Abstract
In this paper, we establish some reversed dynamic inequalities of Hilbert type on time scales nabla calculus by applying reversed Hölder’s inequality, chain rule on time scales, and the mean inequality. As particular cases of our results (when \(\mathbb{T}=\mathbb{N}\) and \(\mathbb{T}=\mathbb{R}\)), we get the reversed form of discrete and continuous inequalities proved by Chang-Jian, Lian-Ying and Cheung (Math. Slovaca 61(1):15–28, 2011).
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1 Introduction
David Hilbert proved Hilbert’s double-series inequality without exact determination of the constant in his lectures (see [14]) and proved that if \(\{a_{m}\}_{m=1}^{\infty }\) and \(\{b_{n}\}_{n=1}^{\infty }\) are two real sequences such that \(0<\sum_{m=1}^{\infty }a_{m}^{2}<\infty \) and \(0<\sum_{n=1}^{\infty }b_{n}^{2}<\infty \), then
In 1911, Schur [27] discovered the integral analogue of (1.1), which became known as the Hilbert integral inequality
for real functions f and g such that \(0<\int _{0}^{\infty }f^{2}(x)\,dx< \infty \) and \(0<\int _{0}^{\infty }g^{2}(y)\,dy<\infty \). The constant π in (1.1) and (1.2) is the best possible constant factor.
In 1925, by introducing a pair of conjugate exponents \((p,q)\) (\(p,q>1\) with \(1/p+1/q=1\)) Hardy [13] gave an extension of (1.1) as follows. If p, \(q>1\) and \(a_{m}\), \(b_{n}\geq 0\) are such that \(0<\sum_{m=1}^{\infty }a_{m}^{p}<\infty \) and \(0<\sum_{n=1}^{\infty }b_{n}^{q}<\infty \), then
Hardy and Reisz [14] proved the equivalent integral analogue of (1.3)
for nonnegative functions f and g such that \(0<\int _{0}^{\infty }f^{p}(x)\,dx<\infty \) and \(0<\int _{0}^{\infty }g^{q}(y)\,dy<\infty \). The constant factor \(\pi /\sin (\pi /p)\) in (1.3) and (1.4) is the best possible. In 1998, Pachpatte [17] gave a new inequality close to that of Hilbert: Let \(a(s)\): \(\{ 0,1,2,\dots ,p \} \subset \mathbb{N}\rightarrow \mathbb{R}\) and \(b(\vartheta )\): \(\{ 0,1,2,\dots ,q \} \subset \mathbb{N}\rightarrow \mathbb{R}\) with \(a(0)=b(0)=0\). Then
where
In 2002, Kim et al. [15] proved that if λ, \(\mu >1 \) and \(a(s)\): \(\{ 0,1,2,\dots ,p \} \subset \mathbb{N}\rightarrow \mathbb{R}\) and \(b(\vartheta )\): \(\{ 0,1,2,\dots ,q \} \subset \mathbb{N}\rightarrow \mathbb{R}\) with \(a(0)=b(0)=0\), then
where
Also, Kim and Kim [15] proved the continuous analogue of (1.6): Let λ, \(\mu >1\), and let f and g be real continuous functions on the intervals \(( 0,x ) \) and \(( 0,y )\), respectively, with \(f(0)=g(0)=0\). Then
for \(x,y\in ( 0,\infty ) \), where
In 2011, Chang-Jian et al. [8] generalized (1.5) as follows. Let \(p_{i}>1\) and \(1/p_{i}+1/p_{i}^{\ast }=1\), and let \(a_{i}(s_{i})\) be real sequences defined for \(s_{i}=0,1,2,\dots ,m_{i}\) such that \(a_{i} ( 0 ) =0\), \(i=1,2,\dots ,n\). Define the operator ∇ by \(\nabla a_{i}(s_{i})=a_{i}(s_{i})-a_{i}(s_{i}-1)\) for any function \(a_{i}(s_{i})\), \(i=1,2,\dots ,n\). Then
Also, the authors of [8] proved that if \(h_{i}\geq 1\) and \(p_{i}>1\) are constants, \(1/p_{i}+1/p_{i}^{\ast }=1\), and \(f_{i}(s_{i})\) are real-valued differentiable functions on \([0,x_{i})\), where \(x_{i}\in ( 0,\infty ) \), such that \(f_{i} ( 0 ) =0\) for \(i=1,2,\dots ,n\), then
In the last decades, a new theory has been discovered to unify the continuous and discrete calculi. It is called a time scale theory. A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real numbers \(\mathbb{R}\). The results in this paper contain the classical continuous and discrete inequalities as particular cases where \(\mathbb{T}=\mathbb{R}\) and \(\mathbb{T}=\mathbb{N}\), respectively. In addition, these inequalities can be extended to the corresponding inequalities on various time scales such as \(\mathbb{T}=h\mathbb{N}\), \(h>0\), and \(\mathbb{T}=q^{\mathbb{N}}\) for \(q>1\). Many authors studied the dynamic inequalities on time scales. For more details about the dynamic inequalities on time scales, see [4–6, 16, 18–26, 28–30], and for applications of Hilbert-type inequalities, see [2, 9–11].
The aim of this paper is to determine assumptions for establishing some new reversed forms of inequalities (1.8) and (1.9) on time scales by establishing some new Hilbert-type inequalities on time scales nabla calculus. Our results will be proved by applying the integration by parts, reverse Hölder’s inequality on time scales, and the reverse of mean inequality.
The organization of the paper as follows. In Sect. 2, we present some definitions, properties, and some lemmas on time scales needed in Sect. 3, where we prove our results. These results as particular cases where \(\mathbb{T}=\mathbb{N}\) and \(\mathbb{T}=\mathbb{R}\) give the reversed of inequalities (1.8) and (1.9), respectively.
2 Preliminaries and basic lemmas
In 2001, Bohner and Peterson [7] defined the time scale \(\mathbb{T}\) as an arbitrary nonempty closed subset of the real numbers \(\mathbb{R}\). Also, they defined the backward jump operator as \(\rho (\tau ):=\sup \{s\in \mathbb{T}:s<\tau \}\). For any function \(f:\mathbb{T}\rightarrow \mathbb{R}\), \(f^{\rho }(\tau )\) denotes \(f(\rho (\tau ))\). We define the time scale interval \([a,b]_{\mathbb{T}}\) by \([a,b]_{\mathbb{T}}:=[a,b]\cap \mathbb{T}\).
Definition 2.1
([3]) A function \(\lambda :\mathbb{T}\rightarrow \mathbb{R}\) is left-dense continuous or ld-continuous if it is continuous at left-dense points in \(\mathbb{T}\) and its right-sided limits exist at right-dense points in \(\mathbb{T}\). The space of ld-continuous functions is denoted by \(\mathrm{C}_{ld}(\mathbb{T}, \mathbb{R})\).
Definition 2.2
([3]) A function \(\lambda :\mathbb{T\rightarrow R}\) is said to be ∇-differentiable at \(t\in \mathbb{T}\) if ψ is defined in a neighborhood U of t and there exists a unique real number \(\psi ^{\nabla }(t)\), called the nabla derivative of ψ at t, such that for each \(\epsilon >0\), there exists a neighborhood N of t with \(N\subseteq U\), and
Lemma 2.1
(Chain rule for nabla derivative [12]) Let \(\psi :\mathbb{R}\rightarrow \mathbb{R}\) be continuously differentiable and suppose that \(\chi :\mathbb{T}\rightarrow \mathbb{R}\) is continuous and nabla differentiable. Then \(\psi \circ \chi :\mathbb{T}\rightarrow \mathbb{R}\) is nabla differentiable, and there exists d in the real interval \([\rho (t),t]\) such that
Definition 2.3
([3]) A function \(\Lambda :\mathbb{T}\rightarrow \mathbb{R}\) is called a nabla antiderivative of \(\psi :\mathbb{T}\rightarrow \mathbb{R}\) if \(\Lambda ^{\nabla }(t)=\psi (t)\) for all \(t\in \mathbb{T}\). We then define the nabla integral of ψ by
Theorem 2.1
([3]) Let a, b, \(c\in \mathbb{T}\) and α, \(\beta \in \mathbb{R}\), and let ψ, \(\chi :\mathbb{T}\rightarrow \mathbb{R}\) be ld-continuous. Then we have the following properties:
-
(1)
\(\int _{a}^{b} [ \alpha \psi (t)+\beta \chi (t) ] \nabla t=\alpha \int _{a}^{b}\psi (t)\nabla t+\beta \int _{a}^{b}\chi (t)\nabla t\),
-
(2)
\(\int _{a}^{b}\psi (t)\nabla t=-\int _{b}^{a} \psi (t)\nabla t\),
-
(3)
\(\int _{a}^{c}\psi (t)\nabla t=\int _{a}^{b}\psi (t) \nabla t+\int _{b}^{c}\psi (t)\nabla t\),
-
(4)
\(\vert \int _{a}^{b}\psi (t)\nabla t \vert \leq \int _{a}^{b} \vert \psi (t) \vert \nabla t\),
-
(5)
If \(\psi (t)\geq 0\) for all \(t\in {}[ a,b]_{\mathbb{T}}\), then \(\int _{a}^{b}\psi (t)\nabla t\geq 0\),
-
(6)
If \(\psi (t)\geq \chi (t)\) for all \(t\in {}[ a,b]_{ \mathbb{T}}\), then \(\int _{a}^{b}\psi (t)\nabla t\geq \int _{a}^{b}\chi (t) \nabla t\).
Lemma 2.2
(The integration by parts on time scales [3]) If a, \(b \in \mathbb{T}\) and f, \(g:\mathbb{T}\rightarrow \mathbb{R}\) are ld-continuous, then
Lemma 2.3
(Reverse Hölder’s inequality [1]) If a, \(b\in \mathbb{T}\), f, \(g\in \mathrm{C}_{ld}(\mathbb{T},\mathbb{R})\), \(\gamma <0\), and \(1/\gamma +1/\nu =1\), then
Lemma 2.4
Let \(a_{i},b_{i}\in \mathbb{T}\), and let either \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},(-\infty ,0])\) be nonincreasing functions or \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},[0,\infty ))\) be nondecreasing functions with \(\psi _{i}(a_{i})=0\), \(i=1,2,\dots ,n\). Then
Proof
Firstly, if \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},(-\infty ,0])\) is a nonincreasing function with \(\psi _{i}(a_{i})=0\), then we see that \(\psi _{i}^{\nabla }(t_{i})\leq 0\), and then
Secondly, if \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},[0,\infty ))\) is a nondecreasing function with \(\psi _{i}(a_{i})=0\), then we observe that \(\psi _{i}^{\nabla }(t_{i})\geq 0\), and then
which is (2.4). The proof is complete. □
Lemma 2.5
(Mean inequality [14]) If \(\alpha _{i}\), \(\beta _{i}>0\) for \(i=1,2,\dots ,n\), then
Lemma 2.6
Let \(q_{i}<0\) with \(1/p_{i}+1/q_{i}=1\), and let \(s_{i}>0\), \(i=1,2,\dots ,n\). Then
Proof
Applying Lemma 2.5 with \(\alpha _{i}=s_{i}\) and \(\beta _{i}=-1/q_{i}\), we observe that
Since \(1/q_{i}=1-1/p_{i}\), we have that
and then inequality (2.9) becomes
which is (2.8). The proof is complete. □
3 Main results
Throughout the paper, we will assume that the integrals considered exist.
Now we can state and prove our results.
Theorem 3.1
Let \(a_{i},b_{i}\in \mathbb{T}\), let \(0< p_{i}<1\) and \(q_{i}<0\) be such that \(1/p_{i}+1/q_{i}=1\), and let either \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},(-\infty ,0])\) be nonincreasing functions or \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},[0,\infty ))\) be nondecreasing functions with \(\psi _{i}(a_{i})=0\), \(i=1,2,\dots ,n\). Then
Proof
From (2.4) we have that for \(\xi _{i}\in [ a_{i},b_{i} ] _{\mathbb{T}}\),
and then
Applying (2.3) to \(\int _{a_{i}}^{\xi _{i}} \vert \psi _{i}^{\nabla }(t_{i}) \vert \nabla t_{i}\), we observe that
and then
Substituting (3.3) into (3.2), we see that
Applying Lemma 2.6 with \(s_{i}=\xi _{i}-a_{i}\), we have that
and so inequality (3.4) becomes
Multiplying (3.5) by \(( n-\sum_{i=1}^{n}1/p_{i} ) ^{n-\sum _{i=1}^{n}1/p_{i}}/ ( \sum_{i=1}^{n} ( \xi _{i}-a_{i} ) /q_{i} ) ^{ \sum _{i=1}^{n}1/q_{i}}\) and integrating over \(\xi _{i}\) from \(a_{i}\) to \(b_{i}\), \(i=1,2,\dots ,n\), we observe that
Applying (2.3) to \(\int _{a_{i}}^{b_{i}} ( \int _{a_{i}}^{\xi _{i}} \vert \psi _{i}^{\nabla }(t_{i}) \vert ^{p_{i}}\nabla t_{i} ) ^{ \frac{1}{p_{i}}}\nabla \xi _{i}\), we have that
and then
Substituting (3.7) into (3.6), we see that
Applying (2.2) on \(\int _{a_{i}}^{b_{i}} ( \int _{a_{i}}^{\xi _{i}} \vert \psi _{i}^{\nabla }(t_{i}) \vert ^{p_{i}}\nabla t_{i} ) \nabla \xi _{i}\), we get
where \(g_{1}(\xi _{i})=\xi _{i}-b_{i}\). Substituting (3.9) into (3.8), we obtain
and then
which is (3.1). The proof is complete. □
Remark 3.1
As a particular case of Theorem 3.1, if \(\mathbb{T}=\mathbb{N}_{0}\), \(\rho (\xi _{i})=\xi _{i}-1\), and \(a_{i}=0\) for \(i=1,2,\dots ,n\), then we get the reversed form of (1.8).
Corollary 3.1
As a particular case of Theorem 3.1, if \(\mathbb{T}=\mathbb{R}\), \(\rho (\xi _{i})=\xi _{i}\), \(a_{i}=0\), \(0< p_{i}<1\) and \(q_{i}<0\) are such that \(1/p_{i}+1/q_{i}=1\), and either \(\psi _{i}\) is a nonpositive nonincreasing function, or \(\psi _{i}\) is a nonnegative nondecreasing function with \(\psi _{i}(0)=0\), \(i=1,2,\ldots,n\), then
where \(K= ( n-\sum_{i=1}^{n}1/p_{i} ) ^{n-\sum _{i=1}^{n}1/p_{i}}/ \prod_{i=1}^{n}b_{i}^{\frac{1}{q_{i}}}\).
In the following theorem, we generalize Theorem 3.1 by replacing the function \(\psi _{i}(\xi _{i})\) with \(\psi _{i}^{h_{i}}(\xi _{i})\), \(h_{i}\geq 1\).
Theorem 3.2
Let \(a_{i},b_{i}\in \mathbb{T}\), \(0< h_{i}\leq 1\), let \(0< p_{i}<1\) and \(q_{i}<0\) be such that \(1/p_{i}+1/q_{i}=1\), and let \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{ \mathbb{T}},\mathbb{R}^{+}\cup \{0\})\) be increasing functions with \(\psi _{i}(a_{i})=0\) for \(i=1,2,\ldots,n\). Then
Proof
Applying (2.1) to \(\psi _{i}^{h_{i}}(t_{i})\), we get
where \(\zeta _{i}\in [ \rho (t_{i}),t_{i} ] \). Since \(\psi _{i}\) is an increasing function, \(0< h_{i}\leq 1\), and \(\zeta _{i} \leq t_{i}\), we have from (3.12) that
(note that this statement holds with equality for \(h_{i}=1\)), and then integrating the last inequality over \(t_{i}\) from \(a_{i}\) to \(\xi _{i}\), where \(\psi _{i}(a_{i})=0\), we observe that
Thus
Applying (2.3) to \(\int _{a_{i}}^{\xi _{i}}\psi _{i}^{h_{i}-1}(t_{i})\psi _{i}^{\nabla }(t_{i}) \nabla t_{i}\), we have that
and then
Substituting (3.14) into (3.13), we get
Applying Lemma 2.6 with \(s_{i}=\xi _{i}-a_{i}\), we have that
Substituting (3.16) into (3.15), we see that
Multiplying (3.17) by \(( n-\sum_{i=1}^{n}1/p_{i} ) ^{n-\sum _{i=1}^{n}1/p_{i}}/ ( \sum_{i=1}^{n} ( \xi _{i}-a_{i} ) /q_{i} ) ^{ \sum _{i=1}^{n}1/q_{i}}\) and integrating over \(\xi _{i}\) from \(a_{i}\) to \(b_{i}\), \(i=1,2,\ldots,n\), we observe that
Applying (2.3) to \(\int _{a_{i}}^{b_{i}} ( \int _{a_{i}}^{ \xi _{i}} ( \psi _{i}^{h_{i}-1}(t_{i})\psi _{i}^{\nabla }(t_{i}) ) ^{p_{i}}\nabla t_{i} ) ^{\frac{1}{p_{i}}}\nabla \xi _{i}\), we have that
and then
Substituting (3.19) into (3.18), we see that
Applying (2.2) to \(\int _{a_{i}}^{b_{i}} ( \int _{a_{i}}^{\xi _{i}} ( \psi _{i}^{h_{i}-1}(t_{i}) \psi _{i}^{\nabla }(t_{i}) ) ^{p_{i}}\nabla t_{i} ) \nabla \xi _{i}\), we get
where \(g_{2}(\xi _{i})=\xi _{i}-b_{i}\). Substituting (3.21) into (3.20), we observe that
which satisfies (3.11). The proof is complete. □
Remark 3.2
As a particular case of Theorem 3.2, if \(h_{i}=1\) for \(i=1,2,\ldots,n\), then we observe that Theorem 3.1 holds.
Remark 3.3
If \(\mathbb{T}=\mathbb{R}\) and \(a_{i}=0\), then we get the reverse of inequality (1.9) under the following conditions: \(0< h_{i}\leq 1\), \(q_{i}<0\), and \(\psi _{i}\in \mathrm{C}( [ a_{i},b_{i} ] ,\mathbb{R}^{+} \cup \{0\})\) are increasing functions with \(\psi _{i}(0)=0\) for \(i=1,2,\ldots,n\).
Theorem 3.3
Let \(a_{i},b_{i}\in \mathbb{T}\), \(h_{i}\geq 1\), \(q_{i}<0\), \(p_{i}=q_{i}/ ( q_{i}-1 ) \), and let \(\psi _{i}\in \mathrm{C}_{ld}( [ a_{i},b_{i} ] _{\mathbb{T}},\mathbb{R}^{+}\cup \{0\})\) be decreasing functions with \(\psi _{i}(b_{i})=0\) for \(i=1,2,\ldots,n\). Then
Proof
Applying the chain rule formula (2.1) to the term \(\psi _{i}^{h_{i}}(t_{i})\), \(h_{i}\geq 1\), we get
where \(\zeta _{i}\in [ \rho (t_{i}),t_{i} ] \). Since \(\psi _{i}\) is a decreasing function, \(h_{i}\geq 1\), and \(\zeta _{i} \leq t_{i}\), we have from (3.23) that
and then by integrating the last inequality over \(t_{i}\) from \(\xi _{i}\) to \(b_{i}\), where \(\psi _{i}(b_{i})=0\), we see that
Therefore
Applying reverse Hölder’s inequality (2.3) to the term
with \(q_{i}<0\), \(p_{i}=q_{i}/ ( q_{i}-1 ) \), \(f(t_{i})= \psi _{i}^{h_{i}-1}(t_{i}) [ -\psi _{i}^{\nabla }(t_{i}) ] \), and \(g(t_{i})=1\), we have that
and then
Substituting (3.25) into (3.24), we get
Applying Lemma 2.6 with \(s_{i}=b_{i}-\xi _{i}\), we have that
Substituting (3.27) into (3.26), we see that
Multiplying (3.28) by the term
and integrating over \(\xi _{i}\) from \(a_{i}\) to \(b_{i}\), \(i=1,2,\ldots,n\), we observe that
Applying reverse Hölder’s inequality (2.3) to the term
with \(q_{i}<0\), \(p_{i}=q_{i}/ ( q_{i}-1 ) \),
we have that
and then
Substituting (3.30) into (3.29), we see that
Applying the integration-by-parts formula (2.2) to the term
with \(f_{3}(\xi _{i})=\int _{\xi _{i}}^{b_{i}} ( \psi _{i}^{h_{i}-1}(t_{i}) [ -\psi _{i}^{\nabla }(t_{i}) ] ) ^{p_{i}}\nabla t_{i}\) and \(g_{3}^{\nabla }(\xi _{i})=1\), we get
where \(g_{3}(\xi _{i})=\xi _{i}-a_{i}\). Since \(g_{3}(a_{i})=0\), we have from (3.32) that
Substituting (3.33) into (3.31), we observe that
which is (3.22). The proof is complete. □
Corollary 3.2
As a particular case of Theorem 3.3, if \(\mathbb{T}=\mathbb{R}\), \(a_{i}\), \(b_{i}\in \mathbb{T}\), \(h_{i}\geq 1\), \(q_{i}<0\), \(p_{i}=q_{i}/ ( q_{i}-1 ) \), and \(\psi _{i}\in \mathrm{C}( [ a_{i},b_{i} ] ,\mathbb{R}^{+}\cup \{0\})\) is a decreasing function with \(\psi _{i}(b_{i})=0\) for \(i=1,2,\ldots,n\), then \(\rho (\xi _{i})=\xi _{i}\), and
Corollary 3.3
As a particular case of Theorem 3.3, if \(\mathbb{T}=\mathbb{N}_{0}\), \(a_{i}\), \(b_{i}\in \mathbb{T}\), \(h_{i}\geq 1\), \(q_{i}<0\), \(p_{i}=q_{i}/ ( q_{i}-1 ) \), and \(\psi _{i}\) is a nonnegative decreasing sequence with \(\psi _{i}(b_{i})=0\), \(i=1,2,\ldots,n\), then \(\rho (\xi _{i})=\xi _{i}-1 \), and
where \(\nabla \psi _{i}(\xi _{i})=\psi _{i}(\xi _{i})-\psi _{i}(\xi _{i}-1)\).
4 Conclusion
In this paper, we establish some reversed dynamic inequalities of Hilbert type on time scales nabla calculus by applying reversed Hölder’s inequality, chain rule on time scales, and the mean inequality. Also, we can get the reversed form of discrete and continuous inequalities proved by Chang-Jian, Lian-Ying, and Cheung.
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Saied, A.I. A study on reversed dynamic inequalities of Hilbert-type on time scales nabla calculus. J Inequal Appl 2024, 75 (2024). https://doi.org/10.1186/s13660-024-03091-8
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DOI: https://doi.org/10.1186/s13660-024-03091-8