Abstract
Some new Hilbert-type inequalities involving Fenchel–Legendre transform are introduced. These inequalities give more general forms of some previously proved inequalities.
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1 Introduction
The form of the established classical discrete Hilbert-type inequality is given as follows [1]:
If \(a_{n} ,b_{n} \geq 0\), \(0< \sum_{n=1}^{\infty } a_{n}^{2} < \infty \), and \(0< \sum_{n=1}^{\infty } b_{n}^{2} < \infty \), then we have
The integral analogue of inequality (1) is given by
unless \(f\equiv 0\) or \(g\equiv 0\), where \(p>1\), \(p^{*}=p/(p-1)\). The constant \(\pi \operatorname{cosec} (\pi /p) \) in (1) and (2) is optimal, see [1].
Inequalities (1) and (2) have many generalizations, see for instance [2–4] and the references therein, these refinements and ameliorations of the original inequality lead to an important development and improvement of many advanced mathematical branches, see for example [5–7].
In [8] the author gave inequalities that can be considered as an extension to inequality (1), containing a series of positive terms as follows.
Theorem 1
Let\(q\geq 1\), \(p\geq 1\), and let\((a_{n})\)and\((b_{m})\)be two positive sequences of real numbers defined for\(n=1,2,\ldots,k\)and\(m=1,2,\ldots,r\), where\(k,r\in \mathbb{N}\), and define\(A_{n} = \sum_{s=1}^{n} a_{s} \), \(B_{m} = \sum_{t=1}^{m} b_{t} \). Then
unless\((a_{n})\)or\((b_{m})\)is null, where\(C(p,q,k,r) = \frac{1}{2} pq \sqrt{kr} \).
In [7], the author gave an improvement of the inequality given in Theorem 1 as follows.
Theorem 2
Let\(q\geq 1\), \(p\geq 1\), and let\((a_{n})\)and\((b_{m})\)be two positive sequences of real numbers defined for\(n=1,2,\ldots,k\)and\(m=1,2,\ldots,r\), where\(k,r \in \mathbb{N}\), and define\(A_{n} = \sum_{s=1}^{n} a_{s} \), \(B_{m} = \sum_{t=1}^{m} b_{t} \). Then, for\(\alpha >0\),
unless\((a_{n})\)or\((b_{m})\)is null, where\(C(p,q,k,r;\alpha ) = ( \frac{1}{2})^{1/ \alpha } pq \sqrt{kr} \).
In this paper, through Fenchel–Legendre transform and by utilizing Jensen’s and Schwarz’s inequalities, we give some improvements of the inequalities given in Theorems 1 and 2. In addition, some new Hilbert-type inequalities are obtained alongside some applications.
2 Preliminaries
In this section we introduce the Fenchel–Legendre transform, which will have an important role in later sections. For more details, we refer, for instance, to [9–11].
Definition 1
Let \(h: \mathbb {R}^{n} \longrightarrow \mathbb {R}\cup \lbrace +\infty \rbrace \) be a function such that \(h \not \equiv +\infty \), i.e., \(\operatorname{dom}(h)= \lbrace x\in \mathbb {R}^{n} \vert h(x) < +\infty \rbrace \neq \emptyset \). Then the Fenchel–Legendre transform is defined as follows:
where \(\langle \cdot, \cdot \rangle\) denotes the scalar product on \(\mathbb {R}^{n}\). The mapping \(h \longrightarrow h^{*}\) will often be called the conjugate operation.
In addition, the domain of \(h^{*}\), i.e., \(\operatorname{dom}(h^{*})\) is the set of slopes of all the affine functions minorizing the function h over \(\mathbb {R}^{n}\).
With more hypotheses on h we can give, in the next corollary, an equivalent formula for (5) called Legendre transform.
Corollary 1
Let\(h: \mathbb {R}^{n} \longrightarrow \mathbb {R}\)be strictly convex, differentiable, and 1-coercive function. Then
for all\(y \in \operatorname{dom}(h^{*}) \), where\(\langle \cdot, \cdot \rangle\)denotes the scalar product on\(\mathbb {R}^{n}\).
Lemma 1
(Fenchel–Young inequality [11])
Lethbe a function and\(h^{*}\)be its Fenchel–Legendre transform, then
for all\(x \in \operatorname{dom}(h)\)and\(y \in \operatorname{dom}(h^{*}) \).
Corollary 2
(Jensen’s inequality [12, 13])
Let\(\varPhi : U\subseteq \mathbb {R}\longrightarrow \mathbb {R}\)be a convex function on a convex setU, with\(x_{i} \in U\), \(i=1,2,\ldots,n\), and\(P_{n} = \sum_{i=1}^{n} p _{i}> 0\)for\(p_{i} \geq 0 \), then
Definition 2
A function Φ is called a submultiplicative function on \([0, \infty )\) if
3 Main results
We begin this section by proving the following simple and useful lemma.
Lemma 2
Forxand\(y\in \mathbb {R}\). Assume that\(x+y \geq 1\), then
Proof
First, we use \(x + y \geq 1\) and \(\frac{\alpha }{\beta }\geq 1\) to write \((x+y)^{\frac{1}{2}} \leq (x+y)^{\frac{\alpha }{2\beta }} \). Then the well-known inequality \(\forall n \geq 1\), \((\vert x \vert + \vert y \vert )^{\frac{1}{n}} \leq \vert x \vert ^{\frac{1}{n}} + \vert y \vert ^{\frac{1}{n}} \) gives the result for \(\alpha \geq \beta \geq \frac{1}{2}\). □
Theorem 3
Let\(q\geq 1\), \(p\geq 1\), \(\alpha \geq \beta \geq \frac{1}{2}\)and\((a_{n})_{1\leq n \leq k} \), \((b_{m})_{1\leq m \leq r} \)be two positive sequences of real numbers where\(k,r\in \mathbb{N}\). Define\(A_{n} = \sum_{s=1}^{n} a_{s} \), \(B_{m} = \sum_{t=1}^{m} b_{t}\). Then the following inequalities hold:
and
unless\((a_{n})\)or\((b_{m})\)is null, where
Proof
By exploiting the following inequality [14, 15]
where \(z_{i} \geq 0\) and \(\gamma \geq 1\) is a constant, we have
Using (11), (12), and the Schwarz inequality, we observe that
squaring both sides of inequality (13) gives
Using (6) (for nonnegative real numbers x and y) in (13) and (14) produces
Let us divide both sides of (15) by \((h(n) + h^{*}(m))^{ \frac{1}{2}}\), take the sum over n from 1 to k afterwards and the sum over m from 1 to r subsequently. Besides, we use the Schwarz inequality, and then we interchange the order of the summations (see[14, 15]). We obtain
Thus,
Now apply Lemma 2 on L.H.S. of (17) to obtain (10). To prove (9), divide both sides of (16) by \(h(n) + h^{*}(m)\), take the sum over n from 1 to k afterwards, then the sum over m from 1 to r, and then interchange the order of the summations to obtain
Therefore,
which is (9). This completes the proof. □
Theorem 4
Under the hypotheses of Theorem 3, for\(\sqrt{n}\in \operatorname{dom}(h)\), \(\sqrt{m}\in \operatorname{dom}(h^{*})\), the following inequality holds:
unless\((a_{n})\)or\((b_{m})\)is null, where
Proof
By the hypothesis that \(\sqrt{n}\in \operatorname{dom}(h)\), \(\sqrt{m} \in \operatorname{dom}(h^{*})\), inequality (6) gives
Complete the proof as we did to obtain inequality (10) in Theorem 3 with appropriate changes. □
Corollary 3
Let\((a_{n})\), \((b_{m})\), \(A_{n}\), and\(B_{m}\)be as defined in Theorem 3. Then the inequalities
and
hold.
Proof
Put \(p=q=1\) in (9) and (10). This completes the proof. □
The following theorem treats the further generalization of the inequality obtained in Corollary 3. Furthermore, suppose that Φ and Ψ are nonnegative, convex, and submultiplicative functions on \([0,\infty )\).
Theorem 5
Let\((a_{n})\), \((b_{m})\), \(A_{n}\), and\(B_{m}\)be as defined in Theorem 3, and\((p_{n})_{1\leq n \leq k}\), \((q_{m})_{1\leq m \leq r}\)be positive sequences. Define\(P_{n} = \sum_{s=1}^{n} p_{s}\), \(Q_{m} = \sum_{t=1}^{m} q_{t} \). Then the following inequality holds:
where
Proof
Using the fact that Φ is a submultiplicative function, we have
then by Jensen’s and Schwarz’s inequalities we have that
similarly, we can get
From inequalities (24), (25) and the Fenchel–Young inequality (for nonnegative reals x and y), we have
Let us divide both sides of (26) by \((h(n) + h^{*}(m))^{ \frac{1}{2}}\), take the sum over n from 1 to k afterwards, then take the sum over m from 1 to r. Additionally, use the Schwarz inequality and then interchange the order of the summations to have
Now define \(M_{1}(k,r)\) as
Therefore,
Now apply Lemma 2 on the L.H.S. of (28) to obtain (22). This completes the proof. □
Lemma 3
Under the hypotheses of Theorem 5, the following inequality holds:
where
Proof
From inequalities (24), (25) and the Fenchel–Young inequality (for nonnegative reals x and y), we have
Now divide both sides of (30) by \(h(n) + h^{*}(m)\), then take the sum over n from 1 to k first and the sum over m from 1 to r, then use the Schwarz inequality to obtain
where
Therefore, if we interchange the order of the summations in (31), we obtain (29). This completes the proof. □
We believe that the inequalities in the next theorem are new to the literature.
Theorem 6
Under the hypotheses of Theorems3and5, the following inequalities hold:
and
Proof
Using Fenchel–Young inequality (6) in (9), (10), (22), and (29) produces inequalities (32), (33), (34), and (35) respectively. This completes the proof. □
The following theorem deals with slight changes of the inequality given in Theorem 9.
Theorem 7
Let\((a_{n})_{1\leq n \leq k}\), \((b_{m})_{1\leq m \leq r}\), \((p_{n})_{1\leq n \leq k}\), and\((q_{m})_{1\leq m \leq r}\)be nonnegative sequences of real numbers where\(k,r\in \mathbb{N}\). Suppose thatΦandΨare nonnegative, convex, and submultiplicative functions on\([0,\infty )\). Let\(A_{n}\), \(B_{m}\)be defined as follows:
where\(P_{n} = \sum_{s=1}^{n} p_{s}\)and\(Q_{m} = \sum_{t=1}^{m} q _{t} \). Then
where
Proof
Using Jensen’s and Schwarz’s inequalities, we observe that
similarly,
The rest of the proof is similar to the proof of Theorems 3 and 5 with suitable changes. □
Corollary 4
Under the hypotheses of Theorem 7, the following inequality holds:
Proof
To prove this result, take \(p_{s} = q_{t} = 1\) for all \(s\geq 1\), \(t \geq 1\), then \(P_{n}=n\), \(Q_{m}=m\) and use the fact that
□
4 Some applications
In this section we try to show the beauty behind our results. We achieve this by utilizing inequality (10) and inequality (19) through substituting \(h(x)\) and \(h^{*}(y)\) by suitable functions. In what follows recall that \(\alpha \geq \beta \geq \frac{1}{2}\).
Example 1
We can derive inequality (3) from inequality (19). To attain this purpose, choose \(h(x)=\frac{x ^{2}}{2}\); then \(h^{*}(y)=\frac{y^{2}}{2}\) for \(x,y \in \mathbb{R}\) (see [10]), then inequality (19) gives
Consequently,
which is inequality (3) as desired.
Example 2
If we take \(h(x)=\frac{x^{s}}{s}\), \(s>1\), then \(h^{*}(y)= \frac{y^{t}}{t}\), \(t>1\), where \(\frac{1}{s}+\frac{1}{t}=1\) and \(x,y \in \mathbb{R}_{+}\) (see [10]), then inequality (10) gives
Clearly,
When \(\beta = \frac{1}{2 \alpha }\), inequality (42) becomes
It is obvious that, if \(\alpha =\beta =1\), inequality (42) yields
If in addition \(s=t=2\), inequality (44) produces
Example 3
We put \(h(x)= e^{x}\) and \(h^{*}(y)= y \log (y) - y\), see [10], in inequality (10) to get
5 Conclusion
Using Fenchel–Young inequality (6) helped in obtaining some inequalities that cover a wide range of Hilbert-type inequalities through choosing the functions \(h(x)\) and \(h^{*}(x)\) suitably.
Although the left-hand sides in inequalities (10) and (22) depend on some parameters (α and β), we obtained upper bounds that are free of those parameters. The effect of these parameters appears on the right-hand side only if the chosen functions have some constant component.
Some results proved in this paper are generalizations of previously proved results. For example, inequality (19) is a generalization of inequality (3).
Integral analogues to all results in this paper can be obtained following the same spirit of the proofs mentioned here with slight changes. For instance, the integral version of Theorem 4 has been proved in [16].
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Hamiaz, A., Abuelela, W. Some new discrete Hilbert’s inequalities involving Fenchel–Legendre transform. J Inequal Appl 2020, 39 (2020). https://doi.org/10.1186/s13660-020-02310-2
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DOI: https://doi.org/10.1186/s13660-020-02310-2