Some Reverse Inequalities for Scalar Birkhoff Weak Integrable Functions

Abstract

The Minkowski and Hölder inequalities play an important role in many areas of pure and applied mathematics, such as Convex Analysis, Probabilities, Control Theory, Fixed Point theorems, and Mathematical Economics. Also, non-additive measures, non-additive integrals and set-valued integrals are useful tools in several areas of theoretical and applied mathematics. In this paper we present and prove some Hölder and Minkowski inequality (or reverse inequality) types obtained for Birkhoff weak integrable functions with respect to a non-additive measure. Then, we apply these results to the interval-valued case.

1. Introduction

The classical Hölder and Minkowski inequalities are among the most powerful tools in modern mathematical analysis. Their influence extends from functional and convex analysis [1,2] to probability theory, control theory, optimization, and mathematical economics [3,4,5]. In particular, these inequalities characterize the geometry of $L^p$-spaces and have numerous implications in the study of convergence, compactness, and stability of solutions to functional and integral equations [6,7].

In recent years, there has been a growing interest in generalizing such fundamental results to the setting of non-additive measures and non-linear integrals motivated by the increasing role of uncertainty modeling and multi-criteria decision analysis. Non-additive frameworks, including fuzzy, submodular, or monotone set functions, offer a natural mathematical foundation for problems where the additivity assumption is no longer realistic or desirable. These ideas have found applications in computer science, artificial intelligence, image analysis, decision theory, and statistics. Numerous examples are presented in [8,9], illustrating the necessity of not assuming countable additivity; there are also applications in the field of medical diagnostics [10], or image analysis [11,12,13], or [14] for a review of the fractal image coding literature, or in the search of equilibria [15].

When $0<p<1$, the classical $L^p$-spaces cease to be normed, and the corresponding inequalities take a reversed or quasi-norm form. Investigating reverse Hölder and Minkowski-type inequalities in this sublinear regime requires nontrivial modifications of the standard additive framework. In particular, the Birkhoff weak integral, which is an extension of the Lebesgue integral to vector-valued or non-additive settings, introduced and studied in [16,17], provides a suitable tool to explore this situation.

Moreover, reverse inequalities are not only a purely mathematical interest but also provide useful tools in applied mathematics, physics, engineering, and other scientific fields where classical normed-space assumptions may fail. There are examples of their applications in analysis on metric measure spaces and geometric analysis on complex manifolds (see [18]); reverse inequalities have been applied to modified unified generalized fractional integral operators. This framework is relevant in modeling viscoelastic media, anomalous diffusion, and systems with memory effects (see for example, [5,6,7,19,20,21]) or in matrix analysis and Operator Theory: here reverse Hölder and Minkowski inequalities have been extended to matrices and quasinorms (“antinorms” for $p<1$). These results have applications in signal processing, quantum information theory, and other areas where operators or matrices are not naturally normed.

The aim of this paper is to establish new inequalities and reverse forms of Hölder and Minkowski inequalities for Birkhoff weak integrable scalar functions with respect to non-additive measures. These results are not only novel in the general non-additive context, but also remain new even in finitely additive settings, thus extending the scope of classical inequalities to broader integration theories. Such developments enrich the structure of generalized $L^p$-type spaces and open perspectives for applications in information aggregation, fuzzy control, and mathematical economics, where non-additive integration plays a key conceptual role.

The paper is organized as follows: after the Introduction, Section 2 is devoted to preliminaries. Section 3 contains some definitions, basic results regarding the Birkhoff weak integrability and we establish some inequalities for the Birkhoff weak integral of a real function relative to a non-additive measure, such as reverses of Hölder and Minkowski inequalities and other types of inequalities. Some applications to interval-valued or vector-valued integrals are presented in Section 4. Finally, we provide a Conclusion section.

2. Preliminaries

Let T be a non-empty set and $\mathcal{E}$ a $\sigma$-algebra of subsets of T. The integrability we consider in this paper is related to the partitions of the whole space T. We begin with some definitions on set functions defined on $\mathcal{E}$ and on partitions of T. As usual, $\mathbb{N}$ denotes the integer numbers, starting from 1.

Definition 1.

A set function $\nu :\mathcal{E}\to [0,\infty )$, with $\nu (\varnothing )=0$, is called:

(i) 

subadditive if $\nu (A\cup B)\le \nu \left(A\right)+\nu \left(B\right)$, for every disjoint sets $A,B\in \mathcal{E}$;

(ii) 

continuous from below if for every ${\left(B_n\right)}_{n\in \mathbb{N}}\subset \mathcal{E}$, with $B_n\subset B_{n+1}$, for all $n\in \mathbb{N}$:

$$\nu \left({\cup }_{n=1}^{\infty }{B}_n\right)=\underset{n\to \infty }{lim}\nu \left(B_n\right).$$

We denote by ${\mathcal{M}}_s$ the class of set functions $\nu :\mathcal{E}\to [0,\infty )$, with $\nu (\varnothing )=0$, which are subadditive.

Example 1.

Let $T=\mathbb{N}$ and $\mathcal{E}$ be its power set.

(i) 

Let $\nu :\mathcal{P}\left(\mathbb{N}\right)\to [0,1]$ defined, for every $A\subset \mathbb{N}$, by $\nu \left(A\right):={sup}_{n\in A}{2}^{-n},$ if $A\ne \varnothing$ and $\nu (\varnothing )=0.$ Obviously if $A\subseteq B$, then $\nu \left(A\right)={sup}_{n\in A}{2}^{-n}\le {sup}_{n\in B}{2}^{-n}=\nu \left(B\right).$ Moreover, for all $A,B\subseteq \mathbb{N}$,

$$\nu (A\cup B)=\underset{n\in A\cup B}{sup}{2}^{-n}=max\left(\underset{n\in A}{sup}{2}^{-n},\underset{n\in B}{sup}{2}^{-n}\right)=max\{\nu \left(A\right),\nu \left(B\right)\}.$$

Since $max\{x,y\}\le x+y$ for all positive $x,y$, we get $\nu (A\cup B)\le \nu \left(A\right)+\nu \left(B\right).$ Finally, ν is continuous from below, in fact, let ${\left(A_k\right)}_{k\in \mathbb{N}}$ be an increasing sequence of subsets of $\mathbb{N}$, and let $A={\bigcup }_k{A}_k$. Then,

$$\nu \left(A\right)=\underset{n\in A}{sup}{2}^{-n}=\underset{k}{sup}\underset{n\in A_k}{sup}{2}^{-n}=\underset{k}{sup}\nu \left(A_k\right)=\underset{k\to \infty }{lim}\nu \left(A_k\right).$$

The function ν is not additive in general; for instance, $\nu \left(\left\{1\right\}\right)={\displaystyle \frac{1}{2}}$, $\nu \left(\left\{2\right\}\right)={\displaystyle \frac{1}{4}}$, but $\nu \left(\{1,2\}\right)={\displaystyle \frac{1}{2}}\ne {\displaystyle \frac{3}{4}}$. Thus, ν is a simple example of a monotone, subadditive, continuous from below set function.

(ii) 

For every $A\subseteq \mathbb{N}$ define $\nu (\varnothing )=0$ and $\nu \left(A\right):=min\{1,{\sum }_{n\in A}{\displaystyle \frac{1}{n}}\}$, if $A\ne \varnothing$.

By construction $\nu \left(A\right)\in [0,1]$ for every $A\subseteq \mathbb{N}$. If $\varnothing \ne A\subseteq B$ then $\nu \left(A\right)=min\{1,{\sum }_{n\in A}{\displaystyle \frac{1}{n}}\}\le min\{1,{\sum }_{n\in B}{\displaystyle \frac{1}{n}}\}=\nu \left(B\right).$

For any nonvoid $A,B\subseteq \mathbb{N}$ set $s_A:={\sum }_{n\in A}{\displaystyle \frac{1}{n}}$ and $s_B:={\sum }_{n\in B}{\displaystyle \frac{1}{n}}$. Then, $s_{A\cup B}\le s_A+s_B$, and $min\{1,s_{A\cup B}\}\le min\{1,s_A+s_B\}$ holds. Therefore,

$$\begin{array}{ccc}\hfill \nu (A\cup B)& =& min\{1,\sum _{n\in A\cup B}{\displaystyle \frac{1}{n}}\}\le min\{1,\sum _{n\in A}{\displaystyle \frac{1}{n}}+\sum _{n\in B}{\displaystyle \frac{1}{n}}\}\le \nu \left(A\right)+\nu \left(B\right),\hfill \end{array}$$

so ν is subadditive. Finally, let ${\left(A_k\right)}_{k\in \mathbb{N}}$ be an increasing sequence of subsets of $\mathbb{N}$ (i.e., for every $k\in \mathbb{N}$, $A_k\subseteq A_{k+1}$) and set $A={\bigcup }_k{A}_k$. Then, the sequence ${\left({\sum }_{n\in A_k}{\displaystyle \frac{1}{n}}\right)}_k$ is nondecreasing with limit $s_A={\sum }_{n\in A}{\displaystyle \frac{1}{n}}$ (possibly $+\infty$). Since the map $x\mapsto min\{1,x\}$ is continuous on $[0,\infty ]$, we get

$$\nu \left(A\right)=min\{1,s_A\}=\underset{k\to \infty }{lim}min\{1,s_{A_k}\}=\underset{k\to \infty }{lim}\nu \left(A_k\right),$$

so ν is continuous from below.

Definition 2.

A property $\left(P\right)$ holds ν-almost everywhere (denoted by ν-a.e.) if there exists $B\in \mathcal{E}$, with $\nu \left(B\right)=0$, so that the property $\left(P\right)$ is valid on $T\setminus B.$

Definition 3.

Suppose $card\left(T\right)\ge {\aleph }_0$ (where $card\left(T\right)$ is the cardinality of T).

(i) 

A countable family of nonvoid sets $P={\left\{B_n\right\}}_{n\in \mathbb{N}}$$\subset \mathcal{E}$ such that $\bigcup _{n\in \mathbb{N}}{B}_n=T$ with $B_i\cap B_j=\varnothing ,$ when $i\ne j,i,j\in \mathbb{N}$, is called a (measurable) countable partition of T.

Denote by $\mathcal{C}$ the set of all countable partitions of T and by ${\mathcal{C}}_B$ the set of countable partitions of $B\in \mathcal{E}$.

(ii) 

For every P and $P^{\prime }\in \mathcal{C}$, $P^{\prime }$ is called finer than P (denoted by $P^{\prime }⩾P$ or $P⩽P^{\prime }$) if every set of $P^{\prime }$ is included in some set of P.

(iii) 

For every P and $P^{\prime }\in \mathcal{C}$, $P=\left\{B_n\right\},P^{\prime }=\left\{C_m\right\}$, the common refinement of P and $P^{\prime }$ is defined to be the countable partition $\{B_n\cap C_m\}$, denoted by $P\wedge P^{\prime }$.

3. Birkhoff Weak Integrability and Related Inequalities

This section contains definitions and basic results on the Birkhoff weak integrability and new results on reverse inequalities. In the sequel T is a non-empty set, with $\mathrm{card}\left(T\right)\ge {\aleph }_0$, $\mathcal{E}$ is a $\sigma$-algebra of subsets of T and $\nu :\mathcal{E}\to [0,\infty )$ is a non-negative set function, such that $\nu (\varnothing )=0.$

We recall the following definition:

Definition 4

([22]). A function $u:T\to \mathbb{R}$ is said to be Birkhoff weakly integrable on T with respect to ν ($B_w-\nu$-integrable), if $b\in \mathbb{R}$ exists such that for every $\epsilon >0$, there are $P_{\epsilon }\in \mathcal{C}$ and $n_{\epsilon }\in \mathbb{N}$ such that for every $P\in \mathcal{C}$, $P={\left(B_n\right)}_{n\in \mathbb{N}}$, $P⩾P_{\epsilon }$ and every $t_n\in B_n,n\in \mathbb{N}$:

$$|\sum _{k=1}^{n}u\left(t_k\right)\nu \left(B_k\right)-b|<\epsilon ,for every n\ge n_{\epsilon }.$$

b is denoted by $\left(B_w\right){\int }_{T}ud\nu$ or simply ${\int }_{T}ud\nu$ and is called the Birkhoff weak integral of u on T with respect to ν.

We denote by $B_w(\nu ,T)$ the family of all $B_w-\nu$-integrable functions on T. The Birkhoff weak integrability on every set $E\in \mathcal{E}$ is defined in the usual way.

In particular, by [16] (Theorem 4.2) u is $B_w-\nu$-integrable on $E\in \mathcal{E}$ if and only if $u·{\mathbf{1}}_E\in B_w(\nu ,T)$ and $\left(B_w\right){\int }_{E}ud\nu =\left(B_w\right){\int }_{T}u·{\mathbf{1}}_{E}d\nu$, where ${\mathbf{1}}_E$ is the characteristic function of E.

With the symbol $B_w\left(\nu \right)$ we denote the family of scalar functions that are $B_w-\nu$-integrable on every $E\in \mathcal{E}$. The family of $B_w-\nu$-integrable functions is closed with respect to the order of $\mathbb{R}$, in fact:

Theorem 1

([23] (Corollary 3.3)). Let $u,v:T\to \mathbb{R}$, such that $u,v\in B_w\left(\nu \right)$. Then, $min\{u,v\}$ and $max\{u,v\}$ are in $B_w\left(\nu \right).$

For other results on this topic we refer to [22,23,24,25], from which this research originates. Now, in order to approach the inequalities of Minkowski and Hölder and their reverse inequalities we consider

Definition 5

([23] (Definition 3.5)). A set function $\nu :\mathcal{E}\to [0,\infty )$ is called $\mathcal{E}$-integrable if for all $B\in \mathcal{E}$, ${\mathbf{1}}_B\in B_w(\nu ,T)$.

Obviously, any measure $\nu :\mathcal{E}\to [0,\infty )$ is $\mathcal{E}$-integrable; see, for example, ([23,26,27]). Moreover, if we consider the set functions given in Example 1, we have

Example 2.

As in Example 1, let $T=\mathbb{N}$ and $\mathcal{E}$ be its power set.

(i) 

Let ν as in Example 1(i) and take $u:\mathbb{N}\to \mathbb{R}$. Since, by [22] (Example II),

$$\left(B_w\right){\int }_{T}ud\nu =\sum _{n=1}^{\infty }u\left(n\right)\nu \left(\left\{n\right\}\right)$$

if the involved series is absolutely convergent, we have that, for every $A\subset \mathbb{N}$,

$$\begin{array}{c}\hfill \nu \left(A\right)\le \sum _{n\in A}\nu \left(\left\{n\right\}\right)and\left(B_w\right){\int }_T{\mathbf{1}}_{A}d\nu =\sum _{n\in A}\nu \left(\left\{n\right\}\right)=\sum _{n\in A}{\left({\displaystyle \frac{1}{2}}\right)}^n<+\infty .\end{array}$$

So ν is integrable in the sense of Definition 5.

(ii) 

In general, it is also possible to construct monotone, subadditive, continuous from below set functions that are not integrable. Consider the set function given in Example 1(ii). For each singleton $\left\{n\right\}$ we have $\nu \left(\left\{n\right\}\right)=min\{1,{\displaystyle \frac{1}{n}}\}={\displaystyle \frac{1}{n}}.$ Hence, for every $A\subset \mathbb{N}$ with $\mathrm{card}A={\aleph }_0$, then

$$\nu \left(A\right)<\sum _{n\in A}\nu \left(\left\{n\right\}\right)and\left(B_w\right){\int }_T{\mathbf{1}}_{A}d\nu =\sum _{n\in A}{\displaystyle \frac{1}{n}}=+\infty .$$

From now on, ${\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ is the set of all set functions $\nu :\mathcal{E}\to [0,\infty )$, with $\nu (\varnothing )=0$, which are $\mathcal{E}$-integrable, continuous from below and subadditive.

In general, for the gauge integrals, like Henstock, McShane, Birkhoff integrals, no measurability condition is asked a priori; see, for example, [24,28,29,30,31]. For the study of the inequalities object of this research, we sometimes need the measurability of functions $u:T\to \mathbb{R}$; we will specify when this is necessary. We denote by $\mathcal{F}(T,\mathbb{R})$ the space of all measurable functions from T to $\mathbb{R}$.

It is known that, by [22] (Theorem 5.1), every bounded function that has zero value almost everywhere has null integral, whereas for the converse relation, we need the measurability of integrands, for $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$, by [23] (Theorem 3.7). We give an example showing that, even for a non-negative function u, $\left(B_w\right)\int ud\nu =0$ does not imply $u=0$ $\nu$-a.e, when $\nu \notin {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$.

Example 3.

Let $\nu :\mathcal{P}\left(\mathbb{N}\right)\to \{0,1\}$ be defined by $\nu \left(A\right)=0$ if A is finite, and $\nu \left(A\right)=1$ otherwise. The set function ν is non negative, monotone and subadditive but not continuous from below. In fact, if $A_n=\{1,2,\dots ,n\}$ for every n, then $\nu \left(\mathbb{N}\right)=1\ne {lim}_{k\to \infty }\nu \left(A_k\right)=0.$ Since, for every $n\in \mathbb{N}$, $\nu \left(\right\{n\left\}\right)=0$, then ν is integrable, since for every $A\in \mathcal{P}\left(\mathbb{N}\right)$ $\left(B_w\right){\int }_{\mathbb{N}}{\mathbf{1}}_{A}d\nu ={\sum }_{n\in A}u\left(n\right)\nu \left(\left\{n\right\}\right)=0.$ So ${\mathbf{1}}_{\mathbb{N}}$ has null integral, but it is not zero ν-a.e.

In particular, on countable sets T endowed with the $\sigma$-algebra of all the parts (so every function is measurable) we are able to prove the following.

Proposition 1.

Let T be a countable set and let $\nu :\mathcal{P}\left(T\right)\to [0,+\infty )$ be a set function which is monotone, subadditive, continuous from below and integrable. Let $u:T\to [0,+\infty )$ be such that $\left(B_w\right){\int }_{\mathbb{N}}ud\nu =0.$ Then, $u=0\nu$-almost everywhere.

Proof. 

It is enough to prove it for $T=\mathbb{N}$. Since $\nu$ is integrable and continuous from below, for every nonnegative function $u:\mathbb{N}\to [0,+\infty )$ one has $\left(B_w\right){\int }_{\mathbb{N}}ud\nu ={\sum }_{n=1}^{\infty }u\left(n\right)\nu \left(\left\{n\right\}\right),$ with the series converging in $[0,+\infty ]$. Assume $\left(B_w\right){\int }_{\mathbb{N}}ud\nu =0.$ Because all terms of the series are nonnegative, it follows that $u\left(n\right)\nu \left(\right\{n\left\}\right)=0$ for every $n\in \mathbb{N}.$ Let $E=\{n\in \mathbb{N}:u(n)>0\}.$ For every $n\in E$ we have $\nu \left(\right\{n\left\}\right)=0$. By the subadditivity and the monotonicity of $\nu$, for every finite $F\subseteq E$, it is $\nu \left(F\right)\le {\sum }_{n\in F}\nu \left(\left\{n\right\}\right)=0.$

Since $E={\bigcup }_{k=1}^{\infty }{F}_k$, where $F_k\subseteq E$ is an increasing sequence of finite sets, by continuity from below we obtain $\nu \left(E\right)={lim}_{k\to \infty }\nu \left(F_k\right)=0.$ Therefore, $\nu \left(\right\{n\in \mathbb{N}:u\left(n\right)\ne 0\left\}\right)=0,$ which proves that $u=0$   $\nu$-almost everywhere. □

So, we cannot avoid the measurability of the integrands for general measurable spaces $(T,\mathcal{E})$.

If $p\in (0,\infty )$ and $u:T\to \mathbb{R}$ is a function with ${\left|u\right|}^p\in B_w(\nu ,T)$, we denote, as usual,

$${\parallel u\parallel }_p={\left(\left(B_w\right){\int }_T{\left|u\right|}^{p}d\nu \right)}^{{\displaystyle \frac{1}{p}}}.$$

(1)

Recall that two indices $p,q>0$ are said to be conjugate if $p^{-1}+q^{-1}=1.$ This requires $p>1$, which forces $q>1$ as well. Here, however, we allow $0<p<1$. In this case the condition $p^{-1}+q^{-1}=1$ implies that $q<0$. We will refer to such pairs $p,q$ again as conjugate indices and we will show later that the function $\parallel ·\parallel :B_w(\nu ,T)\to \mathbb{R}$ satisfies the triangle inequality when $p\ge 1$, under suitable conditions, whereas it fails to satisfy it when $0<p<1$. Consequently, in this latter case it is not a norm, and the vector space $B_w^p(\nu ,T)$ is not normed. In ref. [23], the following result was given that shows the inequalities of Hölder and Minkowski.

Theorem 2

([23] (Theorems 3.8 and 3.9)). Let $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and $u,v:T\to \mathbb{R}$ be measurable functions.

(a) 

Let $p,q$ conjugate indices with $p>1$. If ${\left|u\right|}^p{,\left|v\right|}^q,\left|uv\right|\in B_w(\nu ,T)$, then

$${\parallel uv\parallel }_1\le {\parallel u\parallel }_p·{\parallel v\parallel }_q.(\mathrm{H}ö\mathrm{lder}\mathrm{Inequality})$$

(b) 

Let $p\ge 1$. Suppose that ${\left|u\right|}^p{,\left|v\right|}^p{,|u+v|}^p{,\left|u\right|·|u+v|}^{p-1}{,\left|v\right|·|u+v|}^{p-1}\in B_w(\nu ,T)$. Then

$${\parallel u+v\parallel }_p\le {\parallel u\parallel }_p+{\parallel v\parallel }_p.(\mathrm{Minkowski}\mathrm{Inequality})$$

As in the classical case, the inequalities of Hölder and Minkowski are very important in the definition of the norm of the spaces of integrable functions. The following result is a consequence of Theorem 1.

Proposition 2.

Let $u:T\to \mathbb{R}$ be a real function such that $u\in B_w\left(\nu \right)$. Then, $\left|u\right|\in B_w(\nu ,T).$

Remark 1.

$B_w\left(\nu \right)\cap \mathcal{F}(T,\mathbb{R})$ is a vector space. In fact if $u\in B_w\left(\nu \right)\cap \mathcal{F}(T,\mathbb{R})$, then by construction and Proposition 2, for every $E\in \mathcal{E}$, $u, \left|u\right|\in B_w(\nu ,E)$. So if $u,v\in B_w\left(\nu \right)$, then $\alpha u+\beta v\in B_w\left(\nu \right)$ by [22] (Theorems 4.3 and 4.5) for every $\alpha ,\beta \in \mathbb{R}$. Again by Proposition 2, $|\alpha u+\beta v|\in B_w\left(\nu \right)$. Denote by $L_{B_w}^1(\nu ,T)$ the quotient space of $B_w(\nu ,T)\cap \mathcal{F}(T,\mathbb{R})$ with respect to the usual equivalence relation "∼": for every $u,v\in B_w(\nu ,T)\cap \mathcal{F}(T,\mathbb{R}),u\sim v$ if and only if $u=v$ ν -a.e. If $p\in (1,\infty )$, we can define analogously the space $L_{B_w}^p(\nu ,T)$.

Theorem 3.

Suppose $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$. Then, the function ${\parallel ·\parallel }_1$ is a norm on the space $L_{B_w}^1(\nu ,T).$

Proof. 

The proof is analogous to the classic one, using the properties of the Birkhoff weak integral of [22,23]. □

As said before, the Minkowski and Hölder inequalities fail when $0<p<1$. We give now some examples in the non-additive case.

Example 4.

Let $T=\mathbb{N}$ and $\mathcal{E}$ be its power set. Take $p\in (0,1)$ (e.g., $p={\displaystyle \frac{1}{2}}$) and consider $u={\mathbf{1}}_{\left\{1\right\}},v={\mathbf{1}}_{\left\{2\right\}}$. The supports of u and v are disjoint.

(i) 

Let ${\nu }_1$ as in Examples 1(i) and 2(i) and

$$\begin{array}{ccc}& & {\parallel u\parallel }_{p,{\nu }_1}^p={\nu }_1\left(\left\{1\right\}\right)=2^{-1}{\parallel v\parallel }_{p,{\nu }_1}^p={\nu }_1\left(\left\{2\right\}\right)=2^{-2},\hfill \\ & & {\parallel u+v\parallel }_{p,{\nu }_1}^p={\parallel u\parallel }_{p,{\nu }_1}^p+{\parallel v\parallel }_{p,{\nu }_1}^p={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{4}}={\displaystyle \frac{3}{4}}.\hfill \end{array}$$

For $p={\displaystyle \frac{1}{2}}$ we have

$${\left({\displaystyle \frac{3}{4}}\right)}^2={\displaystyle \frac{9}{16}}>{\left({\displaystyle \frac{1}{2}}\right)}^2+{\left({\displaystyle \frac{1}{4}}\right)}^2={\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{16}},$$

so ${\parallel u+v\parallel }_{p,{\nu }_1}\le {\parallel u\parallel }_{p,{\nu }_1}+{\parallel v\parallel }_{p,{\nu }_1}$.

(ii) 

Let ${\nu }_2$ as in Examples 1(ii) and 2(ii). Now

$$\begin{array}{ccc}& & {\parallel u\parallel }_{p,{\nu }_2}^p={\nu }_2\left(\left\{1\right\}\right)={1,\parallel v\parallel }_{p,{\nu }_2}^p={\nu }_2\left(\left\{2\right\}\right)={\displaystyle \frac{1}{2}},{\parallel u+v\parallel }_{p,{\nu }_2}^p={\displaystyle \frac{3}{2}}.\hfill \end{array}$$

For $p={\displaystyle \frac{1}{2}}$ this yields ${\left({\displaystyle \frac{3}{2}}\right)}^2>1^2+{\left({\displaystyle \frac{1}{2}}\right)}^2,$ so again

$${\parallel u+v\parallel }_{p,{\nu }_2}\le {\parallel u\parallel }_{p,{\nu }_2}+{\parallel v\parallel }_{p,{\nu }_2}$$

and then the Minkowski inequality fails.

Remark 2.

The failure of the Minkowski inequality is a consequence of the fact that $p<1$ makes the map $t\mapsto t^p$ concave. What we are able to prove is that Minkowski’s inequality reverses its direction under suitable assumptions when $0<p<1$, and therefore the usual subadditivity of the $L^p$-quasinorm fails and is replaced by a superadditivity inequality.

Regarding concave maps we can also observe that

Lemma 1

([19] (Corollary 2.1)). Let $0<p<1$ be fixed and set $q:={\displaystyle \frac{p}{p-1}}<0.$ Then,

$$bc\ge {\displaystyle \frac{b^p}{p}}+{\displaystyle \frac{c^q}{q}},for allb>0,c>0.$$

We introduce now the main results of this paper: the reverse Hölder and Minkowski inequalities for $0<p<1$ and other inequalities for Birkhoff weak integrable scalar functions.

Theorem 4.

Let $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and $u,v\in \mathcal{F}(T,\mathbb{R})$. Let $p\in (0,1)$ and q is its conjugate. If

(a) 

${\left|uv\right|,\left|u\right|}^p,{\left|v\right|}^q\in B_w(\nu ,T)$ and ${\int }_T{\left|v\right|}^{q}d\nu >0$, then

$${\phantom{a}\parallel uv\parallel }_1\ge {\parallel u\parallel }_p·{\parallel v\parallel }_q.(\mathrm{Reverse}\mathrm{H}ö\mathrm{lder}\mathrm{Inequality})$$

(b) 

$\left(\right|u|+{\left|v\right|)}^p,{\left|u\right|}^p$, ${\left|v\right|}^p,\left|u\right|\left(\right|u|+{\left|v\right|)}^{p-1},\left|v\right|\left(\right|u|+{\left|v\right|)}^{p-1}$$\in B_w(\nu ,T)$, then

$$\phantom{aa}\phantom{a}\parallel \left|u\right|+{\left|v\right|\parallel }_p\ge {\parallel u\parallel }_p+{\parallel v\parallel }_p.(\mathrm{Reverse}\mathrm{Minkowski}\mathrm{Inequality})$$

Proof. 

(a)

If ${\int }_T{\left|u\right|}^{p}d\nu =0$, then by [23] (Theorem 3.7) it results $uv=0$  $\nu -a.e.$ Therefore, the inequality of Theorem 4(a) is true.

Suppose ${\int }_T{\left|u\right|}^{p}d\nu >0$. Let

$$\begin{array}{c}\hfill b=\left|u\right|·{\left({\int }_T{\left|u\right|}^{p}d\nu \right)}^{-{\displaystyle \frac{1}{p}}},c=\left|v\right|·{\left({\int }_T{\left|v\right|}^{q}d\nu \right)}^{-{\displaystyle \frac{1}{q}}}.\end{array}$$

Since for every $b,c\in (0,\infty ),$ it is $bc\ge {\displaystyle \frac{b^p}{p}}+{\displaystyle \frac{c^q}{q}}$, by Lemma 1, then, in our setting, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{\left|uv\right|}{({\int }_T{\left|u\right|}^p{d\nu )}^{\frac{1}{p}}({\int }_T{\left|v\right|}^q{d\nu )}^{\frac{1}{q}}}}\ge {\displaystyle \frac{{\left|u\right|}^p}{p\left({\int }_T\right|u{|}^{p}d\nu )}}+{\displaystyle \frac{{\left|v\right|}^q}{q\left({\int }_T\right|v{|}^{q}d\nu )}}.\end{array}$$

According to [22] (Theorems 5.5 and 6.1) we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\int }_T\left|uv\right|d\nu }{({\int }_T{\left|u\right|}^p{d\nu )}^{\frac{1}{p}}({\int }_T{\left|v\right|}^q{d\nu )}^{\frac{1}{q}}}}& \ge & {\displaystyle \frac{{\int }_T{\left|u\right|}^{p}d\nu }{p\left({\int }_T{\left|u\right|}^{p}d\nu \right)}}+{\displaystyle \frac{{\int }_T{\left|v\right|}^{q}d\nu }{q\left({\int }_T{\left|v\right|}^{q}d\nu \right)}}=1\hfill \end{array}$$

and the assertion holds.

(b)

From Theorem 4(a), it follows that

$$\begin{array}{ccc}\hfill {\int }_T\left(\right|u|+{\left|v\right|)}^{p}d\nu & =& {\int }_T\left(\right|u|+{\left|v\right|)}^{p-1}\left(\right|u|+\left|v\right|)d\nu \hfill \\ & \ge & {\left({\int }_T\left(\right|u|+{\left|v\right|)}^{q(p-1)}d\nu \right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_T{\left|u\right|}^{p}d\nu \right)}^{{\displaystyle \frac{1}{p}}}+\ge \hfill \\ & +& {\left({\int }_T\left(\right|u|+{\left|v\right|)}^{q(p-1)}d\nu \right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_T{\left|v\right|}^{p}d\nu \right)}^{{\displaystyle \frac{1}{p}}}=\hfill \\ & =& {\left({\int }_T\left(\right|u|+{\left|v\right|)}^{p}d\nu \right)}^{{\displaystyle \frac{1}{q}}}{(\parallel u\parallel }_p+{\parallel v\parallel }_p).\hfill \end{array}$$

(2)

Now we divide (2) by ${\displaystyle {\left({\int }_T\left(\right|u|+{\left|v\right|)}^{p}d\nu \right)}^{\frac{1}{q}}}$ and the reverse Minkowski inequality is obtained.

□

Moreover,

Corollary 1.

Let $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and let $u,v\in \mathcal{F}(T,\mathbb{R})$ with $v\left(t\right)\ne 0$ for every $t\in T$. Let $p\in (0,\infty )\setminus \left\{1\right\}$ and let q be its conjugate index. Assume that ${\left|u\right|, \left|v\right|, \left|u\right|}^p{\left|v\right|}^{-p/q}\in B_w(\nu ,T)$. Then, the form of the Hölder inequality depends on the value of p: when $p>1$ we have

$$\begin{array}{c}\hfill {\left({\int }_T\left|u\right|d\nu \right)}^p\le \left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p/q}}}d\nu \right)·{\left({\int }_T\left|v\right|d\nu \right)}^{p/q},\end{array}$$

(3)

whereas for $p\in (0,1)$ the inequality is reversed:

$$\begin{array}{c}\hfill {\left({\int }_T\left|u\right|d\nu \right)}^p\ge \left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p/q}}}d\nu \right)·{\left({\int }_T\left|v\right|d\nu \right)}^{p/q}.\end{array}$$

(4)

Proof. 

Assume $p>1$. Applying Hölder’s inequality, given in Theorem 2(a), for conjugate indices $p,q$ to the functions ${\displaystyle \frac{\left|u\right|}{{\left|v\right|}^{1/q}}}$ and ${\left|v\right|}^{1/q}.$ Since, by hypotheses, ${\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p/q}}},\left|v\right|\in B_w(\nu ,T),$ we obtain

$${\int }_T\left|u\right|d\nu \le {\left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p/q}}}d\nu \right)}^{1/p}{\left({\int }_T\left|v\right|d\nu \right)}^{1/q},$$

and raising both sides to the power p yields

$${\left({\int }_T\left|u\right|d\nu \right)}^p\le \left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p/q}}}d\nu \right){\left({\int }_T\left|v\right|d\nu \right)}^{p/q}.$$

Applying Theorem 4(a) to the same functions, when $0<p<1$, we obtain the reverse inequality. □

Remark 3.

Observe that, when $p>1$, $\left|u\right|\in B_w(\nu ,T)$ in Corollary 1, is a consequence of the two others integrability assumptions via the inequality we prove. The hypothesis plays a role in the case $0<p<1$.

Moreover, by Corollary 1, we obtain the following inequalities:

$$\begin{array}{ccc}& & {\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p-1}}}d\nu \ge {\displaystyle \frac{{\left({\int }_T\left|u\right|d\nu \right)}^p}{{\left({\int }_T\left|v\right|d\nu \right)}^{p-1}}},for every p>1;\hfill \\ & & {\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p-1}}}d\nu \le {\displaystyle \frac{{\left({\int }_T\left|u\right|d\nu \right)}^p}{{\left({\int }_T\left|v\right|d\nu \right)}^{p-1}}},for every p\in (0,1).\hfill \end{array}$$

In fact for $p>1$, taking ${\displaystyle \frac{\left|u\right|}{{\left|v\right|}^{1/q}}},{\left|v\right|}^{1/q}$, in (3), we have

$${\int }_T\left|u\right|d\nu \le {\left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p-1}}}d\nu \right)}^{1/p}{\left({\int }_T\left|v\right|d\nu \right)}^{1/q}$$

and then

$${\left({\int }_T\left|u\right|d\nu \right)}^p\le \left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p-1}}}d\nu \right){\left({\int }_T\left|v\right|d\nu \right)}^{p-1},$$

While, for $0<p<1$, applying (4), we obtain

$${\int }_T\left|u\right|d\nu \ge {\left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p-1}}}d\nu \right)}^{1/p}{\left({\int }_T\left|v\right|d\nu \right)}^{1/q}$$

and so

$${\left({\int }_T\left|u\right|d\nu \right)}^p\ge \left({\int }_T{\displaystyle \frac{{\left|u\right|}^p}{{\left|v\right|}^{p-1}}}d\nu \right){\left({\int }_T\left|v\right|d\nu \right)}^{p-1},$$

Finally,

Theorem 5.

Let $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and conjugate indices $p,q\in (1,\infty )$. Suppose that $u,v:T\to (0,\infty )$ are measurable functions and that there exist $\alpha ,\beta \in (0,\infty )$ such that

(a) 

$\alpha \le {\displaystyle \frac{u\left(t\right)}{v\left(t\right)}}\le \beta$, for every $t\in T.$ If $u,v,u^{{\displaystyle \frac{1}{p}}}{v}^{{\displaystyle \frac{1}{q}}}\in B_w(\nu ,T)$, then

$$\begin{array}{c}\hfill {\left({\int }_{T}ud\nu \right)}^{{\displaystyle \frac{1}{p}}}·{\left({\int }_{T}vd\nu \right)}^{{\displaystyle \frac{1}{q}}}\le {\left({\displaystyle \frac{\beta }{\alpha }}\right)}^{{\displaystyle \frac{1}{pq}}}·{\int }_T{u}^{{\displaystyle \frac{1}{p}}}{v}^{{\displaystyle \frac{1}{q}}}d\nu .\end{array}$$

(b) 

$\alpha \le {\displaystyle \frac{u^p\left(t\right)}{v^q\left(t\right)}}\le \beta$, for every $t\in T.$ If $uv$, $u^p$, $v^q\in B_w(\nu ,T)$, then

$$\begin{array}{c}\hfill {\left({\int }_T{u}^{p}d\nu \right)}^{{\displaystyle \frac{1}{p}}}·{\left({\int }_T{v}^{q}d\nu \right)}^{{\displaystyle \frac{1}{q}}}\le {\left({\displaystyle \frac{\beta }{\alpha }}\right)}^{{\displaystyle \frac{1}{pq}}}·{\int }_{T}uvd\nu .\end{array}$$

Proof. 

(a)

For every $t\in T$, it is ${\displaystyle \frac{u\left(t\right)}{v\left(t\right)}}\le \beta$; therefore, $v^{{\displaystyle \frac{1}{q}}}\left(t\right)\ge {\beta }^{-{\displaystyle \frac{1}{q}}}{u}^{{\displaystyle \frac{1}{q}}}\left(t\right).$

Then we have, for every $t\in T$,

$$u^{{\displaystyle \frac{1}{p}}}\left(t\right)v^{{\displaystyle \frac{1}{q}}}\left(t\right)\ge {\beta }^{-{\displaystyle \frac{1}{q}}}{u}^{{\displaystyle \frac{1}{p}}}\left(t\right)u^{{\displaystyle \frac{1}{q}}}\left(t\right)={\beta }^{-{\displaystyle \frac{1}{q}}}u\left(t\right).$$

By [22] (Theorems 5.5 and 6.1) it follows that

$${\left({\int }_T{u}^{{\displaystyle \frac{1}{p}}}{v}^{{\displaystyle \frac{1}{q}}}d\nu \right)}^{{\displaystyle \frac{1}{p}}}\ge {\beta }^{-{\displaystyle \frac{1}{pq}}}{\left({\int }_{T}ud\nu \right)}^{{\displaystyle \frac{1}{p}}}.$$

(5)

Since ${\displaystyle \frac{u\left(t\right)}{v\left(t\right)}}\ge \alpha$, for every $t\in T,$ we have $u^{{\displaystyle \frac{1}{p}}}\left(t\right)\ge {\alpha }^{{\displaystyle \frac{1}{p}}}{v}^{{\displaystyle \frac{1}{p}}}\left(t\right)$ and

$$u^{{\displaystyle \frac{1}{p}}}\left(t\right)v^{{\displaystyle \frac{1}{q}}}\left(t\right)\ge {\alpha }^{{\displaystyle \frac{1}{p}}}{v}^{{\displaystyle \frac{1}{p}}}\left(t\right)v^{{\displaystyle \frac{1}{q}}}\left(t\right)={\alpha }^{{\displaystyle \frac{1}{p}}}v\left(t\right),for every t\in T.$$

By [22] (Theorems 5.5 and 6.1), it results

$${\left({\int }_T{u}^{{\displaystyle \frac{1}{p}}}{v}^{{\displaystyle \frac{1}{q}}}d\nu \right)}^{{\displaystyle \frac{1}{q}}}\ge {\alpha }^{{\displaystyle \frac{1}{pq}}}{\left({\int }_{T}vd\nu \right)}^{{\displaystyle \frac{1}{q}}}.$$

(6)

According to (5) and (6), the inequality given in Theorem 5(a) follows.

(b)

This holds if we consider $u^p$ and $v^q$ instead of u and v in Theorem 5(a).

□

4. Applications

In this section we quote some applications and some future fields of research.

4.1. Vector-Valued Case

We can extend our results to vector functions $u:T\to X$ where X is a Banach space. The definition of $B_w-\nu$ integrability is the same as in Definition 4, where we consider the ${\parallel ·\parallel }_X$ instead of $|·|$. In this case we will use the symbol $B_w^X(\nu ,T)$.

Some results have already been obtained concerning integrability and convergence results in [16,22,23].

Let X be a Banach space, $p\in (0,+\infty )$ and $u:T\to X$ be a function satisfying the condition ${\parallel u\parallel }_X^p\in B_w(\nu ,T)$. Denote

$${\parallel u\parallel }_{X,p}={\left({\int }_T{\parallel u\parallel }_X^{p}d\nu \right)}^{1/p}.$$

The following result can be obtained.

Theorem 6.

Consider $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and $u,v:T\to X$ measurable functions, $p\in (0,1)$ and its conjugate q. Suppose

(a) 

${\parallel u\parallel }_X·{\parallel v\parallel }_X{, \parallel u\parallel }_X^p, {\parallel v\parallel }_X^q\in B_w(\nu ,T)$ and ${\int }_T{\parallel v\parallel }_X^{q}d\nu >0$, then

$$\phantom{a}{\int }_T{\parallel u\parallel }_X·{\parallel v\parallel }_{X}d\nu \ge {\parallel u\parallel }_{X,p}·{\parallel v\parallel }_{X,q}.(\mathrm{Reverse}\mathrm{H}ö\mathrm{lder}\mathrm{Inequality})$$

(b) 

${(\parallel u\parallel }_X+{\parallel v\parallel }_X{)}^p, {\parallel u\parallel }_X^p·{\parallel v\parallel }_X^p, {\parallel u\parallel }_X{(\parallel u\parallel }_X+{\parallel v\parallel }_X{)}^{p-1}, {\parallel v\parallel }_X{(\parallel u\parallel }_X+{\parallel v\parallel }_X{)}^{p-1}\in B_w(\nu ,T)$, then

$${\phantom{aa}\phantom{a}|\parallel u\parallel }_X+{\parallel v\parallel }_X{|}_p\ge {\parallel u\parallel }_{X,p}+{\parallel v\parallel }_{X,p}.(\mathrm{Reverse}\mathrm{Minkowski}\mathrm{Inequality})$$

Remark 4.

Similar results to Corollary 1, Remark 3 and Theorem 5 can be obtained.

4.2. Applications in Interval Analysis

An important field with many applications is Interval Analysis. In 1966, Moore [32] used for the first time elements of Interval Analysis in numerical analysis and computer science. Interval-valued functions have many applications in uncertainty theory, signal and image processing or in edge detection algorithms (e.g., [11,12,13,14]).

4.2.1. Interval-Valued Functions and Scalar Set Functions ($F,\nu$)

In refs. [16,27,33] the authors defined the Birkhoff weak (simple) integral of multifunctions and presented some of its properties, making use of the Hausdoff distance and of the Rådström embedding. Integral inequalities of interval-valued functions were obtained, for example, in [25], with respect to different types of integrability. Integral inequalities are important tools in computing deviations or measuring actions.

Let $\left(ck\left({\mathbb{R}}_0^{+}\right),\oplus ,·,d_H,⪯\right)$ be the complete (not linear) metric space (see, for example, [34]) consisting of the family of all non-empty bounded and closed intervals of ${\mathbb{R}}_0^{+}$ with the Minkowski addition ⊕, the multiplication by scalars and the Hausdorff distance $d_H$, where $d_H(A;B):={sup}_{x\in A}{inf}_{y\in B}|x-y|$. The hyperspace $ck\left(\right[0,1\left]\right)$ is used, for example, in decision theory.

• By convention $\left\{0\right\}=[0,0]$ and $\parallel A\parallel :=sup\left\{\right|x|:x\in A\}$.
• In particular, if $A=[a^{-},a^{+}]\in ck\left({\mathbb{R}}_0^{+}\right)$ then $\parallel A\parallel =a^{+}$, $d_H([0,a^{+}],[0,b^{+}])=|b^{+}-a^{+}|$ and $d_H([a^{-},a^{+}],[b^{-},b^{+}])=max\left\{\right|a^{-}-b^{-}|,|a^{+}-b^{+}\left|\right\}$. Moreover, $d_H(A\oplus B,C\oplus D)\le d_H(A,C)+d_H(B,D).$
• Finally, ⪯ is the weak interval order, namely $[a^{-},a^{+}]⪯[b^{-},b^{+}]$, if and only if $a^{-}\le b^{-}$ and $a^{+}\le b^{+}$.
• In order to work in this setting, and in particular to study the Hölder inequality, we also consider the following operation:

  $$[a^{-},a^{+}]\otimes [b^{-},b^{+}]=[a^{-}·b^{-},a^{+}·b^{+}]=[a^{-}{b}^{-},a^{+}{b}^{+}].$$

Following [24], let $F:T\to ck\left({\mathbb{R}}_0^{+}\right)$ be the interval-valued function defined by $F\left(t\right)=[u^{-}\left(t\right),u^{+}\left(t\right)]$, with $u^{-}\left(t\right)\le u^{+}\left(t\right)$ for every $t\in T$. The functions $u^{\pm }$ are the so-called endpoints functions of the interval $F\left(t\right)$.

Definition 6.

$F:T\to ck\left({\mathbb{R}}_0^{+}\right)$ is said to be Birkhoff weakly integrable (on T) with respect to ν, if there exists an interval $I\in ck\left({\mathbb{R}}_0^{+}\right)$ such that for every $\epsilon >0$ there are $P_{\epsilon }\in \mathcal{C}$ and $n_{\epsilon }\in \mathbb{N}$, such that for every $P\in \mathcal{C}$, $P={\left(B_n\right)}_{n\in \mathbb{N}}$, $P⩾P_{\epsilon }$ and every $t_n\in B_n,n\in \mathbb{N}$:

$$\begin{array}{c}\hfill d_H({\oplus }_{k=1}^{n}F\left(t_k\right)\nu \left(B_k\right),I)<\epsilon ,for every n\ge n_{\epsilon }.\end{array}$$

The interval I is called the Birkhoff weak integral of F on T with respect to ν, and is denoted by $I:={\int }_{T}Fd\nu$. Obviously, if it exists, it is unique.

Analogously to the scalar case, we denote by ${\mathbf{B}}_w(\nu ,T)$ the family of all interval-valued functions that are integrable in the Birkhoff weak sense on T with respect to $\nu$.

Remark 5.

Given $P={\left(B_k\right)}_{k\in \mathbb{N}}\in \mathcal{C}$, and $t_k\in B_k$, we said that ${\left\{(B_k,t_k)\right\}}_{k\in \mathbb{N}}$ is a tagged partition. For every $n\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill {\sigma }_{F,n}\left(P\right)& :=& {\oplus }_{k=1}^{n}F\left(t_k\right)\nu \left(B_k\right)={\oplus }_{k=1}^n[u^{-}t\left(t_k\right),u^{+}\left(t_k\right)]\nu \left(B_k\right)=\hfill \\ & =& \{\sum _{k=1}^n{y}_k,y_k\in [u^{-}\left(t_k\right),u^{+}\left(t_k\right)]\nu \left(B_k\right)\}.\hfill \end{array}$$

Since the set ${\sigma }_{F,n}\left(P\right)\in ck\left({\mathbb{R}}_0^{+}\right)$, it is an interval $[u_n^{{-}^{\left(P\right)}},u_n^{{+}^{\left(P\right)}}]$.

Theorem 7.

$F=[u^{-},u^{+}]\in {\mathbf{B}}_w(\nu ,T)$ if and only if $u^{\pm }\in B_w(\nu ,T)$ and

$$\begin{array}{c}\hfill {\int }_{T}Fd\nu =\left[{\int }_T{u}^{-}\left(t\right)d\nu ,{\int }_T{u}^{+}\left(t\right)d\nu \right].\end{array}$$

(7)

Proof. 

Let $F=[u^{-},u^{+}]\in {\mathbf{B}}_w(\nu ,T)$. So, there exists $I=[a,b]\in ck\left({\mathbb{R}}_0^{+}\right)$ such that for every $\epsilon >0$, there exist $n_{\epsilon }\in \mathbb{N}$ and a countable partition $P_{\epsilon }$ of T, so that for every tagged partition $P={\left\{(A_n,t_n)\right\}}_{n\in \mathbb{N}}$ of T with $P⩾P_{\epsilon }$, by Remark 5, we have

$$d_H([u_n^{{-}^{\left(P\right)}},u_n^{{+}^{\left(P\right)}}],[a,b]):=max\left\{|u_n^{{-}^{\left(P\right)}}-a|,|u_n^{{+}^{\left(P\right)}}-b|\right\}<\epsilon ,foreveryn\ge n_{\epsilon }.$$

So, in particular, for every tagged partition $P={\left\{(A_n,t_n)\right\}}_{n\in \mathbb{N}}$ of T with $P⩾P_{\epsilon }$ and for every $n\ge n_{\epsilon }$, we have

$$max\left\{|\sum _{k=1}^n{u}^{-}\left(t_k\right)\nu \left(A_k\right)-a|,|\sum _{k=1}^n{u}^{+}\left(t_k\right)\nu \left(A_k\right)-b|\right\}\le \epsilon ,$$

and then $u^{\pm }\in B_w(\nu ,T)$. The equality (7) follows from the definition of the integral.

For the converse implication, for every $\epsilon >0$, let $n_{\epsilon }$ and let $P_{\epsilon }$ be a countable partition so that the definition of Birkhoff weak integrability is verified on T for both $u^{\pm }$. Then, for every partition $P:=\{B_n,n\in \mathbb{N}\}⩾P_{\epsilon }$, for every $t_n\in B_n$ we have, for every $n\ge n_{\epsilon }$,

$$\begin{array}{c}\hfill max\left\{|\sum _{k=1}^n{u}^{-}\left(t_k\right)\nu \left(B_k\right)-{\int }_T{u}^{-}d\nu |,|\sum _{k=1}^n{u}^{+}\left(t_k\right)\nu \left(B_k\right)-{\int }_T{u}^{+}d\nu |\right\}<\epsilon .\end{array}$$

Since $u^{\pm }\left(t\right)\in F\left(t\right)$, for all $t\in T$, this implies

$$d_H\left([u_n^{{-}^{\left(P\right)}},u_n^{{+}^{\left(P\right)}}],\left[{\int }_T{u}^{-}d\nu ,{\int }_T{u}^{+}d\nu \right]\right)\le \epsilon for every n\ge n_{\epsilon }$$

and then the assertion follows. □

Moreover, the Birkhoff weak integrability is hereditary on subsets $A\in \mathcal{E}$. In fact,

Proposition 3.

$F\in {\mathbf{B}}_w(\nu ,A)$ for every $A\in \mathcal{E}$ if and only if $F{\mathbf{1}}_A\in {\mathbf{B}}_w(\nu ,T)$ and

$${\int }_{A}Fd\nu ={\int }_{T}F{\mathbf{1}}_{A}d\nu .$$

Proof. 

Assume that $F\in {\mathbf{B}}_w(\nu ,T)$ and denote by $[a,b]$ its integral. Then, for every $\epsilon >0$, there exists a countable partition $P_{\epsilon }$ of T, such that for every finer countable partition $P:={\left\{B_n\right\}}_{n\in \mathbb{N}}$ and for every $t_n\in B_n$ we have

$$d_H\left({\sigma }_{F,n}\left(P\right),[a,b]\right)\le \epsilon ,for every n\ge n_{\epsilon }.$$

Let $A\in \mathcal{E}$ and $P_0⩾P_{\epsilon }\wedge \{A,T\setminus A\}$; so, $P_0$ can be split into two parts that are partitions of A and $T\setminus A$, respectively. Let $P_A\subset P_0$ be a partition of A.

Let ${\Pi }_A⩾P_A$ be a partition of A and extend it with a common partition of $T\setminus A$ in such a way that the new partition is finer than $P_{\epsilon }$.

It is possible to prove that ${\sigma }_{F,n}\left({\Pi }_A\right)$ is Cauchy in $ck\left({\mathbb{R}}_0^{+}\right)$, and so the first claim follows by the completeness of the space. The equality follows from Proposition 7. □

Analogously to [24] and using Theorem 7 and Proposition 3, it is possible to prove some properties of the Birkhoff weak integral of an interval-valued function:

• The Birkhoff weak integral is additive with respect to the Minkowski addition. In fact, given $F:=[u^{-},u^{+}]$ and $G:=[v^{-},v^{+}]$ in ${\mathbf{B}}_w(\nu ,T)$, we can write

  $$\begin{array}{ccc}\hfill {\int }_T(F\oplus G)d\nu & =& {\int }_T[u^{-}\left(t\right)+v^{-}\left(t\right),u^{+}\left(t\right)+v^{+}\left(t\right)]d\nu =\hfill \\ & =& \left[{\int }_T(u^{-}\left(t\right)+v^{-}\left(t\right))d\nu ,{\int }_T(u^{+}\left(t\right)+v^{+}\left(t\right))d\nu \right]=\hfill \\ & =& \left[{\int }_T{u}^{-}\left(t\right)d\nu ,{\int }_T{u}^{+}\left(t\right)d\nu \right]\oplus \left[{\int }_T{v}^{-}\left(t\right)d\nu ,{\int }_T{v}^{+}\left(t\right)d\nu \right]=\hfill \\ & =& {\int }_{T}Fd\nu \oplus {\int }_{T}Gd\nu .\hfill \end{array}$$
• Using the same notation as before, if $p>0$ we have

  $${\left(F\oplus G\right)}^p={\left([u^{-}+v^{-},u^{+}+v^{+}]\right)}^p=\left[{(u^{-}+v^{-})}^p,{(u^{+}+v^{+})}^p\right].$$
  In fact, on ${\mathbb{R}}_0^{+}$ the map $x\mapsto x^p$ is increasing. Since $F\oplus G=[u^{-}+v^{-},u^{+}+v^{+}]$, the image of $F\oplus G$ under the map $x\mapsto x^p$ is the interval

  $$\left\{x^p:x\in [u^{-}+v^{-},u^{+}+v^{+}]\right\}=\left[{(u^{-}+v^{-})}^p,{(u^{+}+v^{+})}^p\right].$$
  .
• If $F^p\in {\mathbf{B}}_w(\nu ,T)$

  $${\left\{{\int }_T{F}^{p}d\nu \right\}}^{1/p}:=\left[{\left({\int }_T{\left(u^{-}\right)}^{p}d\nu \right)}^{1/p},{\left({\int }_T{\left(u^{+}\right)}^{p}d\nu \right)}^{1/p}\right]=\left[\parallel u^{-}{\parallel }_p,{\parallel u^{+}\parallel }_p\right].$$

  We denote by

  $${\parallel F\parallel }_p:=d_H\left({\left\{{\int }_T{F}^{p}d\nu \right\}}^{1/p},\left\{0\right\}\right)={\parallel u^{+}\parallel }_p.$$

  $$\begin{array}{ccc}\hfill {\left\{{\int }_T{\left(F\oplus G\right)}^{p}d\nu \right\}}^{1/p}& =& {\left\{{\int }_T[{(u^{-}+v^{-})}^p,{(u^{+}+v^{+})}^p]d\nu \right\}}^{1/p}=\hfill \\ & =& \left[{\left\{{\int }_T{(u^{-}+v^{-})}^{p}d\nu \right\}}^{1/p},{\left\{{\int }_T{(u^{+}+v^{+})}^p]d\nu \right\}}^{1/p}\right].\hfill \end{array}$$

So, if we assume for $u^{\pm },v^{\pm }$ the hypotheses of Theorem 2(b) or Theorem 4(b), we are able to obtain a Minkowski or a reverse Minkowski inequality, depending on the value of $p>0$.

Theorem 8.

Let $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and let $F=[u^{-},u^{+}]$ and $G=[v^{-},v^{+}]$ with $u^{\pm }, v^{\pm }\in \mathcal{F}(T,{\mathbb{R}}_0^{+})$. Let $p>0$. If ${\left(u^{\pm }\right)}^p,{\left(v^{\pm }\right)}^p,{(u^{-}+u^{+})}^p,{(v^{-}+v^{+})}^p,u^{\pm }{(u^{-}+u^{+})}^{p-1},v^{\pm }{(v^{-}+v^{+})}^{p-1}\in B_w(\nu ,T)$, then the Minkowski, for $p\ge 1$ (or the reverse Minkowski, for $0<p<1$) inequality holds.

Proof. 

By hypotheses $F\oplus G\in {\mathbf{B}}_w(\nu ,T)$ and

$$\begin{array}{ccc}\hfill {\parallel F\oplus G\parallel }_p& =& \{{\int }_T{\left({F\oplus G)}^{p}d\nu \right\}}^{1/p}={\left\{{\int }_T[{(u^{-}+v^{-})}^p,{(u^{+}+v^{+})}^p]d\nu \right\}}^{1/p}=\hfill \\ & =& \left[{\left\{{\int }_T{(u^{-}+v^{-})}^{p}d\nu \right\}}^{1/p},{\left\{{\int }_T{(u^{+}+v^{+})}^p]d\nu \right\}}^{1/p}\right]=\hfill \\ & =& \left[\parallel u^{-}+v^{-}{\parallel }_p,{|u^{+}+v^{+}\parallel }_p\right]\hfill \end{array}$$

We now split the proof into two cases depending on the values of p.

Case ($\mathit{p}\ge \mathbf{1}$)

By the Minkowski inequality given in Theorem 2(b), applied to $u^{\pm },v^{\pm }$, we have

$$\parallel u^{-}+v^{-}{\parallel }_p \le \parallel u^{-}{\parallel }_p+\parallel v^{-}{\parallel }_p,\parallel u^{+}+v^{+}{\parallel }_p \le \parallel u^{+}{\parallel }_p+{\parallel v^{+}\parallel }_p.$$

and so, in $ck\left({\mathbb{R}}_0^{+}\right)$ with the weak interval order ⪯ we have

$$\begin{array}{ccc}\hfill \left[\parallel u^{-}+v^{-}{\parallel }_p,{\parallel u^{+}+v^{+}\parallel }_p\right]& ⪯& \left[\parallel u^{-}{\parallel }_p+\parallel v^{-}{\parallel }_p,\parallel u^{+}{\parallel }_p+{\parallel v^{+}\parallel }_p\right]=\hfill \\ & =& \left[\parallel u^{-}{\parallel }_p,{\parallel u^{+}\parallel }_p\right]\oplus \left[\parallel v^{-}{\parallel }_p,{\parallel v^{+}\parallel }_p\right]\hfill \\ \hfill {\parallel F\oplus G\parallel }_p& =& \parallel u^{+}+v^{+}{\parallel }_p \le \parallel u^{+}{\parallel }_p+{\parallel v^{+}\parallel }_p=\hfill \\ & =& {\parallel F\parallel }_p+{\parallel G\parallel }_p.\hfill \end{array}$$

Case ($\mathbf{0}<\mathit{p}<\mathbf{1}$)

We proceed in the same way, this time using the reverse Minkowski inequality given in Theorem 4(b), applied to $u^{\pm },v^{\pm }$.

□

We are ready now to examine the Hölder inequality for $p>1$.

Theorem 9.

Let $\nu \in {\mathcal{M}}_{cs}\left(\mathcal{E}\right)$ and let $F=[u^{-},u^{+}]$ and $G=[v^{-},v^{+}]$ with $u^{\pm },v^{\pm }\in \mathcal{F}(T,{\mathbb{R}}_0^{+})$ be such that ${\left(u^{\pm }\right)}^p,{\left(v^{\pm }\right)}^q,\left(u^{-}{u}^{+}\right),\left(v^{-}{v}^{+}\right)\in B_w(\nu ,T)$. Then, if $p\in {\mathbb{R}}^{+}\setminus \left\{1\right\}$ and q is the conjugate index, we have

$$\begin{array}{ccc}& & {\parallel F\otimes G\parallel }_1\le {\parallel F\parallel }_p·{\parallel G\parallel }_q,when p>1;\hfill \\ & & {\parallel F\otimes G\parallel }_1\ge {\parallel F\parallel }_p·{\parallel G\parallel }_q,when 0<p<1and{\int }_T{\left(u^{+}\right)}^{q}d\nu >0,{\int }_T{\left(v^{+}\right)}^{q}d\nu >0.\hfill \end{array}$$

Proof. 

Let $F=[u^{-},u^{+}]$ and $G=[v^{-},v^{+}]$. By the definition of the integral, the definition of ⊗, the weak interval order ⪯, and Theorems 7 and 2(a) we have

$$\begin{array}{ccc}\hfill {\int }_{T}F\otimes Gd\nu & =& {\int }_T\left[u^{-}{v}^{-},u^{+}{v}^{+}\right]d\nu =\left[{\int }_T{u}^{-}{v}^{-}d\nu ,{\int }_T{u}^{+}{v}^{+}d\nu \right]=\hfill \\ & =& \left[\parallel u^{-}{v}^{-}{\parallel }_1,{\parallel u^{+}{v}^{+}\parallel }_1\right]⪯\left[\parallel u^{-}{\parallel }_p·\parallel v^{-}{\parallel }_q,\parallel u^{+}{\parallel }_p·{\parallel v^{+}\parallel }_q\right]\hfill \\ \hfill {\parallel F\otimes G\parallel }_1& \le & \parallel u^{+}{\parallel }_p·\parallel v^{+}{\parallel }_q={\parallel F\parallel }_p·{\parallel G\parallel }_q.\hfill \end{array}$$

For the second inequality we apply Theorem 4(a) instead of Theorem 2(a) to the components of F and G. □

4.2.2. Scalar Functions and Interval-Valued Set Functions ($u,M$)

M is said to be an interval submeasure if $M:\mathcal{E}\to ck\left({\mathbb{R}}_0^{+}\right)$ is defined by $M\left(A\right)=[{\nu }^{-}\left(A\right),{\nu }^{+}\left(A\right)]$, where ${\nu }^{\pm }:\mathcal{E}\to {\mathbb{R}}_0^{+}$ are two monotone and subadditive set functions with ${\nu }^{-}\left(A\right)\le {\nu }^{+}\left(A\right)$ for every $A\in \mathcal{E}$. An interval submeasure M satisfies:

• $M(\varnothing )=\left\{0\right\};$
• $M\left(A\right)⪯M\left(B\right)$ for every $A,B\in \mathcal{E}$ with $A\subseteq B$ (monotonicity);
• $M(A\cup B)⪯M\left(A\right)\oplus M\left(B\right)$ for every $A,B\in \mathcal{E}$ with $A\cap B=\varnothing$ (subadditivity).

An example of an interval-valued submeasure was given in [35,36], where a mathematical theory of evidence was proposed using the non-additive measures called Belief and Plausibility; in such a way, an interval was associated with every set A:

$$[{\nu }^{-}\left(A\right),{\nu }^{+}\left(A\right)]:=[Bel\left(A\right),Pl\left(A\right)].$$

For results on this subject see, for example, [8,37].

M is an additive multimeasure if $M(A\cup B)=M\left(A\right)\oplus M\left(B\right)$ for every disjoint set $A,B\in \mathcal{E}$. By [37] (Remark 3.6), $M\left(A\right)=[{\nu }^{-}\left(A\right),{\nu }^{+}\left(A\right)]$ is a multisubmeasure with respect to “⪯” if and only if ${\nu }^{\pm }$ are monotone and subadditive.

Now, the Birkhoff weak integrability of a scalar function with respect to a interval-valued multisubmeasure follows easily:

Definition 7.

$u:T\to \mathbb{R}$ is said to be Birkhoff weakly integrable (on T) with respect to $M:\mathcal{E}\to ck\left({\mathbb{R}}_0^{+}\right)$, if we can find an interval $I\in ck\left({\mathbb{R}}_0^{+}\right)$ such that for every $\epsilon >0$ there exist $P_{\epsilon }\in \mathcal{C}$ and $n_{\epsilon }\in \mathbb{N}$ such that for every $P\in \mathcal{C}$ satisfying $P={\left(B_n\right)}_{n\in \mathbb{N}}$ and $P⩾P_{\epsilon }$, and every $t_n\in B_n$, $n\ge n_{\epsilon },$ we have

$$\begin{array}{c}\hfill d_H({\oplus }_{k=1}^{n}u\left(t_k\right)M\left(B_k\right),I)<\epsilon .\end{array}$$

The interval $I:={\int }_{T}udM$ and is called the Birkhoff weak integral of u on T with respect to M. Obviously, if it exists, it is unique. Let ${\mathbf{B}}_w(M,T)$ be the family of all scalar functions that are integrable in the Birkhoff weak sense on T with respect to M.

Remark 6.

It is easy to see that if $u\in {\mathbf{B}}_w(M,T)$, then

$${\int }_{T}udM=\left[{\int }_{T}ud{\nu }^{-},{\int }_{T}ud{\nu }^{+}\right]$$

and

$$\begin{array}{ccc}& & d_H\left({\oplus }_{k=1}^{n}u\left(t_k\right)M\left(B_k\right),\left[{\int }_{T}ud{\nu }^{-},{\int }_{T}ud{\nu }^{+}\right]\right)=\hfill \\ & & max\left\{|\sum _{k=1}^{n}u\left(t_k\right){\nu }^{-}\left(B_k\right)-{\int }_{T}ud{\nu }^{-}|,|\sum _{k=1}^{n}u\left(t_k\right){\nu }^{+}\left(B_k\right)-{\int }_{T}ud{\nu }^{+}|\right\}.\hfill \end{array}$$

So, the linearity follows from the scalar–scalar ($u,\nu$) case; in fact

$$\begin{array}{ccc}\hfill {\int }_T(u+v)dM& =& \left[{\int }_T(u+v)d{\nu }^{-},{\int }_T(u+v)d{\nu }^{+}\right]=\hfill \\ & =& \left[{\int }_{T}ud{\nu }^{-}+{\int }_{T}vd{\nu }^{-},{\int }_{T}ud{\nu }^{+}+{\int }_{T}vd{\nu }^{+}\right]=\hfill \\ & =& {\int }_{T}udM\oplus {\int }_{T}vdM.\hfill \end{array}$$

At this point the Minkowski inequality or its reverse are obtained, as in the interval-valued–scalar case ($F,\nu$), thanks to the scalar–scalar ($u,\nu$) case. For the Hölder inequality we can observe that if the hypotheses of the Theorem 2(a) or Theorem 4(a) are verified, then

$$\begin{array}{ccc}\hfill {\int }_{T}u\left(t\right)v\left(t\right)dM& =& \left[{\int }_{T}u\left(t\right)v\left(t\right)d{\nu }^{-},{\int }_{T}u\left(t\right)v\left(t\right)d{\nu }^{+}\right]\mathrm{and}\hfill \\ \hfill \parallel {\int }_{T}u\left(t\right)v\left(t\right)dM\parallel & =& \left|{\int }_{T}u\left(t\right)v\left(t\right)d{\nu }^{+}\right|,\hfill \end{array}$$

i.e.,

$$\begin{array}{ccc}\hfill {\parallel uv\parallel }_{1,M}& =& {\parallel uv\parallel }_{1,{\nu }^{+}},\hfill \end{array}$$

and so Hölder and reverse Hölder inequalities immediately follow from the scalar–scalar ($u,\nu$) case, when $\nu ={\nu }^{+}$.

Remark 7.

Interval-valued functions provide a natural modeling framework in image processing when uncertainty, quantization effects, or measurement errors cannot be neglected. In fact, representing images as interval-valued functions allows one to explicitly encode this uncertainty at each spatial location, rather than treating it as an additive perturbation. For example, in applications such as fractal image coding for compression and edge detection, representing images as interval-valued objects allows one to explicitly account for round-off errors and discretization inaccuracies arising in the conversion of analog signals into digital data. More generally, when pixel values cannot be assigned with full precision, an interval-valued approach becomes preferable. This situation is common in images acquired through physical measurement processes, such as biomedical imaging (e.g., CT or MR imaging), where sensor sensitivity and data conversion introduce intrinsic uncertainty. In this setting, interval-valued functions offer a unified formalism to encode noise and imprecision at the pixel level, enabling robust image analysis and processing methods that preserve relevant structural information while explicitly handling uncertainty. For results in this subject see, for example, [11,13,38,39].

4.3. A Sparsity-Promoting Reconstruction Model

We can model a gray-scale image as a measurable function $f:T\subset {\mathbb{R}}^2\to \mathbb{R},$ such that $f^p\in B_w(T,\nu )$. In practical acquisition systems, the observed image is often corrupted by noise, which can be written as $\left|g\right(x)-f(x\left)\right|\le \left|\eta \right(x\left)\right|$ for a.e. $x\in T$, where ${\eta }^p\in B_w(T,\nu )$ represents a noise and $g:T\to \mathbb{R}$ denotes the observed image, i.e., the data provided by the acquisition process.

This condition is natural in many acquisition models, including: $g=f+\eta$, $g=f+\psi (f,\eta )$ with $\left|\psi \right(f,\eta \left)\right|\le \left|\eta \right|$. A sparsity-promoting reconstruction model promotes compact representation since only a few coefficients carry significant information; this preserves edges and textures, and fine details are maintained while suppressing noise.

While standard models are known in the cases of functions belonging to Lebesgue spaces $L^p$, it can be useful to consider a sparsity-promoting reconstruction model in the quasi-Banach regime $0<p<1$ as the variational problem in $B_w(T,\nu )$ given by

$$\underset{f^p\in B_w(T,\nu )}{min}\left\{\mathcal{D}(f,g)+\lambda {\int }_{\Omega }{\left|f\left(x\right)\right|}^{p}d\mu \left(x\right)\right\},$$

where $\mu$ is a submeasure modeling spatial interaction or uncertainty, $\mathcal{D}(f,g)$ is a data fidelity term that measures the discrepancy between the candidate reconstruction f and the observed data g, and $\lambda >0$ is a regularization parameter that balances fidelity and sparsity.

In a non-additive setting, a natural choice for the data fidelity term could be a Birkhoff weak-type functional of the form

$$\mathcal{D}(f,g):={\int }_{\Omega }{|f\left(x\right)-g\left(x\right)|}^{r}d\mu \left(x\right),r>0.$$

The regularization functional

$$\mathcal{R}\left(f\right):={\int }_{\Omega }{\left|f\left(x\right)\right|}^{p}d\mu \left(x\right),0<p<1,$$

is nonconvex and strongly promotes sparsity, in the sense that it penalizes small nonzero coefficients more heavily than convex $B_w(T,\nu )$-type penalties. In other words, the $0<p<1$ regularization naturally produces images with few nonzero components, which is useful in problems such as compression, denoising, or reconstruction from incomplete data.

In general, r and p are independent. For example, $r=2$ corresponds to a least-squares type fidelity (classical in additive noise). For sparsity-promoting reconstruction models in the standard setting $L^p$ see, for example, [40,41].

5. Conclusions

We have proved several inequalities and reverse forms of the classical Hölder and Minkowski inequalities for Birkhoff weakly integrable functions when the underlying set function with respect to which we integrate is non-additive. Moreover, these results have also been extended to vector-valued integrands and to the multivalued setting, where the integrands, or the set functions involved, are allowed to take interval-valued outputs.

Hölder and Minkowski inequalities remain an open problem when both the function F and the set functions M are interval-valued. In this latter setting, results concerning integrability, convergence, and properties of the integral have been established, while the corresponding inequalities are currently the subject of ongoing investigation. Also, applications in image reconstruction, in the spirit of Remark 7 and of Section 4.3, will be investigated in a future study.