Inequalities in Riemann–Lebesgue Integrability

Abstract

In this paper, we prove some inequalities for Riemann–Lebesgue integrable functions when the considered integration is obtained via a non-additive measure, including the reverse Hölder inequality and the reverse Minkowski inequality. Then, we generalize these inequalities to the framework of a multivalued case, in particular for Riemann–Lebesgue integrable interval-valued multifunctions, and obtain some inequalities, such as a Minkowski-type inequality, a Beckenbach-type inequality and some generalizations of Hölder inequalities.

1. Introduction

Inequalities have important contributions and applications in many domains of pure and applied mathematics, such as convex analysis, probabilities, control theory, fixed point theorems and mathematical economics [1,2,3,4,5]. Inequalities are used in the theory of local fractional calculus, with applications in physics, control theory, communications engineering and random walk processes. In many papers, the notion of a convex function on fractal sets has been generalized to study Hermite–Hadamard inequalities. For example, in [6], some new identities of generalized fractional integral Hermite–Hadamard-type inequalities via generalized $({\tilde{h}}_1,{\tilde{h}}_2)$-preinvex functions are obtained using integral operators of fractional order with a Mittag–Leffler kernel on Yang’s fractal sets. An integral inequality corresponding to control moments of continuous random variables is established. In recent decades, some generalizations of different classic inequalities have been obtained, such as Cauchy, Hölder, Minkowski, Jensen, Hardy, Steffensen, Jensen–Steffensen, Cebyshev, Clarkson, Riemann–Liouville, etc. (for instance, [4,7,8,9,10,11,12,13,14]). Furthermore, some integral inequalities have been established for multifunctions in different integral settings (for example, [15,16,17,18]).

These inequalities in the non-additive setting have applications, for instance, in decision theory, applied statistics and in many problems of engineering and social choice where the aggregation of data is required. Non-additive integrals can be useful tools from two points of view: decisions under uncertainty and multi-criteria decision making. In fact, the additivity constraint of a measure in some applications is a request that is too strong. In multi-criteria decisions, for example, the hypothesis of additivity implies the non-interactivity of the considered criteria, which does not correspond to a large portion of applications in economical and sociological problems. At the same, time non-additive integrals are useful tools in several areas of applied mathematics that have been built on non-additive measures [19,20,21,22].

In this paper, some inequalities or reverses of classic inequalities (Hölder, Minkowski) are obtained in the framework of the Riemann–Lebesgue integral, both in single-valued and interval-valued cases, where the integration is given with respect to non-additive measures. We were motivated to explore some classic inequalities for Riemann–Lebesgue integrals in non-additive settings for potential applications in generalizations of convexity and preinvexity to set-valued cases by considering interval-valued multifunctions.

The structure of the paper is as follows: Section 2 is for preliminaries. In Section 3, we present some inequalities for Riemann–Lebesgue integrable single-valued functions, including the reverse Hölder inequality and the reverse Minkowski inequality. We emphasize that Riemann–Lebesgue integrability is considered with respect to a non-additive measure. In Section 4, we derive some inequalities for the multivalued case, in particular for Riemann–Lebesgue integrable interval-valued multifunctions, such as the Minkowski-type inequality, the Beckenbach-type inequality and some generalizations of the Hölder inequality.

2. Preliminaries

As usual, let ${\mathbb{R}}^{+},{\mathbb{R}}_0^{+}$ be the halflines of positive real numbers or non-negative real numbers. For a set A, by $cardA$, we denote the cardinality of A.

Let $(S,\mathcal{C})$ be a measurable space, i.e., S is a non-empty set and $\mathcal{C}$ is a $\sigma$-algebra of subsets of S. All the non-additive measures m we consider are finite, non-negative, monotone and subadditive; namely, $m:\mathcal{C}\to [0,\infty )$ is a set function such that $m(\varnothing )=0,$ $m\left(A\right)\le m\left(B\right)$ if $A\subset B$ and $A,B\in \mathcal{C}$ and $m(E\cup F)\le m\left(E\right)+m\left(F\right)$ if $E\cap F=\varnothing ,E,F\in \mathcal{C}$. We denote by $\mathcal{M}$ the class of such set functions.

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

$(ck\left(\mathbb{R}\right),+,d_H)$ denotes the family of all non-empty convex compact subsets of $\mathbb{R}$ endowed with the Minkowski addition, the standard multiplication by scalars and the Hausdorff–Pompeiu distance $d_H$. It is a complete metric space [17]. By convention, $\left\{0\right\}=[0,0]$.

If $A=[x,y]$, then $\parallel A\parallel \hspace{0.17em}=\max \left\{\right|x|,|y\left|\right\}$. Moreover,

$$\begin{array}{ccc}& & d_H([x,y],[w,z])=\max \left\{\right|x-w|,|y-z\left|\right\},\mathrm{for} \mathrm{every}x,y,w,z\in \mathbb{R}\hfill \\ & & d_H([0,x],[0,y])=|y-x|,\mathrm{for} \mathrm{every}x,y\in {\mathbb{R}}_0^{+}.\hfill \end{array}$$

In $ck\left({\mathbb{R}}_0^{+}\right)$, we consider the following operations: multiplication (·), inclusion (⊆), an order relation (⪯ weak interval order) and the supremum and infimum $\vee ,\wedge$ defined for every $a,b,c,d\in {\mathbb{R}}_0^{+}$ in the following way:

(i) 

$[a,b]·[c,d]:=[a·c,b·d]$;

(ii) 

$[a,b]\subseteq [c,d]⟺c\le a\le b\le d$;

(iii) 

$[a,b]⪯[c,d]⟺a\le c$ and $b\le d$ (weak interval order);

(iv) 

$[a,b]\wedge [c,d]=\left[\min \right\{a,c\},\min \{b,d\left\}\right]$;

(v) 

$[a,b]\vee [c,d]=\left[\max \right\{a,c\},\max \{b,d\left\}\right]$;

(vi) 

${[a,b]}^p=[a^p,b^p]$ for every $p\in (0,\infty );$

(vii) 

${\displaystyle \frac{[a,b]}{[c,d]}}=\left[\min \{{\displaystyle \frac{a}{c}},{\displaystyle \frac{a}{d}},{\displaystyle \frac{b}{c}},{\displaystyle \frac{b}{d}}\},\max \{{\displaystyle \frac{a}{c}},{\displaystyle \frac{a}{d}},{\displaystyle \frac{b}{c}},{\displaystyle \frac{b}{d}}\}\right],\mathrm{for} 0\notin [c,d].$

There is no relation between the inclusion and the weak interval order on $ck\left({\mathbb{R}}_0^{+}\right)$, but they coincide on $\{[0,x],x\in {\mathbb{R}}_0^{+}\}$. We recall from [17] the following result.

Remark 1 

([17] Remark 2). If $[a,b],[c,d],[s,t]\in ck\left({\mathbb{R}}^{+}\right)$, then

1. (a) 

$[a,b]⪰[s,t]⟺{\displaystyle \frac{[a,b]}{[c,d]}}⪰{\displaystyle \frac{[s,t]}{[c,d]}}.$

1. (b) 

$[c,d]⪯[s,t]⟺{\displaystyle \frac{[a,b]}{[c,d]}}⪰{\displaystyle \frac{[a,b]}{[s,t]}}.$

Definition 1. 

Let S be a set, with $cardS\ge {\aleph }_0.$

1.(i) 

A measurable countable partition of S is a countable family of nonvoid sets $\Pi ={\left\{E_n\right\}}_{n\in \mathbb{N}}$$\subset \mathcal{C}$ such that ${\displaystyle \bigcup _{n\in \mathbb{N}}}{E}_n=S$ with $E_i\cap E_j=\varnothing ,$ when $i\ne j,i,j\in \mathbb{N}$. $\mathcal{P}$ will be the set of all countable partitions of S, and ${\mathcal{P}}_E$ will be the set of countable partitions of $E\in \mathcal{C}$.

1.(ii) 

For every Π and ${\Pi }^{\prime }\in \mathcal{P}$, it is said that ${\Pi }^{\prime }$ is finer than Π (denoted by ${\Pi }^{\prime }\ge \Pi$ or $\Pi \le {\Pi }^{\prime }$) if every set of ${\Pi }^{\prime }$ is included in some set of Π.

1.(iii) 

For every Π and ${\Pi }^{\prime }\in \mathcal{P}$, $\Pi =\left\{E_n\right\},{\Pi }^{\prime }=\left\{C_n\right\}$, the common refinement of Π and ${\Pi }^{\prime }$ is the countable partition $\Pi \wedge {\Pi }^{\prime }:=\{E_n\cap C_m\}$.

3. Inequalities in Riemann–Lebesgue Integrability

In this section, we present some inequalities regarding the Riemann–Lebesgue integral. In the following, S is a set with $cardS\ge {\aleph }_0$, $(X,\parallel ·\parallel )$ is a Banach space and $m:\mathcal{C}\to [0,\infty )$ is a non-additive measure in the class $\mathcal{M}$. For more results on this subject, we refer to [13].

Definition 2 

([23] ([24] Definition 6)). A vector function $g:S\to X$ is called an absolute (respectively, unconditional) Riemann–Lebesgue integrable (on S) with respect to a non-additive measure m, if there exists $a\in X$ such that for every $\epsilon >0$, there exists ${\Pi }_{\epsilon }\in \mathcal{P}$, such that for every $\Pi \in \mathcal{P}$, $\Pi ={\left(E_n\right)}_{n\in \mathbb{N}}$, $\Pi \ge {\Pi }_{\epsilon }$,

• g is bounded on every $E_n\in \mathcal{C}$, with $m\left(E_n\right)>0$;
• For every $s_n\in E_n$, $n\in \mathbb{N}$, the series ${\displaystyle \sum _{n=0}^{+\infty }}g\left(s_n\right)m\left(E_n\right)$ is absolutely (respectively, unconditionally) convergent and

  $$∥\sum _{n=0}^{+\infty }g\left(s_n\right)m\left(E_n\right)-a∥<\epsilon .$$

We call ${\displaystyle a={\int }_{S}gdm}$ the absolute Riemann–Lebesgue $AR{L}_m$ (respectively, $UR{L}_m$) integral of g (on S) with respect to m.

We denote by $AR{L}_m\left(S\right)$ (respectively, $UR{L}_m\left(S\right)$) the set of all absolute (respectively, unconditional) Riemann–Lebesgue integrable functions on S. This set will be denoted by $R{L}_m\left(S\right)$ in the case of finite dimensional spaces X, when $AR{L}_m$-integrability coincides with $UR{L}_m$-integrability. Obviously, if a exists, then it is unique. As shown in [25], $AR{L}_m\left(S\right)$ and $UR{L}_m\left(S\right)$ are linear function spaces.

For the comparison of $AR{L}_m$- and $UR{L}_m$-integrability and the integrability of Bochner and Pettis when m is countably additive, we refer to [23]. Other comparisons are given in [26] and [25] (Theorem 10). Finally, ref. [24] (Remark 2) contains a summary of known results related to the properties of the space X.

Although the measurability of functions $g:S\to X$ is not required in the definition of $AR{L}_m$- and $UR{L}_m$-integrability, this requirement is nevertheless useful for the study of the inequalities that are the object of this research.

If $p\in (0,\infty )$ and $g:S\to \mathbb{R}$ is a function with ${\left|g\right|}^p\in R{L}_m\left(S\right)$, we denote

$${\parallel g\parallel }_p={\left({\int }_S{\left|g\right|}^{p}dm\right)}^{\frac{1}{p}}.$$

According to [24] (Remark 4), for $p\in [1,\infty )$, the function ${\parallel ·\parallel }_p$ is a seminorm on the linear space of measurable $R{L}_m$-integrable functions.

Countable Subadditive and RL-Integrable Non-Additive Measures

From now on, in this subsection, $p,q$ are conjugate indexes, namely $p,q\in (0,\infty )$ and $p^{-1}+q^{-1}=1.$

We assume that the non-additive measures $m\in \mathcal{M}$ are countable subadditive; namely, for every ${\left(A_n\right)}_{n\in \mathbb{N}}\subset \mathcal{C}$ with $A_i\cap A_j=\varnothing$ if $i\ne j$,

$$m\left({\cup }_{n\in \mathbb{N}}{A}_n\right)\le \sum _{n\in \mathbb{N}}m\left(A_n\right).$$

Moreover, following [27], [28] (Definition 3.2) and [24] (Definition 7), we say that $m:\mathcal{C}\to [0,\infty )$ is RL-integrable if for all $E\in \mathcal{C}$, the characteristic function of the set E: ${\chi }_E\in R{L}_m\left(S\right)$ and ${\int }_S{\chi }_{E}dm=m\left(E\right).$

We denote by ${\mathcal{M}}_{\sigma }$ the class of non-additive measures $m\in \mathcal{M}$ which are countable subadditive and RL-integrable. In view of [24] (Theorem 2), this class allows us to link non-additive integrability with finitely additive integrability.

We recall from [24] the following theorem, which shows the inequalities of Hölder and Minkowski for $p\ge 1.$

Theorem 1 

([24] Theorem 4). Let $m\in {\mathcal{M}}_{\sigma }$, $p,q$ br conjugate indexes and $g,h:S\to \mathbb{R}$ be measurable functions with ${\left|g\right|}^p,{\left|h\right|}^q\in R{L}_m\left(S\right)$.

1. (a) 

If $p,q\in (1,\infty )$, and $g·h\in R{L}_m\left(S\right)$, then

$${\parallel g·h\parallel }_1\le {\parallel g\parallel }_p·{\parallel h\parallel }_q(\mathrm{H}\text{ö}\mathrm{lder}\mathrm{inequality}).$$

1. (b) 

If $p\in [1,\infty )$, ${|g+h|}^p,{|g+h|}^{q(p-1)}$ are $R{L}_m$-integrable, then

$${\parallel g+h\parallel }_p\le {\parallel g\parallel }_p+{\parallel h\parallel }_p(\mathrm{Minkowski}\mathrm{inequality}).$$

Reverse Hölder’s and Minkowski’s inequalities for $0<p<1$ are presented in the next result.

Theorem 2. 

Let $m\in {\mathcal{M}}_{\sigma }$ and let $g,h:S\to \mathbb{R}$ be measurable functions. Let $p,q\in (0,\infty )$ be conjugate indexes. The following inequalities hold:

2. (a) 

If $g·{h,\left|g\right|}^p,{\left|h\right|}^q\in R{L}_m\left(S\right)$ and ${\int }_S{\left|h\right|}^{q}dm>0$, then

$${\parallel g·h\parallel }_1\ge {\parallel g\parallel }_p·{\parallel h\parallel }_q(\mathrm{reverse}\mathrm{H}\text{ö}\mathrm{lder}\mathrm{inequality}).$$

2. (b) 

If ${|g+h|}^p{,|g+h|}^{(p-1)q},{\left|g\right|}^p$ and ${\left|h\right|}^p$ are $R{L}_m$-integrable, then

$$\parallel \left|g\right|+{\left|h\right|\parallel }_p\ge {\parallel g\parallel }_p+{\parallel h\parallel }_p(\mathrm{reverse}\mathrm{Minkowski}\mathrm{inequality}).$$

Proof. 

(2. a) 

If ${\int }_S{g}^{p}dm=0$, then according to [24] (Theorem 3), it follows that $gh=0$ $m-a.e.$ In this case, the inequality of integrals is satisfied.

Consider ${\int }_S{g}^{p}dm>0$. We replace $a=\left|g\right|·({\int }_S{\left|g\right|}^p{dm)}^{-\frac{1}{p}}$ and $b=\left|h\right|·({\int }_S{\left|h\right|}^q{d\nu )}^{-\frac{1}{q}}$ in the well-known inequality $ab\ge {\displaystyle \frac{a^p}{p}}+{\displaystyle \frac{b^q}{q}}$, for every $a,b>0$ and for every $0<p<1$ with $\frac{1}{p}+\frac{1}{q}=1.$ Then,

$$\begin{array}{c}\hfill \frac{\left|gh\right|}{({\int }_S{\left|g\right|}^p{dm)}^{\frac{1}{p}}({\int }_S{\left|h\right|}^q{dm)}^{\frac{1}{q}}}\ge \frac{{\left|g\right|}^p}{p\left({\int }_S\right|g{|}^{p}dm)}+\frac{{\left|h\right|}^q}{q\left({\int }_S\right|h{|}^{q}dm)}.\end{array}$$

Applying Theorems 3 and 6 from [25], it holds that

$$\begin{array}{c}\hfill \frac{{\int }_S\left|gh\right|dm}{({\int }_S{\left|g\right|}^p{dm)}^{\frac{1}{p}}({\int }_S{\left|h\right|}^q{dm)}^{\frac{1}{q}}}\ge \frac{{\int }_S{\left|g\right|}^{p}dm}{p\left({\int }_S{\left|g\right|}^{p}dm\right)}+\frac{{\int }_S{\left|h\right|}^{q}dm}{q\left({\int }_S{\left|h\right|}^{q}dm\right)}=\frac{1}{p}+\frac{1}{q}=1\end{array}$$

and the conclusion yields.

(2. b) 

By (2.a), it results that

$$\begin{array}{ccc}\hfill {\int }_S\left(\right|g|+{\left|h\right|)}^{p}dm& =& {\int }_S\left(\right|g|+{\left|h\right|)}^{p-1}\left(\right|g|+\left|h\right|)dm\ge {\left({\int }_S\left(\right|g|+{\left|h\right|)}^{q(p-1)}dm\right)}^{\frac{1}{q}}{\left({\int }_S{\left|g\right|}^{p}dm\right)}^{\frac{1}{p}}+\hfill \\ & +& {\left({\int }_S\left(\right|g|+{\left|h\right|)}^{q(p-1)}dm\right)}^{\frac{1}{q}}{\left({\int }_S{\left|h\right|}^{p}dm\right)}^{\frac{1}{p}}=\hfill \\ & =& {\left({\int }_S\left(\right|g|+{\left|h\right|)}^{q(p-1)}dm\right)}^{\frac{1}{q}}{(\parallel g\parallel }_p+{\parallel h\parallel }_p).\hfill \end{array}$$

Dividing the above inequality by ${\displaystyle {\left({\int }_S\left(\right|g|+{\left|h\right|)}^{q(p-1)}dm\right)}^{\frac{1}{q}}}$, we obtain the reverse Minkowski inequality.

□

Corollary 1. 

Let $m\in {\mathcal{M}}_{\sigma }$ and let $g,h:S\to \mathbb{R}$, with $h\left(s\right)\ne 0$ be measurable functions for every $s\in S$. Let $p,q\in (0,\infty )$ be conjugate indexes. Suppose that ${\left|g\right|}^p$ and ${\left|h\right|}^{\frac{p}{q}}$ are $R{L}_m$-integrable. Then,

$${\int }_S\frac{{\left|g\right|}^p}{{\left|h\right|}^{\frac{p}{q}}}dm\ge {\displaystyle \frac{{\left({\int }_S\left|g\right|dm\right)}^p}{{\left({\int }_S\left|h\right|dm\right)}^{\frac{p}{q}}}},\mathit{for} \mathit{every} p>1$$

and the reverse inequality is satisfied for $p\in (0,1)$.

Proof. 

According to the Hölder inequality (Theorem 1.(a)), it follows that

$$\begin{array}{c}\hfill {\int }_S\left|g\right|dm={\int }_S\frac{\left|g\right|}{{\left|h\right|}^{\frac{1}{q}}}·{\left|h\right|}^{\frac{1}{q}}dm\le {\left({\int }_S\frac{{\left|g\right|}^p}{{\left|h\right|}^{\frac{p}{q}}}dm\right)}^{\frac{1}{p}}·{\left({\int }_S\left|h\right|dm\right)}^{\frac{1}{q}}.\end{array}$$

Now, taking the p-th power on the both sides of this inequality, the conclusion is obtained.

The reverse inequality follows by Theorem 2.(a). □

Remark 2. 

In particular, rewriting the inequality of Corollary 1, the following inequalities hold:

$$\begin{array}{ccc}& & {\int }_S\frac{{\left|g\right|}^p}{{\left|h\right|}^{p-1}}dm\ge {\displaystyle \frac{{\left({\int }_S\left|g\right|dm\right)}^p}{{\left({\int }_S\left|h\right|dm\right)}^{p-1}}},\mathit{for} \mathit{every} p>1;\hfill \\ & & {\int }_S\frac{{\left|g\right|}^p}{{\left|h\right|}^{p-1}}dm\le \frac{{\left({\int }_S\left|g\right|dm\right)}^p}{{\left({\int }_S\left|h\right|dm\right)}^{p-1}},\mathit{for} \mathit{every} p\in (0,1).\hfill \end{array}$$

Theorem 3. 

Let $m\in {\mathcal{M}}_{\sigma }$, $p,q\in (1,\infty )$ be conjugate indexes and let $g,h:S\to (0,\infty )$ be measurable functions such that there exist two positive real numbers $k,K$, verifying the condition:

3. (a) 

$0<k\le {\displaystyle \frac{g\left(s\right)}{h\left(s\right)}}\le K$ for all $s\in S.$ If $g,h,g^{\frac{1}{p}}·h^{\frac{1}{q}}$ are $R{L}_m$-integrable, then

$$\begin{array}{c}\hfill {\left({\int }_{S}gdm\right)}^{\frac{1}{p}}·{\left({\int }_{S}hdm\right)}^{\frac{1}{q}}\le {\left(\frac{K}{k}\right)}^{\frac{1}{pq}}·{\int }_S{g}^{\frac{1}{p}}·h^{\frac{1}{q}}dm.\end{array}$$

3. (b) 

$0<k\le {\displaystyle \frac{g^p\left(s\right)}{h^q\left(s\right)}}\le K$ for all $s\in S.$ If $g·h$, $g^p$ and $h^q$ are $R{L}_m$-integrable, then

$$\begin{array}{c}\hfill {\left({\int }_S{g}^{p}dm\right)}^{\frac{1}{p}}·{\left({\int }_S{h}^{q}dm\right)}^{\frac{1}{q}}\le {\left(\frac{K}{k}\right)}^{\frac{1}{pq}}·{\int }_{S}ghdm.\end{array}$$

Proof. 

3. (a) 

Since ${\displaystyle \frac{g\left(s\right)}{h\left(s\right)}}\le K$ for every $s\in S$, then $h^{\frac{1}{q}}\left(s\right)\ge K^{\frac{-1}{q}}{g}^{\frac{1}{q}}\left(s\right).$ Therefore,

$$g^{\frac{1}{p}}\left(s\right)h^{\frac{1}{q}}\left(s\right)\ge K^{\frac{-1}{q}}{g}^{\frac{1}{p}}\left(s\right)g^{\frac{1}{q}}\left(s\right)=K^{\frac{-1}{q}}g\left(s\right),\mathrm{for} \mathrm{every}s\in S.$$

According to [25] (Theorems 3 and 6), and taking the $\frac{1}{p}$-th power on both sides of the above inequality, it follows that

$${\left({\int }_S{g}^{\frac{1}{p}}{h}^{\frac{1}{q}}dm\right)}^{\frac{1}{p}}\ge K^{\frac{-1}{pq}}{\left({\int }_{S}gdm\right)}^{\frac{1}{p}}.$$

(1)

While from the inequality ${\displaystyle \frac{g\left(s\right)}{h\left(s\right)}}\ge k$, for every $s\in S,$ it follows that $g^{\frac{1}{p}}\left(s\right)\ge k^{\frac{1}{p}}{h}^{\frac{1}{p}}\left(s\right)$ and

$$g^{\frac{1}{p}}\left(s\right)h^{\frac{1}{q}}\left(s\right)\ge k^{\frac{1}{p}}{h}^{\frac{1}{p}}\left(s\right)h^{\frac{1}{q}}\left(s\right)=k^{\frac{1}{p}}h\left(s\right),\mathrm{for} \mathrm{every}s\in S.$$

Applying again Theorems 3 and 6 from [25] and taking the $\frac{1}{q}$-th power on both sides of this inequality, we obtain

$${\left({\int }_S{g}^{\frac{1}{p}}{h}^{\frac{1}{q}}dm\right)}^{\frac{1}{q}}\ge k^{\frac{1}{pq}}{\left({\int }_{S}hdm\right)}^{\frac{1}{q}}.$$

(2)

By inequalities (1) and (2), the conclusion holds.

3. (b) 

The inequality results by replacing g and h with $g^p$ and $h^q$, respectively, in 3.a).

□

4. Interval-Valued Case

Definition 3. 

Let $m_1,m_2:\mathcal{C}\to {\mathbb{R}}_0^{+}$ be non-additive measures in $\mathcal{M}$ such that $m_1\left(A\right)\le m_2\left(A\right)$, for every $A\in \mathcal{C}$. A set multifunction $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$, defined by 

$$\begin{array}{c}\hfill \Gamma \left(A\right)=\left[m_1\left(A\right),m_2\left(A\right)\right], \mathrm{for} \mathrm{every} A\in \mathcal{C},\end{array}$$

(3)

is called an interval-valued set multifunction. In this case, we write $\Gamma :=\left[m_1,m_2\right].$

The interval-valued set multifunction $\Gamma$ is said to be:

• Monotone if $\Gamma \left(A\right)⪯\Gamma \left(B\right)$ for every $A,B\in \mathcal{C}$ with $A\subseteq B$;
• A multisubmeasure if $\Gamma$ is monotone and $\Gamma (A\cup B)⪯\Gamma \left(A\right)+\Gamma \left(B\right)$ for every (disjoint) set $A,B\in \mathcal{C}$.

Observe that $\Gamma (\varnothing )=\left\{0\right\}$ and $\Gamma$ is monotone if and only if $m_1$ and $m_2$ are monotone set functions.

Definition 4. 

Let $\Gamma =\left[m_1,m_2\right]$ be an interval-valued set multifunction. By definition, a property $\left(P\right)$ regarding the points of S holds $\Gamma$-almost everywhere (denoted by Γ-a.e.) if there exists $A\in \mathcal{C}$, with $\Gamma \left(A\right)=\left\{0\right\}$, so that the property $\left(P\right)$ is valid on $S\setminus A.$

If $\Gamma$ is monotone, then the property $\left(P\right)$ holds $\Gamma$-almost everywhere if and only if $\left(P\right)$ holds $m_2$-almost everywhere.

We recall from [29] the definition of the Riemann–Lebesgue integrability of an interval-valued multifunction with respect to an interval-valued set mutifunction $\Gamma =[m_1,m_2]$ as in (3).

Definition 5. 

Let $H:S\to ck\left({\mathbb{R}}_0^{+}\right)$, $H=[u,v]$, where $u,v:S\to {\mathbb{R}}_0^{+}$ and $u\left(s\right)\le v\left(s\right),$ for every $s\in S$. $H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ is called Riemann–Lebesgue integrable with respect to $\Gamma$ on S($R{L}_{\Gamma }$-integrable in short) if there exists $[c,d]\in ck\left({\mathbb{R}}_0^{+}\right)$ such that for every $\epsilon >0$, there exists a countable partition ${\Pi }_{\epsilon }\in \mathcal{P}$, so that for every partition $\Pi ={\left\{A_n\right\}}_{n\in \mathbb{N}}$ of S finer than ${\Pi }_{\epsilon }$, and for every $s_n\in A_n$, the series ${\sigma }_{H,\Gamma }(\Pi )={\sum }_{n=1}^{\infty }H\left(s_n\right)\Gamma \left(A_n\right):={\sum }_{n=1}^{\infty }\left[u\left(s_n\right)m_1\left(A_n\right),v\left(s_n\right)m_2\left(A_n\right)\right]$ is convergent and

$$\begin{array}{c}\hfill d_H\left({\sigma }_{H,\Gamma }(\Pi ),[c,d]\right)<\epsilon .\end{array}$$

(4)

The interval $[c,d]$ is called the Riemann–Lebesgue integral of H with respect to Γ ($R{L}_{\Gamma }$-integral in short) and we denote

$$[c,d]=\left(R{L}_{\Gamma }\right){\int }_{S}Hd\Gamma ={\int }_{S}Hd\Gamma .$$

As for the single-valued case, if it exists, the Riemann–Lebesgue integral is unique.

In view of [24] (Remark 5), the following particular cases are of interest.

($\alpha$)

If $m_1=m_2=m\in \mathcal{M}$ and $H:=[u,v]$, where $u,v$ are $R{L}_m$-integrable, then

$$\begin{array}{c}\hfill {\int }_{S}Hd\Gamma =\left[\left(R{L}_m\right){\int }_{S}udm,\left(R{L}_m\right){\int }_{S}vs.dm\right].\end{array}$$

($\beta$)

If $u\left(s\right)=v\left(s\right)$ for every $s\in S$ and u is $R{L}_{m_i}$-integrable, $i=1,2$, then

$$\begin{array}{c}\hfill {\int }_{S}Hd\Gamma =\left[\left(R{L}_{m_1}\right){\int }_{S}ud{m}_1,\left(R{L}_{m_2}\right){\int }_{S}ud{m}_2\right].\end{array}$$

Proposition 1 

([24] Remark 5.iii). Suppose $H=[u,v]$ and $\Gamma =[m_1,m_2]$ are as in Definition 5. Then, H is $R{L}_{\Gamma }$ -integrable on S if and only if u is $R{L}_{m_1}$-integrable, v is $R{L}_{m_2}$-integrable and

$$\begin{array}{c}\hfill {\int }_{S}Hd\Gamma =\left[\left(R{L}_{m_1}\right){\int }_{S}ud{m}_1,\left(R{L}_{m_2}\right){\int }_{S}vs.d{m}_2\right].\end{array}$$

Definition 6. 

Suppose Γ is as in (3). $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ is called RL-integrable if

• For all $E\in \mathcal{C},{\chi }_E\in R{L}_{m_1}\left(S\right)\bigcap R{L}_{m_2}\left(S\right)$;
• ${\int }_S{\chi }_{E}d\Gamma =\Gamma \left(E\right)=[m_1\left(E\right),m_2\left(E\right)].$

Remark 3. 

The following assumptions hold:

3. (i) 

Γ is RL-integrable if and only if $m_1$ and $m_2$ are RL-integrable.

3. (ii) 

Suppose that $m_2$ is monotone, σ-subadditive and of finite variation. Then, the RL-integrability of Γ coincides with the RL-integrability of $m_2$.

Proposition 2. 

Let $m\in {\mathcal{M}}_{\sigma }$. If $u,v:S\to [0,\infty )$ are measurable $R{L}_m$-integrable functions and ${\int }_{S}Hd[m,m]=\left\{0\right\}$, then $H=\left\{0\right\}$ m-a.e.

Proof. 

Since ${\int }_{S}Hd[m,m]=\left\{0\right\}$, by ($\alpha$), we have ${\int }_{S}udm={\int }_{S}vdm=0$. By [24] (Theorem 3), it results that $u=0$ and $v=0$ m-a.e., which shows that $H=\left\{0\right\}$ m-a.e. □

Proposition 3. 

Let $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ as in (3) be a countable subadditive RL-integrable interval-valued set multifunction. If $u:S\to [0,\infty )$ is a measurable $R{L}_{\Gamma }$-integrable function and ${\int }_{S}ud\Gamma =\left\{0\right\}$, then $u=0$ Γ-a.e.

Proof. 

Since ${\int }_{S}ud\Gamma =\left\{0\right\}$, from ($\beta$), we have ${\int }_{S}ud{m}_1=0$ and ${\int }_{S}ud{m}_2=0$. Again by [24] (Theorem 3), it results that $u=0$ $m_1$-a.e. and $m_2$-a.e., which shows that $u=0$ $\Gamma$-a.e. □

Corollary 2. 

Let $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ as in (3) be a monotone countable subadditive RL-integrable interval-valued set multifunction. If $u,v:S\to [0,\infty )$ are measurable $R{L}_{\Gamma }$-integrable functions and ${\int }_{S}Hd\Gamma =\left\{0\right\}$, then $H=\left\{0\right\}$ Γ-a.e.

Proof. 

By ${\int }_{S}Hd\Gamma =\left\{0\right\}$, it follows that ${\int }_{S}ud{m}_1=0$ and ${\int }_{S}vd{m}_2=0$. By [24] (Theorem 3), we have $u=0$ $m_1$-a.e. and $v=0$ $m_2$-a.e. Since $u\le v$, then $\{s\in S;u(s)\ne 0\}\subseteq \{s\in S;v(s)\ne 0\}$ and the monotonicity of $m_2$ implies that $u=0$ $m_2$-a.e., which means that $u=v=0$ $m_2$-a.e. □

Definition 7. 

If $p\in (0,\infty )$ and $H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ is a multifunction with $H=[u,v]$ and $H^p\in R{L}_{\Gamma }\left(S\right)$, then we denote

$${\left|H\right|}_p={\left({\int }_S{v}^{p}d\Gamma \right)}^{\frac{1}{p}}=\left[{\left(\left(R{L}_{m_1}\right){\int }_S{v}^{p}d{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_S{v}^{p}d{m}_2\right)}^{\frac{1}{p}}\right]$$

and

$${\parallel H\parallel }_p={\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}}=\left[{\left(\left(R{L}_{m_1}\right){\int }_S{u}^{p}d{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_S{v}^{p}d{m}_2\right)}^{\frac{1}{p}}\right].$$

A Minkowski-type inequality in the interval-valued case holds.

Theorem 4 

(A Minkowski-type inequality). Let $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ as in (3) be countable, subadditive and RL-integrable and let $G,H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ be such that $G=[u_1,v_1]$ and $H=[u,v]$, where $u_1,u,v_1,v$ are non-negative measurable functions. For $p,q\in (0,\infty )$, suppose $u_1^p,u^p,{(u+u_1)}^p,{(u+u_1)}^{(p-1)q}\in R{L}_{m_1}\left(S\right)$ and $v_1^p,v^p,{(v+v_1)}^p,{(v+v_1)}^{(p-1)q}\in R{L}_{m_2}\left(S\right)$. Then,

4. (i) 

$${\left({\int }_S{(G+H)}^{p}d\Gamma \right)}^{\frac{1}{p}}⪯{\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}+{\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}},\mathrm{for} p>1;$$

4. (ii) 

$${\left({\int }_S{(G+H)}^{p}d\Gamma \right)}^{\frac{1}{p}}⪰{\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}+{\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}},\mathrm{for} p\in (0,1).$$

Proof. 

4. (i) Using interval operations and Theorem 1, for $p>1$, it results that:

$$\begin{array}{cc}\hfill & {\left({\int }_S{(G+H)}^{p}d\Gamma \right)}^{\frac{1}{p}}={\left({\int }_S{([u_1,v_1]+[u,v])}^{p}d\Gamma \right)}^{\frac{1}{p}}={\left({\int }_S{\left([u_1+u,v_1+v]\right)}^{p}d\Gamma \right)}^{\frac{1}{p}}=\hfill \\ \hfill & ={\left[\left(R{L}_{m_1}\right){\int }_S{(u_1+u)}^{p}d{m}_1,\left(R{L}_{m_2}\right){\int }_S{(v_1+v)}^{p}d{m}_2\right]}^{\frac{1}{p}}=\hfill \\ \hfill & =\left[{\left(\left(R{L}_{m_1}\right){\int }_S{(u_1+u)}^{p}d{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_S{(v_1+v)}^{p}d{m}_2\right)}^{\frac{1}{p}}\right]\stackrel{\text{Theorem1}.\mathrm{b}}{⪯}\hfill \\ \hfill & ⪯\left[{\left(\left(R{L}_{m_1}\right){\int }_S{u}_1^{p}d{m}_1\right)}^{\frac{1}{p}}+{\left(\left(R{L}_{m_1}\right){\int }_S{u}^{p}d{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_S{v}_1^{p}d{m}_2\right)}^{\frac{1}{p}}+{\left(\left(R{L}_{m_2}\right){\int }_S{v}^{p}d{m}_2\right)}^{\frac{1}{p}}\right]=\hfill \\ \hfill & =\left[{\left(\left(R{L}_{m_1}\right){\int }_S{u}_1^{p}d{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_S{v}_1^{p}d{m}_2\right)}^{\frac{1}{p}}\right]+\left[{\left(\left(R{L}_{m_1}\right){\int }_S{u}^{p}d{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_S{v}^{p}d{m}_2\right)}^{\frac{1}{p}}\right]=\hfill \\ \hfill & ={\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}+{\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}}.\hfill \end{array}$$

4. (ii) For the case where $0<p<1$, we apply Theorem 2.(b). □

One of the most interesting generalizations of the Hölder inequality is now known as the Beckenbach inequality, which we have extended in our framework.

Theorem 5 

(Beckenbach’s Inequality). Suppose $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ as in (3) is countable, subadditive and RL-integrable and let $G,H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ be such that $G=[u_1,v_1]$ and $H=[u,v]$, where $u_1,u,v_1,v>0$ are measurable functions. Let $0<p<1$. If ${(u+u_1)}^p,{(u+u_1)}^{(p-1)q},u_1^p,u^p\in R{L}_{m_1}\left(S\right)$ and ${(v+v_1)}^p,{(v+v_1)}^{(p-1)q},v_1^p,v^p\in R{L}_{m_2}\left(S\right)$, then

$$\frac{{\int }_S{(G+H)}^{p+1}d\Gamma }{{\int }_S{(G+H)}^{p}d\Gamma }⪯\frac{{\int }_S{G}^{p+1}d\Gamma }{{\int }_S{G}^{p}d\Gamma }+\frac{{\int }_S{H}^{p+1}d\Gamma }{{\int }_S{H}^{p}d\Gamma }.$$

Proof. 

Denoting $A_1={\left({\int }_S{G}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}$, $B_1={\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}$, $A_2={\left({\int }_S{H}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}$, $B_2={\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}}$ and using the interval Radon inequality established in [17] (Theorem 7), we have

$$\frac{A_1^{p+1}}{B_1^p}+\frac{A_2^{p+1}}{B_2^p}⪰\frac{{(A_1+A_2)}^{p+1}}{{(B_1+B_2)}^p},$$

which means

$$\begin{array}{ccc}& & \frac{{\int }_S{G}^{p+1}d\Gamma }{{\int }_S{G}^{p}d\Gamma }+\frac{{\int }_S{H}^{p+1}d\Gamma }{{\int }_S{H}^{p}d\Gamma }⪰\frac{{\left({\left({\int }_S{G}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}+\left({\int }_S{H}^{p+1}{d\Gamma )}^{\frac{1}{p+1}}\right)\right)}^{p+1}}{{\left({\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}+{\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}})\right)}^p}.\hfill \end{array}$$

Since $0<p<1$, then $1<p+1<2$. According to Theorem 4, it follows that

$${\left({\int }_S{(G+H)}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}⪯{\left({\int }_S{G}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}+{\left({\int }_S{H}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}$$

(5)

and

$${\left({\int }_S{(G+H)}^{p}d\Gamma \right)}^{\frac{1}{p}}⪰{\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}+{\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}}.$$

(6)

By Remarks 1, (5) and (6), we have

$$\begin{array}{c}\hfill \frac{{\left({\left({\int }_S{G}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}+{\left({\int }_S{H}^{p+1}d\Gamma \right)}^{\frac{1}{p+1}}\right)}^{p+1}}{{\left({\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}+{\left({\int }_S{H}^{p}d\Gamma \right)}^{\frac{1}{p}}\right)}^p}⪰\frac{{\int }_S{(G+H)}^{p+1}d\Gamma }{{\int }_S{(G+H)}^{p}d\Gamma },\end{array}$$

which finishes the proof. □

Theorem 6. 

Let $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ as in (3) be countable, subadditive and RL-integrable and let $G,H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ such that $G=[u_1,v_1]$ and $H=[u,v]$, where $u_1,u,v_1,v$ are non-negative measurable functions and $0<k_1\le u\le vs.\le K_1<\infty ,0<k_2\le u_1\le v_1\le K_2<\infty .$

• Let $p\in (1,\infty )$, with $p^{-1}+q^{-1}=1.$ Suppose $u,u_1,u^{\frac{1}{p}}·u_1^{\frac{1}{q}}\in R{L}_{m_1}\left(S\right)$ and $v,v_1,v^{\frac{1}{p}}·v_1^{\frac{1}{q}}\in R{L}_{m_2}\left(S\right)$. Then, it holds that

  $${\left({\int }_{S}Gd\Gamma \right)}^{\frac{1}{p}}·{\left({\int }_{S}Hd\Gamma \right)}^{\frac{1}{q}}⪯{\left(\frac{K_1{K}_2}{k_1{k}_2}\right)}^{\frac{1}{pq}}·{\int }_S{G}^{\frac{1}{p}}{H}^{\frac{1}{q}}d\Gamma .$$

Proof. 

By the hypothesis, it follows that ${\displaystyle \frac{k_1}{K_2}}\le {\displaystyle \frac{u}{u_1}}\le {\displaystyle \frac{K_1}{k_2}}$ and ${\displaystyle \frac{k_1}{K_2}}\le {\displaystyle \frac{v}{v_1}}\le {\displaystyle \frac{K_1}{k_2}}$. Then,

$$\begin{array}{cc}\hfill {\left({\int }_{S}Gd\Gamma \right)}^{\frac{1}{p}}·{\left({\int }_{S}Hd\Gamma \right)}^{\frac{1}{q}}& =\left[{\left(\left(R{L}_{m_1}\right){\int }_{S}ud{m}_1\right)}^{\frac{1}{p}},{\left(\left(R{L}_{m_2}\right){\int }_{S}vd{m}_2\right)}^{\frac{1}{p}}\right]·\hfill \\ & ·\left[{\left(\left(R{L}_{m_1}\right){\int }_S{u}_{1}d{m}_1\right)}^{\frac{1}{q}},{\left(\left(R{L}_{m_2}\right)){\int }_S{v}_{1}d{m}_2\right)}^{\frac{1}{q}}\right]=\hfill \\ \hfill & =\left[{\left(\left(R{L}_{m_1}\right){\int }_{S}ud{m}_1\right)}^{\frac{1}{p}}{\left(\left(R{L}_{m_1}\right){\int }_S{u}_{1}d{m}_1\right)}^{\frac{1}{q}},{\left(\left(R{L}_{m_2}\right){\int }_{S}vd{m}_2\right)}^{\frac{1}{p}}{\left(\left(R{L}_{m_2}\right){\int }_S{v}_{1}d{m}_2\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Now, by applying Theorem 3, we have

$$\begin{array}{cc}\hfill & \left[{\left(\left(R{L}_{m_1}\right){\int }_{S}ud{m}_1\right)}^{\frac{1}{p}}{\left(\left(R{L}_{m_1}\right){\int }_S{u}_{1}d{m}_1\right)}^{\frac{1}{q}},{\left(\left(R{L}_{m_2}\right){\int }_{S}vd{m}_2\right)}^{\frac{1}{p}}{\left(\left(R{L}_{m_2}\right){\int }_S{v}_{1}d{m}_2\right)}^{\frac{1}{q}}\right]⪯\hfill \\ \hfill & ⪯\left[{\left(\frac{K_1{K}_2}{k_1{k}_2}\right)}^{\frac{1}{pq}}·\left(R{L}_{m_1}\right){\int }_S{u}^{\frac{1}{p}}·u_1^{\frac{1}{q}}d{m}_1,{\left(\frac{K_1{K}_2}{k_1{k}_2}\right)}^{\frac{1}{pq}}·\left(R{L}_{m_2}\right){\int }_S{v}^{\frac{1}{p}}·v_1^{\frac{1}{q}}d{m}_2\right]=\hfill \\ \hfill & ={\left(\frac{K_1{K}_2}{k_1{k}_2}\right)}^{\frac{1}{pq}}\left({\int }_S{G}^{\frac{1}{p}}·H^{\frac{1}{q}}d\Gamma \right).\hfill \end{array}$$

Therefore, the proof is finished. □

Replacing G and H with $G^p$ and $H^q$, respectively, in Theorem 6, the following result holds.

Theorem 7. 

Let $\Gamma :\mathcal{C}\to ck\left({\mathbb{R}}_0^{+}\right)$ as in (3) be countable, subadditive and RL-integrable and let $G,H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ such that $G=[u_1,v_1]$ and $H=[u,v]$, where $u_1,u,v_1,v$ are non-negative measurable functions and

$$0<k_1\le u^p\le v^p\le K_1<\infty ,0<k_2\le u_1^q\le v_1^q\le K_2<\infty .$$

Let $p\in (1,\infty )$, with $p^{-1}+q^{-1}=1.$ Suppose $u,u·u_1,u^p,u_1^q\in R{L}_{m_1}\left(S\right)$ and $v,v·v_1,v^p,v_1^q\in R{L}_{m_2}\left(S\right)$. Then, for every $p\in (1,\infty )$, it holds that

$${\left({\int }_S{G}^{p}d\Gamma \right)}^{\frac{1}{p}}·{\left({\int }_S{H}^{q}d\Gamma \right)}^{\frac{1}{q}}⪯{\left(\frac{K_1{K}_2}{k_1{k}_2}\right)}^{\frac{1}{pq}}·\left({\int }_{S}G·Hd\Gamma \right).$$

Theorem 8. 

Let $m\in {\mathcal{M}}_{\sigma }$ and let $G,H:S\to ck\left({\mathbb{R}}_0^{+}\right)$ so that $G=[u_1,v_1]$ and $H=[u,v]$, where $u_1,u,v_1,v$ are non-negative measurable functions.

Let $p,q\in (0,\infty )$, with $p^{-1}+q^{-1}=1.$ If $G^p$ and $H^{\frac{p}{q}}=H^{p-1}$ are $R{L}_m$-integrable, then

$${\int }_S\frac{G^p}{H^{p-1}}dm⪰\frac{{\left({\int }_{S}Gdm\right)}^p}{{\left({\int }_{S}Hdm\right)}^{p-1}},\mathit{for} \mathit{every} p>1$$

and

$${\int }_S\frac{G^p}{H^{p-1}}dm⪯{\displaystyle \frac{{\left({\int }_{S}Gdm\right)}^p}{{\left({\int }_{S}Hdm\right)}^{p-1}}},\mathit{for} \mathit{every} p\in (0,1).$$

Proof. 

For $p>1$, using interval operations and Remark 2, we have

$$\begin{array}{cc}\hfill {\int }_S\frac{G^p}{H^{p-1}}dm& =\left[{\int }_S\frac{u_1^p}{v^{p-1}}dm,{\int }_S\frac{v_1^p}{u^{p-1}}dm\right]⪰\hfill \\ \hfill & ⪰\left[\frac{{\left({\int }_S{u}_{1}dm\right)}^p}{{\left({\int }_{S}vdm\right)}^{p-1}},\frac{{\left({\int }_S{v}_{1}dm\right)}^p}{{\left({\int }_{S}udm\right)}^{p-1}}\right]=\frac{{\left({\int }_{S}Gdm\right)}^p}{{\left({\int }_{S}Hdm\right)}^{p-1}}.\hfill \end{array}$$

For the case where $0<p<1$, we apply the reverse inequality by Remark 2. □

5. Conclusions

In Theorems 1 and 2, some inequalities or reverses of classic inequalities (Hölder, Minkowski) are obtained for the Riemann–Lebesgue integral when the set function that we integrate with respect to is non-additive. Then, we apply these results to the multivalued case when both the functions and the multivalued set functions are interval-valued. We also point out some further studies and complementary aspects; in this direction, the following very recent references can be mentioned: In [6], the authors discuss local fractional integral inequalities of Hermite–Hadamard type with Mittag–Leffler kernel operators and construct an inequality to establish central moments of a random variable. In [30], the authors introduce the Bézier variant of a sequence of summation-integral-type operators involving the inverse Pólya–Eggenberger distribution and Păltănea operators, and estimate the approximation behavior and the rate of convergence with a class of functions of derivatives of bounded variation.