- Research
- Open Access
A new look at classical inequalities involving Banach lattice norms
- Ludmila Nikolova^{1},
- Lars-Erik Persson^{2, 3, 4}Email author and
- Sanja Varošanec^{5}
https://doi.org/10.1186/s13660-017-1576-8
© The Author(s) 2017
- Received: 12 September 2017
- Accepted: 28 November 2017
- Published: 8 December 2017
Abstract
Some classical inequalities are known also in a more general form of Banach lattice norms and/or in continuous forms (i.e., for ‘continuous’ many functions are involved instead of finite many as in the classical situation). The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. We already here contribute by discussing some results of this type and also by deriving some new results related to classical Popoviciu’s, Bellman’s and Beckenbach-Dresher’s inequalities.
Keywords
- inequalities
- continuous forms
- Hölder’s inequality
- Minkowski’s inequality
- Popoviciu’s inequality
- Bellman’s inequality
- Beckenbach-Dresher’s inequality
- Milne’s inequality
- Banach function space
- interpolation of families of spaces
MSC
- 26D10
- 26D15
- 46B70
- 46E15
1 Introduction
Let \((Y, \Sigma, \nu)\) be a σ-finite measure space, and let \(L^{0}(Y)\) denote the space of ν-measurable functions defined and being finite a.e. on Y. A Banach subspace \((E, \Vert \cdot \Vert )\) of \(L^{0}(Y)\) is a Banach lattice (Banach function space) on \((Y, \Sigma, \nu)\) if, for every \(x\in E\), \(y \in L^{0}(Y)\), \(\vert y \vert \leq \vert x \vert \), ν-a.e., it follows that \(y\in E\) and \(\Vert y \Vert \leq \Vert x \Vert \).
Some classical inequalities are known to hold also in the frame of such Banach lattice norms. See, for example, [1] and [2].
It is also known that some classical inequalities for finite many functions (like those of Hölder and Minkowski) can be generalized to hold for continuous (infinitely) many functions. For such results in \(L_{p}\) and \(l_{p}\) -spaces, we refer the reader to the recent article [3] and the references therein. We proved there the continuous versions of Popoviciu’s and Bellman’s inequalities.
However, there exists a generalization of Hölder’s inequality in both of these directions simultaneously, see [4] and also Lemma 2.2.
The main aim of this paper is to initiate a more consequent study of classical inequalities in this more general frame. Some known results of this type which we need in this paper can be found in our Section 2. We now shortly discuss some elementary forms of the inequalities we consider to generalize as new contributions in this paper.
We assume that \(a_{i}\), \(b_{i}\), \(i=1,2,\ldots,n\), are positive numbers, \(c_{1}\), \(c_{2}\) are positive numbers and f and g are positive functions on \((Y, \Sigma, \nu)\).
For more general forms, see, e.g., Theorem 3.3 which in particular shows that ‘>’ in the assumptions of (1.1) and (1.2) can be replaced by ‘≥’. Some continuous forms of (1.1) and (1.2) were recently proved in [3].
In Section 3 of this paper we present, prove and apply our main results concerning Popoviciu’s inequality (see Theorems 3.1 and 3.3).
Our main results related to this inequality are proved and discussed in Section 4. We remark that obviously Popoviciu’s and Bellman’s inequalities may be regarded as a type of reversed inequalities of Hölder’s and Minkowski’s inequalities, respectively.
In Section 5 of the present paper, we derive a new version of (1.5) which is both ‘continuous’ (containing infinitely many functions, e.g., sequences) and involving Banach lattice norms (see Theorem 5.1). Moreover, we also derive a type of reversed inequality of the same general form (see Theorem 5.4). Finally, Section 6 is reserved for some concluding remarks and results. Especially, we present new Popoviciu’s inequality in the case of infinite interpolation families (see Theorem 6.1), and the connection to Milne’s inequality is pointed out (see Section 6.2).
2 Preliminaries
It is known that if \(\Vert \cdot \Vert _{E}\) is a Banach function norm, then \(\Vert f(x,\cdot) \Vert _{E}\) need not be a measurable function. But it is also known that if E has the Fatou property, then indeed \(\Vert f(x,\cdot) \Vert _{E}\) is measurable (see [8]). Therefore, for simplicity, we assume that the considered Banach function spaces have the Fatou property. It is also known that in this situation E is a perfect space, i.e., \(E=E''\), where \(E''\) denotes the second associate space of E.
We need the following simple generalization of Hölder’s inequality.
Lemma 2.1
A simple proof of this lemma in an even general symmetric form can be found in [1], p. 369.
We also need the following more general form of Hölder’s inequality (both continuous and involving Banach function norms).
Lemma 2.2
A proof of this result can be found in [4].
We note that (2.3) is an inequality between generalized geometric means. We also need the following inequality.
Lemma 2.3
See, e.g., [9]. Another proof can be done by just using reversed form of suitable generalizations of Beckenbach-Dresher’s inequality with \(p=\alpha/ (\alpha-\beta) \) and letting \(\alpha,\beta\rightarrow 0\) in the corresponding generalized Gini-means \(G(\alpha,\beta)\).
We also need the following analogous version of Minkowski’s inequality.
Lemma 2.4
Since E has the Fatou property and \(p\geq1\), we have that \(E^{p}\) has the Fatou property, i.e., it is a perfect space, and the proof can be found in [10], Chapter 2. Note that in the case \(E=L_{1}(Y)\) this is just the classical integral Minkowski inequality.
The smallest M satisfying the corresponding inequality is called constant of p-convexity, respectively, of q-concavity.
Lemma 2.5
Proof
3 Popoviciu type inequalities involving Banach function norms
Our first main result reads as follows.
Theorem 3.1
Proof
Example 3.2
Next we state the following complementary result.
Theorem 3.3
- (a)Let \(p\geq1\). If \(c_{1}^{p}- \Vert f^{p} \Vert _{E}\geq0\), \(c_{2}^{q}- \Vert g^{q} \Vert _{E}\geq0\), then$$ c_{1}c_{2}- \Vert fg \Vert _{E} - \bigl(c_{1}^{p}- \bigl\Vert f^{p} \bigr\Vert _{E}\bigr)^{1/p} \bigl(c_{2}^{q}- \bigl\Vert g^{q} \bigr\Vert _{E}\bigr)^{1/q}\geq0. $$(3.3)
- (b)
Let \(0< p<1\). If \(\Vert g^{q} \Vert _{E}>0\), \(c_{2}^{q}- \Vert g^{q} \Vert _{E}>0\), then reverse inequality (3.3) holds.
- (c)
Let \(p<0\). If \(\Vert f^{p} \Vert >0\), \(c_{1}^{p}- \Vert f^{p} \Vert _{E} >0\), then reverse inequality (3.3) holds.
Proof
(a) Let \(p,q>0\). Let \(c_{1}^{p}- \Vert f^{p} \Vert _{E}\), \(g_{0}^{q}- \Vert g^{q} \Vert _{E}\) be strictly positive. Let \(X_{1} \cup X_{2}=[0,b)\), let \(X_{1} \cap X_{2}\) be empty, \(\int_{X_{1}}u(x) \,dx =\frac {1}{p}\), \(\int_{X_{2}}u(x) \,dx =\frac{1}{q}\). We can get the result like a corollary from inequality (3.2), by taking \(f_{0}(x)=c_{1}^{p}\) for \(x\in X_{1}\) and \(f_{0}(x)=c_{2}^{q}\) and \(f(x,y)=f^{p}(y)\) for \(x\in X_{1}\), \(f(x,y)=g^{q}(y)\) for \(x\in X_{2}\).
(c) The case \(p<0\) can be proved similarly (just interchange the roles of f and g and p and q, respectively). □
Remark 3.4
Note that Theorem 3.3 in particular means that inequalities (1.1) and (1.2) hold also if ‘>’ in these inequalities are replaced by ‘≥’.
We also state a generalization of Theorem 3.3(a).
Corollary 3.5
Proof
The proof is complete. □
Remark 3.6
If we compare inequalities (3.4) and (3.3), we can see that in the case \(0< r,s<1\) inequality (3.4) is better than inequality (3.3). Moreover, since in the case \(r,s>1\) inequality (3.6) holds in the reversed direction when \(r>1\), in the case \(r,s>1\) inequality (3.3) is stronger than inequality (3.4).
2. Note that here we do not need the condition \(E''=E\) because of Remark after Theorem 4.1 from [4].
4 Bellman type inequalities involving Banach function norms
Our first main result in this case reads as follows.
Theorem 4.1
If E is 1-concave with constant of concavity 1, \(0< p<1\) or \(p<0\) and \(\Vert f^{p}(x,\cdot) \Vert _{E}>0\), then inequality (4.1) holds in the reverse direction.
Proof
If \(0< p<1\), then first we use reverse to inequality (4.2) and then instead of (2.5) we use (2.8) for \(M=1\).
The proof in the case \(p<0\) is similar. □
Remark 4.2
If \(E=L_{1}\), then we get the result of the first part of the continuous Bellman inequality for \(p\geq1\) proved in [3], Theorem 3.1.
Next, we state the following Bellman type inequalities.
Theorem 4.3
- (a)Let E be a Banach function space, let \(f,g>0\), \(p\geq1\), \(c_{1}^{p}- \Vert f^{p} \Vert _{E}\geq0\), \(c_{2}^{p}- \Vert g^{p} \Vert _{E}\geq0\). Then$$ \bigl(\bigl(c_{1}^{p}- \bigl\Vert f^{p} \bigr\Vert _{E}\bigr)^{1/p} + \bigl(c_{2}^{p}- \bigl\Vert g^{p} \bigr\Vert _{E}\bigr)^{1/p}\bigr)^{p} \leq (c_{1}+c_{2})^{p}- \bigl\Vert (f+g)^{p} \bigr\Vert _{E}. $$(4.4)
- (b)
If E is an arbitrary 1-concave lattice with constant of concavity 1, \(0< p<1\) or \(p<0\) and \(c_{1},f>0\), \(c_{1}^{p}- \Vert f^{p} \Vert _{E}>0\), \(c_{2}>0\), \(g>0\), \(c_{2}^{p}- \Vert g^{p} \Vert _{E}>0\), then reverse inequality (4.4) holds.
Proof
(b) All inequalities above hold in the reverse direction in this case, and the proof follows by just doing obvious modifications of the proof of (a). □
5 Direct and reverse Beckenbach-Dresher type inequalities involving Banach function norms
The following result concerning Beckenbach-Dresher’s inequality was announced in [13]. For completeness, we give here also the proof.
Theorem 5.1
If \(u>1\), \(q\leq1\leq p\), \(q\neq0\) and F is 1-concave with constant of concavity equal to N, then inequality (5.1) holds in the reverse direction with \(C=N^{u-1}\).
If \(u<0\), \(p \leq1\leq q\), \(p\neq0\) and E is 1-concave with constant of concavity equal to M, then inequality (5.1) holds in the reverse direction with \(C=M^{-u}\).
Proof
In the proof we will use (2.5), (2.8) and Hölder’s or reverse Hölder’s inequalities.
Remark 5.2
Remark 5.3
Next we state a kind of reverse version of Theorem 5.1, reversed in the same way as Popoviciu’s and Bellman’s inequalities may be regarded as reversed versions of Hölder’s and Minkowski’s inequalities, respectively.
Theorem 5.4
Let \(f_{0}(x)> \Vert f(x,\cdot) \Vert _{E^{p}}\) for all \(x\in X\), let \(g(x,z)\) be a positive measurable function on \(X\times Z\) and assume that \(g_{0}(x)\) is a function on X such that \(g_{0}(x)> \Vert g(x,\cdot) \Vert _{F^{q}}\) for all \(x\in X\), where E is a Banach function space on Y for all \(x\in X\) with the Fatou property and F is a Banach function space on Z for all \(x\in X\) with the Fatou property.
If \(u< 0\), \(0< q\leq1\) and \(p\geq1 \) or \(p<0\), then reverse inequality (5.3) holds.
In the cases when \(p<1\), an additional condition on the function space E is that it should be concave with constant of concavity 1; in the cases when \(q<1\), an additional condition on the function space F is that it should be concave with constant of concavity 1.
Proof
Let now \(u\geq1\). If \(0< p\leq1\), then inequality (1) holds in the reverse direction and, therefore, since \(\frac{u}{p}>0\), also (2) holds in the reverse direction. If \(q\geq1\), then inequality (3) holds, and since \(\frac{1-u}{q}\leq 0\) in this case, we conclude that (4) holds in the reverse direction. In the case \(q<0\), inequality (3) holds in the reverse direction, but since \(\frac{1-u}{q}\geq0\), inequality (4) holds in the reverse direction also in this case. The second statement thus follows by using reverse Hölder’s inequality and arguing as in the proof of the first case.
Finally, let \(u<0\). If \(p\geq1\), then again by Theorem 4.1 we have that inequality (1) holds, and since \(\frac {u}{p}<0\) in this case, we find that inequality (2) holds in the reverse direction. Symmetrically, if \(p<0\), inequality (1) holds in the reverse direction, but since \(\frac{u}{p}>0\) in this case, still inequality (2) holds in the reverse direction.
If \(0< q\leq1\), then inequality (3) yields in the reverse direction, and because \(\frac{1-u}{q}\geq0\) in this case, also (4) holds in the reverse direction. The third case is thus proved by just using reverse Hölder’s inequality and arguing as in the first two cases. The proof is complete. □
6 Concluding results
6.1 Popoviciu type result in the case of infinite interpolation families
The result of this subsection was announced in [13] but here we give all the details.
Let D be a suitable simply connected domain in the complex plane with boundary Γ and \(B(\gamma)\in\Gamma\) be an interpolation family on Γ in the sense of [15]. Let for simplicity \(\Gamma=\{ \vert z \vert =1\}\) and \(D=\{ \vert z \vert <1\}\). When we speak about interpolation in the families of Banach spaces (complex or real), we are in the situation when the actual family of Banach spaces is indexed by the points of the unit circle for simplicity \(\Gamma=\{ \vert z \vert =1\}\) in the complex plane, while the interpolation spaces are labeled by the points of the unit disk \(D=\{ \vert z \vert <1\}\). The authors of [15] construct, for each \(z_{0} \in D\), a new space \(B [z_{0}]\), which consists of the values \(f(z_{0})\) at \(z_{0}\) of certain analytic vector-valued functions \(f(z)\) on D whose boundary values \(f(\gamma) \in B(\gamma)\) for a.e. \(\gamma\in\Gamma\), and \(\Vert f \Vert _{F(\gamma)}=\operatorname{ess}\sup_{\gamma} \Vert f(\gamma) \Vert _{B(\gamma)}<\infty\). The space \(B [z_{0}]\) has an interpolation property, i.e., if a linear operator T is bounded on each \(B(\gamma)\) with norm \(M(\gamma)\) and also bounded on a certain space U containing each \(B(\gamma)\), then T is also bounded on \(B [z_{0}]\) with norm \(M(z_{0})\), which can be estimated in terms of the function \(M(\gamma)\). A variant of the construction was suggested independently in [16].
In such terms we are now ready to formulate the following general Popoviciu type inequality.
Theorem 6.1
Proof
6.2 Connection to Milne’s inequality
It is easy to see that this inequality is stronger than (6.2).
Declarations
Acknowledgements
The research of Ludmila Nikolova was partially supported by the Sofia University SRF under contract 80.10-120/2017.
Authors’ contributions
All authors have worked with all parts of this paper on an equal basis. All authors have approved the paper in this final form.
Competing interests
All authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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