Liapounoff type inequality for pseudo-integral of interval-valued function

1.   Introduction

In mathematics, integral inequalities are a useful tool in solving various problems. Some of the best known are Chebyshev, Jensen, Hölder, Minkowski, Cauchy-Schwartz and Liapounoff inequalities. Many generalizations of these types of inequalities (and many others) for different classes of integrals have been investigated. In paper ^[6], are given for fuzzy integral and in ^[2,7,11] for Sugeno integral, in ^[1,3,8] for pseudo-integral.

The main topic of this paper is Liapounoff type inequality. One of its forms is

where $0 < t < s < r$ and $f: [0, 1]\to[0, \infty)$ is an integrable function (see ^[5]).

Pseudo-analysis is a part of mathematical analysis where the field of real numbers is replaced with a closed or semi-closed interval $[a, b]\subset[-\infty, \infty],$ and operations of addition and multiplication of real numbers are replaced with two new operations, pseudo-addition and pseudo-multiplication, defined on the considered interval $[a, b]$. The structure $([a, b], \oplus, \odot),$ where $\oplus$ and $\odot$ are pseudo-addition and pseudo-multiplications, respectively, is called semiring. There are three classes of semirings ^[20,21]. In this paper, the focus is on two classes of semirings on the interval $[0, \infty].$ In the first part of the investigation the focus is on so-called $g$-semirings where pseudo-operations are given by a continuous function $g:[0.\infty] \to [0, \infty].$ The second part of the investigation is dedicated to idempotent semiring $([0, \infty], \sup, \odot)$ with generated pseudo-multiplication. Tools from pseudo-analysis have applications in various fields, such as game theory, nonlinear partial differential equations, probability theory, interval probability theory, etc. (see ^{[13,21,27,28]}).

Two generalizations of the Liapounoff type inequality for pseudo-integral are given in ^[15]. The first generalization is given for the semiring $([0, 1], \oplus, \odot)$ when both pseudo-operations are given by the generator $g: [0, 1] \to [0, 1]$ and the second generalization refers to the semiring $([0, 1], \sup, \odot)$ from the first class when the pseudo-multiplication is given by an increasing generator $g: [0, 1] \to [0, 1].$

Based on results from ^[15] it holds that

where $0 < t < s < r, \; f:[0, 1]\to[0, 1]$ is a measurable function, $\oplus$ and $\odot$ are pseudo-operations given by generator $g:[0, 1]\to[0, 1],$ $\preceq$ is the total order on the considered semiring from the second class and $\mu$ is an $\oplus$-measure.

In the case when the semiring belongs to the first class, pseudo-addition is $\sup$ and pseudo-multiplication is given by an increasing generator $g:[0, 1] \to [0, 1]$ the Liapounoff type inequality has the form

where $0 < t < s < r, \; f:[0, 1]\to[0, 1]$ is a measurable function and $\mu$ is a complete $\sup$-measure.

One generalization of ordinary functions (single-valued functions) are set-valued functions. The theory of set-valued functions has application in the mathematical economy and optimal control ^[12]. The well-known terms for single-valued function, such as continuity, differentiation and integration are investigated in the field of set-valued functions. The first results of the integration of set-valued functions are given in ^[4]. Set-valued functions and their pseudo-integration has been investigated in ^[10]. The investigation of pseudo-integral of set-valued function is focused on a special case of the set-valued function, interval-valued function since the interval-valued function is suitable for applications. Different integral inequalities for interval-valued functions with respect to pseudo-integral are proven in ^[14,29]. Jensen type and Hölder type inequality for interval-valued Choquet integrals are given in ^[14] and inequalities of Liapounoff and Stolarsky type for Choquet-like integrals with respect to nonmonotonic fuzzy measures are given in ^[29]. Interval Minkowski's inequality, interval Radon's inequality, and interval Beckenbach's inequality for Aumann integral are proven in ^[24].

The paper is organized in the following way. The second section contains preliminary notions needed for the investigation. It contains the definition of semiring and some illustrative examples. Definition of $\oplus$-measure and pseudo-integral are also the part of this section as well as definitions of operations on intervals and interval-valued function and its integration. The third section contains the main result of the paper, the Liapounoff type inequality for the pseudo-integral of an interval-valued function and some illustrative examples. The fourth section contains the definition of the interval-valued central moment of order $n$ and an example where the Liapounoff inequality for pseudo-integral of interval-valued function is used for estimation of the interval-valued central moment of order $n$.

2.   Preliminary notions

Some basic notions and definitions from pseudo-analysis are presented in this section.

2.1.   Semiring

Let $[a, b]$ be a closed (or semi-closed) subinterval of $[-\infty, \infty]$ and let $\preceq$ be the total order on the interval $[a, b].$ On the interval $[a, b]$ two operations are considered:

i) commutative and associative binary operation $\oplus$ called pseudo-addition which is non-decreasing with respect to $\preceq$ and has a neutral element denoted by $\bf 0,$

ii) commutative and associative binary operation $\odot$ called pseudo-multiplication which is positively non-decreasing with respect to $\preceq,$ i.e. $x\preceq y$ implies $x\odot z\preceq y\odot z$ for all $z \in [a, b]_+$ where $[a, b]_+ = \{z \in [a, b]: \bf 0 \preceq z\}.$ It has a neutral element denoted by $\bf 1.$

The structure $([a, b], \oplus, \odot)$ is called a semiring if the following conditions are satisfied:

i) $\oplus: [a, b]^2 \to [a, b]$ and $\odot: [a, b]^2 \to [a, b]$ are pseudo-addition and pseudo-multiplication, respectively,

ii) pseudo-multiplication $\odot$ is distributive over pseudo-addition $\oplus,$ i.e. $x \odot (y \oplus z) = x \odot y \oplus x \odot z$ for all $x, y, z \in [a, b],$

iii) for all $x \in [a, b]$ it holds ${\bf{0}} \odot x = {\bf{0}}.$

More about pseudo-operations and semirings can be found in ^[20,21,22].

Depending on other properties that pseudo-operations possess, there are three classes of semirings (see ^[21,22]).

The first class consists of semirings where pseudo-addition is an idempotent operation, and pseudo-multiplication is a non-idempotent operation. The semiring investigated in this paper is $([a, b], \sup, \odot),$ where pseudo-addition is $x\oplus y = \sup(x, y)$ and pseudo-multiplication is given by $x \odot y = g^{-1}(g(x) g(y))$ and $g:[a, b] \to [0, \infty]$ is a continuous increasing function. The function $g$ is called generator of pseudo-multiplication. An example of this type of semiring is $([-\infty, \infty], \sup, +),$ where the generator for pseudo-multiplication is $g(x) = e^x.$

The second class consists semirings where both pseudo-operations are strict and defined by a monotone and continuous function $g:[a, b]\to [0, \infty],$ by

The function $g$ is called generator of pseudo-operations and the semiring is called $g$-semiring. In this case, the pseudo-operations are called $g$-operations.

The third class of semirings are semirings with both idempotent pseudo-operations.

In this investigation the pseudo-multiplication is always defined by a generator $g,$ and in this case the pseudo-power $x^{(n)}$ is defined by

for $x\in[a, b]$ and $n\in \mathbb{N}.$ It can be shown (see ^[3]) that the pseudo-power $x^{(p)}$ is well defined for all $p\in(0, \infty) \cap \mathbb{Q}$ in the same way, i.e.

Due to the continuity of $\odot,$ it holds that (see ^[3])

On the class of $g$-semirings, the total order on the interval $[a, b]$ is given by $x\preceq y$ if and only if $g(x)\leq g(y).$ If $g$ is an increasing generator, the total order $\preceq$ is the usual order $\leq$ on the real line and if $g$ is a decreasing generator, the total order $\preceq$ is opposite to the usual order on the real line. If pseudo-addition is $x\oplus y = \sup(x, y),$ then the total order is defined by $x\preceq y$ if and only if $\sup(x, y) = y,$ and the total order is the usual order $\leq$ on the real line. Similarly, if $\oplus = \min$ the total order $\preceq$ is the order opposite to the usual order on the real line.

2.2.   Pseudo-integral

One generalization of additive measure from the classical measure theory is so-called $\oplus$-measure.

Let $X$ be a non-empty set, and let $\mathcal{A}$ be the $\sigma$-algebra of the subsets of $X.$ A set function $\mu:\mathcal{A} \to [a, b]_+$ is an $\oplus$-measure, or a pseudo-additive measure if

i) $\mu\left(\emptyset\right) = {\bf 0},$

ii) $\mu\left(\bigcup\limits_{i = 1}^\infty A_i\right) = \bigoplus\limits_{i = 1}^\infty \mu(A_i) = \lim\limits_{n\to\infty}\bigoplus\limits_{i = 1}^n \mu(A_i),$ where $\{A_i\}$ is a sequence of pairwise disjoint sets from $\mathcal{A}.$

If $\oplus$ is an idempotent operation, the first condition and disjointness of sets from the second condition can be omitted.

For the $g$-semiring, the second condition has the form

The focus of this paper is on two cases of pseudo-integral (see ^[20]).

Let $([a, b], \oplus, \odot)$ be a $g$-semiring. $g$-integral of a measurable function $f: X \rightarrow [a, b]$ is of the form

where the integral on the right-hand side is the Lebesgue integral. In the special case, when $X = [c, d], \; \mathcal{A} = \mathcal{B}([c, d])$ is a Borel $\sigma$-algebra on $[c, d]$ and $g\circ \mu$ is a Lebesgue measure on $[c, d],$ then

Let $([a, b], \sup, \odot)$ be a semiring where pseudo-multiplication is given by an increasing function $g:[a, b] \to [0, \infty].$ Pseudo-integral of a measurable function $f: X \to [a, b]$ is

where $\Psi: [c, d]\to [a, b]$ is a continuous density which determines $\sup$-decomposable measure $\mu.$

Due to the fact that every semiring $([a, b], \sup, \odot)$ of the first class can be obtained as a limit of a family of $g$-semirings of the second class generated by $g^{\lambda},$ i.e.,

and

where $g$ is an increasing generator of pseudo-multiplication $\odot$ (see ^[19]), this research is focused on the class of $g$-semirings.

Also, in ^[19] it is shown that for the semiring $([0, \infty], \sup, \odot)$ with generated pseudo-multiplication it holds that

where $\mu$ is a $\sup$-decomposable measure on $[0, \infty]$ and $f: [0, \infty] \to [0, \infty]$ is a continuous function.

2.3.   Pseudo-operations on intervals

As this paper deals with interval-valued functions, for further work it is necessary to define pseudo-multiplication on the class

of closed sub-intervals of $[a, b]_+.$

The pseudo-product of two intervals $A = [l_1, r_1]$ and $B = [l_2, r_2]$ is defined in ^[16]. Since this paper only deals with generated pseudo-multiplication, $A \odot B$ when $x \odot y = g^{-1} (g(x) \; g(y))$ is given below.

If the pseudo-multiplication is given by an increasing generator $g,$ then

and when the generator $g$ is a decreasing function, it holds that

For every family $\{ [l_i, r_i]:[l_i, r_i]\in \mathcal{I}, i\in I\}$ of closed sub-intervals of $[a, b]_+,$ where the index set $I$ is a countable set, based on results from ^[9] and ^[25] it holds that

Also,

if $\lim\limits_{n\to\infty} l_n$ and $\lim\limits_{n\to\infty} r_n$ exist.

The pseudo-power $x^{(p)}, \; p \in (0, \infty), \; x \in [0, \infty]$ is extended to the pseudo-power $A^{(p)}$ of a set $A\subset [0, \infty]$ (see ^[16]) as

In the special case, when $A = [c, d],$ the next lemma is shown in ^[16].

Lemma 1. Let $n, m\in \mathbb{N}, \; p\in \mathbb{R}^+\setminus \mathbb{Q}$ and pseudo-multiplication $\odot$ is given by a generator $g$. Then it holds that

i) $[c, d]^{(n)} = \left[c^{(n)}, d^{(n)}\right],$

ii) $[c, d]^{(\frac{1}{n})} = \left[c^{\left(\frac{1}{n}\right)}, d^{\left(\frac{1}{n}\right)}\right],$

iii) $[c, d]^{\left(\frac{m}{n}\right)} = \left[c^{\left(\frac{m}{n}\right)}, d^{\left(\frac{m}{n}\right)}\right].$

iv) $[c, d]^{(p)} = \sup\left\{\left[c^{(r)}, d^{(r)}\right]: r\in (0, p)\cap \mathbb{Q}\right\}.$

Based on (2.1), (2.5) and $iv)$ from Lemma 1 it holds that

2.4.   The relation "less or equal" on $\mathcal{I}$

The relation "less or equal" for the intervals from $\mathcal{I}$ is denoted by $\preceq_S.$

Definition 1. Let $A, B \in \mathcal{I}. \; A \preceq_S B$ if and only if for all $x \in A$ there exists $y \in B$ such that $x \preceq y$ and for all $y \in B$ there exists $x \in A$ such that $x \preceq y.$

In paper ^[17], the relation $\preceq_S$ was defined in a more general manner, on the set of all non-empty subsets of the interval $[a, b].$

The necessity of introducing the relation $\preceq_S$ was illustrated in ^[17] by an example - if the usual subset is used instead of $\preceq_S,$ for $x, y\in [a, b]_+$ such that $x\preceq y$ and $x \neq y,$ for elements $A = [x, x]$ and $B = [y, y]$ neither $A \subseteq B$ nor $B\subseteq A$ holds, but it holds that $A \preceq_S B.$

2.5.   Pseudo-integral of an interval-valued function

Let $X$ be a non-empty set, $([a, b], \oplus, \odot)$ a semiring, ${\mathcal{I}}$ a class of closed sub-intervals of $[a, b]_+$ and $F:X\rightarrow \mathcal{I}$ an interval-valued function.

An interval-valued function $F$ is pseudo-integrably bounded if there exists a function $h\in L^1_{\oplus}(\mu)$ such that

i) $\bigoplus\limits_{\alpha\in F(x)}\alpha\preceq h(x),$ for the non-idempotent pseudo-addition,

ii) $\sup\limits_{\alpha\in F(x)}\alpha\preceq h(x),$ for the pseudo-addition given by an increasing generator $g$,

iii) $\inf\limits_{\alpha\in F(x)}\alpha\preceq h(x),$ for the pseudo-addition given by a decreasing generator $g$

holds, where $L^1_{\oplus}(\mu)$ is the family of functions which are integrable with respect to the pseudo-integral in the sense of the considered semiring.

Based on results from ^[10], the pseudo-integral of a pseudo-integrably bounded interval-valued function $F:X\rightarrow \mathcal{I}$ represented by its border functions $F_l, \, F_r:X\rightarrow [a, b]_+$ by $F(x) = [F_l(x), F_r(x)]$ is defined by

If $F$ is a pseudo-integrably bounded function, then $F$ is a pseudo-integrable function. More about the pseudo-integral of an interval-valued function, its basic properties and application can be found in ^[10].

3.   Liapounoff type pseudo-integral inequality of interval-valued function

The main result of this paper, the Liapounoff type inequality for pseudo-integral of an interval-valued function is presented in this section.

Obviously, the Liapounoff type inequalities from ^[15] hold for a $g$-semiring $([0, \infty], \oplus, \odot)$ with an increasing generating function $g:[0, \infty] \to [0, \infty]$ and a semiring $([0, \infty], \sup, \odot)$ and pseudo-multiplication given by an increasing generator $g: [0, \infty] \to [0, \infty].$ In those cases, Liapounoff type inequalities deal with a measurable function $f: [0, 1] \to [0, \infty)$ and then $g \circ f : [0, 1] \to [0, \infty],$ i.e. (1.1) and (1.2) hold. Based on this fact, in this investigation the interval $[0, \infty]$ is considered, instead of the interval $[0, 1]$ observed in ^[15].

Let us consider a $g$-semiring $([0, \infty], \oplus, \odot)$ with generator $g:[0, \infty]\to[0, \infty]$ or a semiring $([0, \infty], \sup, \odot)$ where $\odot$ is given by an increasing generator $g:[0, \infty]\to[0, \infty]$ and an interval-valued function $F:[0, 1]\rightarrow \mathcal{I}$ represented by its border functions $F_l, \, F_r:[0, 1]\rightarrow [0, \infty)$ as $F(x) = [F_l(x), F_r(x)].$

Lemma 2. Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ or $\beta \in \mathbb{R} \setminus \mathbb{Q}.$ Then it holds that

Proof. If $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ or $\beta \in \mathbb{R} \setminus \mathbb{Q},$ based on (2.5), (2.7) and (2.8) it holds that

Theorem 1. Let $([0, \infty], \oplus, \odot)$ be a $g$-semiring with generator $g:[0, \infty]\to[0, \infty].$ For a pseudo-integrably bounded interval-valued function $F(x) = [F_l(x), F_r(x)],$ where the border functions $F_l, \, F_r:[0, 1]\rightarrow [0, \infty)$ are measurable, $t, s, r\in \mathbb{R}$ and $0 < t < s < r,$ holds

Proof. Let all pseudo-powers in (3.1) be rational numbers.

From (2.8) and Lemma 1 for the left-hand side of inequality (3.1) it holds that

Similarly, for the right-hand side of inequality (3.1) from Lemma 1, definition of pseudo-multiplication on $[0, \infty]$ and definition of pseudo-multiplication on $\mathcal{I}$ it follows that

Let the generator $g$ be an increasing function. Based on (2.3) it holds that

Since the generator $g$ is an increasing function the total order $\preceq$ is the usual order on the real line.

For every $x\in\left(\int\limits_{[0, 1]}^\oplus F^{(s)} \odot d \mu\right)^{(r-t)}$ it holds that $x \leq \left(\int\limits_{[0, 1]}^\oplus F_r^{(s)} \odot d \mu\right)^{(r-t)}$. From inequality (1.1) applied to the function $F_r: [0, 1] \to [0, \infty)$ it follows that

Let

It holds that

Therefore, for every $x \in \left(\int\limits_{[0, 1]}^\oplus F^{(s)} \odot d \mu\right)^{(r-t)}$ holds $x \leq y.$

For every

holds

From (1.1) applied to the function $F_l: [0, 1] \to [0, \infty)$ it follows that

so that for $x = \left(\int\limits_{[0, 1]}^\oplus F_l^{(s)} \odot d \mu\right)^{(r-t)}$ holds $x \in \left(\int\limits_{[0, 1]}^\oplus F^{(s)} \odot d \mu\right)^{(r-t)}$ and $x \leq y.$

Now the inequality (3.1) holds by Definition 1.

Let the generator $g$ be a decreasing function. Based on (2.4) it holds that

Since the generator $g$ is a decreasing function the total order $\preceq$ is the order opposite to the usual order on the real line.

For every $x \in \left(\int\limits_{[0, 1]}^\oplus F^{(s)} \odot d \mu\right)^{(r-t)}$ it holds that $\left(\int\limits_{[0, 1]}^\oplus F_l^{(s)} \odot d \mu\right)^{(r-t)} \leq x.$ Based on inequality (1.1) applied to the function $F_l: [0, 1] \to [0, \infty)$ it follows that

Let

Then it holds that

and for every $x \in \left(\int\limits_{[0, 1]}^\oplus F^{(s)} \odot d \mu\right)^{(r-t)}$ it holds that $y \leq x.$

For every

holds

From inequality (1.1) applied to the function $F_l: [0, 1] \to [0, \infty)$ it holds that

so for $x = \left(\int\limits_{[0, 1]}^\oplus F_l^{(s)} \odot d \mu\right)^{(r-t)}$ holds $x \in \left(\int\limits_{[0, 1]}^\oplus F^{(s)} \odot d \mu\right)^{(r-t)}$ and $y \leq x.$

Now, inequality (3.1) holds by Definition 1.

If at least one of the pseudo-powers in (3.1) is not a rational number, the proof of the inequality (3.1) is similar, using Lemma 2 and $iv)$ from Lemma 1.

Remark 1. For $g(x) = x$ Theorem 1 is Liapounoff type inequality for Aumann integral.

Since the proof of Theorem 1 is based on pseudo-multiplication of sub-intervals from $\mathcal{I}$ and pseudo-power of elements from $\mathcal{I}$, based on results from ^[15] and ^[19] it is obvious that the next theorem holds.

Theorem 2. Let $([0, \infty], \sup, \odot)$ be a semiring from the first class where pseudo-multiplication $\odot$ is given by an increasing generator $g:[0, \infty]\to[0, \infty]$ and $\mu$ is a $\sup$-decomposable measure on $\mathcal{B} ([0, 1]),$ given by $\mu(A) = {\rm{ess}}\sup(\Psi(x): x\in A),$ where $\Psi:[0, 1]\to[0, \infty]$ is a continuous density. For a pseudo-integrably bounded interval-valued function $F = [F_l, F_r],$ where the border functions $F_l, \, F_r:[0, 1]\rightarrow [0, \infty)$ are measurable, $t, s, r\in \mathbb{R}$ and $0 < t < s < r$ holds

Remark 2. Theorem 1 holds if any $g$-semiring $([a, b], \oplus, \odot)$ is considered, where $[a, b]\subseteq[0, \infty],$ for $F(x) = [F_l(x), F_r(x)]$ holds $\mathrm{Range}(F_l)\subseteq[a, b]$ and $\mathrm{Range}(F_r)\subseteq[a, b].$ The similar holds for Theorem 2, for the semirings of the third class with generated pseudo-multipplication.

Examples

For the interval-valued function $F(x) = [F_l(x), F_r(x)],$ from (2.8), the definition of $g$-integral and properties of interval-valued pseudo-integral (see ^[10]), for any generator $g$ follows

Now, the left-hand side of inequality (3.1) has the form

Similarly, for the right-hand side of inequality (3.1) it holds that

Therefore, the inequality (3.1) has the form

Example 1. Since indefinite integral $\int\sin x^2 dx$ cannot be expressed in terms of elementary functions, $\int\limits_0^1 \sin x^2 dx$ will be estimated using the Liapounoff type inequality for the pseudo-integral of an interval-valued function.

Let $([0, \infty], \oplus, \odot)$ be the $g$-semiring with generator $g(x) = x^2.$ The interval-valued function $F(x) = [\sqrt{\sin x^2}, x]$ will be used for estimation of integral $\int\limits_0^1 \sin x^2 dx.$ In this case the left-hand side of the inequality (3.1) has the form

and the right-hand side of the inequality (3.1) has the form

From Definition 1 and Theorem 1 follows

Now, for the chosen parameters $t = \frac 12, s = 1$ and $r = \frac 32$ from the fact that $g$ is an increasing function it follows that

Example 2. Let $([0, \infty], \oplus, \odot)$ be the $g$-semiring with generator $g(x) = \ln(1+x).$ For the interval-valued function $F(x) = [F_l(x), F_r(x)],$ the left side of inequality (3) has the form

and the right side of inequality (3) has the form

Therefore,

for every $\alpha_1, \beta_1, \alpha_2, \beta_2\in[0, 1]$ such that $\alpha_1+ \beta_1 = 1$ and $\alpha_2+ \beta_2 = 1.$

The following example is based on an example from ^[26].

Example 3. Let $([0, \infty], \sup, \odot)$ be the semiring from the first class where pseudo-multiplication is generated by $g(x) = x.$ Let $\mu$ be a $\sup$-measure on $([0, \infty], \mathcal{B}([0, \infty]))$ with density function $\Psi(x) = x.$

Let us consider the interval-valued function $F(x) = [F_l(x), 2x],$ where $F_l (x) \leq 2x, \; x \in [0, 1].$

Similarly,

and

and

For the interval-valued function $F(x) = [F_l(x), 2x],$ where $F_l(x)\leq 2x, \; x\in [0, 1],$ based on (3.2) and (2.6) it holds that

4.   Liapounoff inequality for interval-valued central $g$-moments of order $n$

In this part, the interval Liapounoff inequality is applied for estimation of interval-valued central $g$-moment of order $n$ for interval-valued functions in a $g$-semiring.

Let $([a, b], \oplus, \odot)$ be a $g$-semiring such that $\mu([a, b]) = {\bf{1}},$ where ${\bf{1}}$ is neutral element for the given pseudo-multiplication and $\mu$ is an $\oplus$-measure. It is known that in the case when $\mu([a, b]) = {{\bf{1}}},$ pseudo-additive measure $\mu$ is called pseudo-probability measure and it is given by $\mu = g^{-1}\circ P,$ where $g$ is a generator of pseudo-operations and $P$ is a probability measure (see ^[18]).

Based on the result from ^[18] the central $g$-moment of order $n > 0$ for a measurable function $f:[0, 1]\to[a, b]$ is given by

One representation of the Liapounoff inequality for pseudo-integral (1.1) in terms of the central $g$-moment of order $n$ is given in the following Lemma.

Lemma 3. Let $([a, b], \oplus, \odot)$ be a $g$-semiring such that $\mu([a, b]) = {\bf{1}}.$ For a measurable function $f:[0, 1]\to[0, \infty)$ and central $g$-moment of order $n$ it holds that

Proof. For a measurable function $f$, the left-hand side of the Liapounoff inequality for pseudo-integral given in (1.1) is

The right-hand side of the same inequality is

so that the inequality (4.2) holds.

The following definition is one new generalization of the central $g$-moment of order $n$ in the sense of the interval-valued functions.

Definition 2. Let $([0, \infty], \oplus, \odot)$ be a $g$-semiring and $F = [F_l, F_r]$ be an interval-valued function where the border functions $F_l, \, F_r:[0, 1]\rightarrow [0, \infty)$ are measurable. The interval-valued central $g$-moment of order $n > 0$ for the interval-valued function $F = [F_l, F_r],$ is

Theorem 3. Let $([0, \infty], \oplus, \odot)$ be a $g$-semiring and let $F = [F_l, F_r]$ be an interval-valued function where the border functions $F_l, \, F_r:[0, 1]\rightarrow [0, \infty)$ are measurable. For interval-valued central $g$-moment of order $n$ it holds that

Proof. From (4.1) $0 < t < s < r,$ for the left-hand side of inequality (3.1) holds

and for the right-hand side of the same inequality holds

Now, from (3) follows the inequality (4.3).

Example 4. Let $([0, \infty), \oplus, \odot)$ be a $g$-semiring with generator $g(x) = x^{\frac1n}, \; n > 1.$ The inverse function is $g^{-1}(x) = x^n,$ and the pseudo operation are given by $x\oplus y = (\sqrt[n]x+\sqrt[n]y)^n$ and $x\odot y = xy.$

Let $F = [F_l, F_r]$ be an interval-valued function with measurable border functions, $t = n-1, \; s = n$ and $r = n+1, \; n > 1.$

Since $g$ is an increasing generator from (4.2) for function $F_l$ holds

Analogously, for function $F_r$ it follows that

For all $x \in [E^{g, n}[F_l], E^{g, n}[F_r]]$ holds $x \leq E^{g, n}[F_r].$ From (4.5) holds

Also, for all $y \in \left[\sqrt{E^{g, n-1}[F_l]\cdot E^{g, n+1}[F_l]}, \sqrt{E^{g, n-1}[F_r]\cdot E^{g, n+1}[F_r]}\right]$ holds the inequality $\sqrt{E^{g, n-1}[F_l]\cdot E^{g, n+1}[F_l]} \leq y,$ and from (4.4) it follows that

From Definition 1 it follows that

so one estimation of interval-valued central $g$-moment of order $n$ is

Note that in inequality (4.6), the estimation of interval-valued central $g$-moment of order $n$ is obtained using interval-valued central $g$-moment of order $n-1$ and interval-valued central $g$-moment of order $n+1.$

5.   Conclusions

In this paper, we have proven two generalizations of the Liapounoff inequality for pseudo-integral of interval-valued function. Also, the Liapounoff inequality for central $g$-moment of order $n$ for a function $f$ and the Liapounoff inequality for interval-valued central $g$-moment of order $n$ for an interval-valued function $F$ are proven.

The first step in the future investigation will be the generalization of theorems about the convergence of a sequence of random variables using the inequality (4.2) for the central $g$-moment of order $n$ in the pseudo-probability space. The second step will be the generalization of theorems about the convergence of a sequence of interval-valued random sets using the inequality (4.3) for interval-valued central $g$-moment of order $n,$ in the pseudo-probability space.

Acknowledgments

This work was supported by the Department of Fundamental Sciences, Faculty of Technical Sciences, University of Novi Sad, through the project "Teorijska i primenjena matematika u tehničkim i informatičkim naukama".

Conflict of interest

The authors declare that there are no conflicts of interest.