with \(w= ( h_{q,r,a}g ( x ) ) ^{p}\) and \(D_{q}v ( x ) =x^{-p}\) noting that \(v(x)=\frac{x^{1-p}}{ [ 1-p ] _{q}}\) we get
We notice that from (2.2)
is negative since \(p-1\in \mathbb{N} \), \(p-1>0\) and \(h_{q,r,a}g ( b ) >0\) with \(b>0\). Also, from the definition of \(h_{q,r,a}g ( x ) \) we have
Hence, by \([ 1-p ] _{q}=-\frac{1}{q^{ ( p-1 ) }} [ ( p-1 ) ] _{q}\)
or equivalently
Finally, by using (4.6) and the q-Hölder inequality, we have
and
and this completes the proof. □
In Theorem 7 if we take the limit \(q\rightarrow 1^{-}\) we obtain the following result, proved by N. Levinson in 1964 [13] on continuous analysis.
Remark 4
Assume that f is a nonnegative function on \(( 0,\infty ) \), u is absolutely continuous function on \(( 0,\infty )\), and \(p>1\). Also assume \(\int _{a}^{b} ( f ( x ) ) ^{p}\,dx\) is convergent, and
for almost every \(x>0\) and for some \(\lambda >0\). Then we have the following inequality:
where
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Alp, N., Sarikaya, M.Z. q-Hardy type inequalities for quantum integrals. Adv Differ Equ 2021, 355 (2021). https://doi.org/10.1186/s13662-021-03514-6
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DOI: https://doi.org/10.1186/s13662-021-03514-6