On interval-valued $\mathbb{K}$-Riemann integral and Hermite-Hadamard type inequalities

1.   Introduction

Interval analysis was first proposed in order to reduce errors during mathematical computation. Since the first monograph was written by Moore^[2], interval analysis plays an important role in many other fields such as engineering, economics, statistics, and so on. Especially in the engineering field, the dynamics model can solve many dynamic problems, and such systems usually involve multiple uncertain parameters or interval coefficients.

Based on the works of Moore, many researchers hope to establish a complete theoretical system for interval analysis like the real analysis. In 1979, Markov first presented the concept of g$H$-difference and differentiability for interval-valued functions in ^[6], which addressed the problems in subtraction between two intervals. In 2009, Stefanini gave some more research on g$H$-derivative and interval differential equations in ^[7]. In 2015, Lupulescu studied the fractional calculus for interval-valued functions in ^[5]. Last year, Chalco-Cano dealt with the algebra of g$H$-differentiable interval-valued functions in ^[9]. This year, Ghosh proposed the notions of g$H$-directional derivative, g$H$-Gâteaux derivative and g$H$-Fréchet derivative for interval-valued functions in ^[18]. The generalization of the theory for interval analysis drew more and more attentions in the past decade.

In 2006, Boros give some notions of a subfield $\mathbb{K}$ of $\mathbb{R}$ and presented the ideas of $\mathbb{Q}$-subdifferential of Jensen-convex functions in ^[22]. After this, in 2017, Olbryś introduced the concepts of $\mathbb{K}$-Riemann integral and gave a new generalization of classical Hermite-Hadamard inequality in ^[4]. Motivated by their works, one of the main purposes of this paper is to introduce a new generalization for interval-valued $\mathbb{K}$-Darboux integral and $\mathbb{K}$-Riemann integral.

On the other hand, convex functions take effects in optimal control theory, economic analysis, probability estimate, and many other fields. For the past decades, classical convex has been generalized to many other types. After Costa gave the concepts of interval-valued convex in ^[1], some classical inequalities for convex functions have been extended to the form of interval-valued especially the Hermite-Hadamard inequality. In fact, if a function is convex then it satisfies the Hermite-Hadamard inequality (More details, see ^[10,11,12], ^{[14,15,16,17]}, ^[19]). Motivated by this, the other purpose of this paper is to generalize some Hermite-Hadamard type inequalities with respect to $\mathbb{K}$-Riemann integral for $\mathbb{K}$-convex and $log$-$\mathbb{K}$-convex interval-valued functions. In addition, the results of this paper may be used as a powerful tool in fuzzy-valued function, interval optimization, and interval-valued differential equations.

In section 2, we provide the basic theory of interval analysis and some properties. The definition of interval-valued $\mathbb{K}$-Darboux, $\mathbb{K}$-Riemann integral and some basic properties are given in section 3. Then we propose the definition of radial $\mathbb{K}$-g$H$-derivative and the relationship between $\mathbb{K}$-Riemann integral in section 4. In section 5, we give some statements for Hermite-Hadamard type inequalities of $\mathbb{K}$-convex and log-$\mathbb{K}$-convex interval-valued functions. Some examples are also presented.

2.   Preliminaries

Let $\mathcal{K}_c = \{I = [a, b]\; |\; a, b \in \mathbb{R}, a \leq b\}$. The length of interval $I = [a, b] \in \mathcal{K}_c$ can be defined by $\ell(I): = b-a$. Moreover, we say $I$ is positive if $a > 0$, and we denote $\mathcal{K}_c^+$ by all positive intervals belong to $\mathcal{K}_c$.

Some basic properties of algebra operations between two intervals can be found in ^[2]. For two intervals $A = [a^-, a^+]$ and B$= [b^-, b^+]$ belong to $\mathcal{K}_c$, we now give the following important properties.

Definition 2.1. For any $A, B \in \mathcal{K}_c$, the g$H$-difference between $A, B$ can be defined as

Specially, if $B = b \in \mathbb{R}$ is a constant, then

More properties of g$H$-difference can be found in ^[8].

The partial order relationship "$\preceq$'' between $A$ and $B$ can be defined as

The Hausdorff-Pompeiu distance $\mathcal{H} : \mathcal{K}_c \times \mathcal{K}_c \rightarrow[0, \infty)$ between $A$ and $B$ is defined by $\mathcal{H}(A, B) = \max \left\{\left|a^{-}-b^{-}\right|, \left|a^{+}-b^{+}\right|\right\}$. Then $(\mathcal{K}_c, \mathcal{H})$ is a complete and separable metric space (see ^[13]).

Base on this, define a map $\|\cdot\|$: $\mathcal{K}_c \rightarrow[0, \infty)$, $\|A\|$: = $\max\{|a^-|, |a^+|\} = \mathcal{H}(A, \{0\})$. It's easy to see that $\|\cdot\|$ is a norm on $\mathcal{K}_c$. Hence, $(\mathcal{K}_c, \|\cdot\|)$ is a normed quasi-linear space (see ^[5]).

In this paper, we use symbols $F$ and $G$ refer to interval-valued functions. For any $F :[a, b] \rightarrow \mathcal{K}_c$ where $F = [f^-, f^+]$, we say that $F$ is $\ell$-increasing(or $\ell$-decreasing) on $[a, b]$ if $\ell(F):[a, b] \rightarrow[0, \infty)$ is strictly increasing(or decreasing) on $[a, b]$. If $\ell(F)$ is strictly monotone on $[a, b]$, then we say $F$ is $\ell$-monotone on $[a, b]$.

Definition 2.2. $F :[a, b] \rightarrow \mathcal{K}_c$ is said to be continuous at $x_0 \in [a, b]$, if

We denote $C([a, b], \mathcal{K}_c)$ by the set of all continuous interval-valued function on $[a, b]$.

Definition 2.3. (Stefanini^[7]) Let $F :[a, b] \rightarrow \mathcal{K}_c$, we say that $F(x)$ is g$H$-differential at $x_0 \in (a, b)$, if there exist $\mathcal{D} \in \mathcal{K}_c$ such that

$\mathcal{D}$ defined in this fashion is denoted the symbol $F^{\prime}(x_0)$ and $F^{\prime}(x_0)$ is said to be the g$H$-derivative of $F(x)$ at $x_0$.

More properties of g$H$-derivative can be found in ^[9].

Now, we introduced the concept of the field $\mathbb{K}$ and number set $[a, b]_\mathbb{K}$ given by Olbryś in ^[4]. Let $\mathbb{Q} \subseteq \mathbb{K} \subseteq \mathbb{R}$, the set $[a, b]_\mathbb{K}$ denotes by

It's obvious that $[a, b]_\mathbb{K} \subseteq [a, b]$. Also, we can regard $[a, b]_\mathbb{K}$ as a convex combination of $\{a, b\}$ with coefficient from $\mathbb{K}$, i.e. $[a, b]_\mathbb{K} = conv_{\mathbb{K}}(\{a, b\})$. For the case $\mathbb{K} = \mathbb{R}$, we use $[a, b] = conv(\{a, b\})$ instead of $[a, b]_\mathbb{K} = conv_{\mathbb{K}}(\{a, b\}).$ Actually, such subfield $\mathbb{K}$ of the real numbers $\mathbb{R}$ always exists. Since $\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset...\subset \mathbb{Q}(\sqrt{2}, \sqrt{3}, ..., \sqrt{p}) \subset... \subset \mathbb{R}$, where $p$ is a prime. There are infinitely many subfields $\mathbb{K}$ satisfies $\mathbb{Q} \subseteq \mathbb{K} \subseteq \mathbb{R}$. Meanwhile, $\mathbb{Q}$ is a subfield of any number field.

Now we can give the following definitions.

Definition 2.4. Let $F :[a, b] \rightarrow \mathcal{K}_c$, we say that $F$ is $\mathbb{K}$-linear if $F$ satisfies two properties:

(1) Additive:

(2) $\mathbb{K}$-Homogeneity:

Example 2.5. Let $\mathbb{K} = \mathbb{Q}(\sqrt{2}) = \{a+\sqrt{2}b\; |\; a, b\in \mathbb{Q}\}$ and let $F :\mathbb{K} \rightarrow \mathcal{K}_c$ be $F(x) = [-\sqrt{2}x, \sqrt{2}x]$. We can check that $F$ isn't linear in the usual sense but is $\mathbb{K}$-linear.

Definition 2.6. Let $F :[a, b] \rightarrow \mathcal{K}_c$, we say that $F$ is $\mathbb{K}$-convex if for any $\alpha \in \mathbb{K} \cap (0, 1)$,

We denote $CX_{\mathbb{K}}([a, b], \mathcal{K}_c)$ by the set of all $\mathbb{K}$-convex interval-valued function on $[a, b]$.

Definition 2.7. Let $F :[a, b] \rightarrow \mathcal{K}_c$, we say that $F$ is radial $\mathbb{K}$-continuous at $x_0 \in [a, b]$ if

where $\alpha \in \mathbb{K}_+$ and $\mathbb{K}_+$ denotes the all positive numbers of $\mathbb{K}$. We denote $C_\mathbb{K}([a, b], \mathcal{K}_c)$ by the set of all radial $\mathbb{K}$-continuous interval-valued function on $[a, b]$.

3.   Interval-valued $\mathbb{K}$-Riemann integral

We firstly present the concept of bounded, then the definition of supremum and infimum intervals for interval-valued functions can be given.

Definition 3.1. Let $F :[a, b] \rightarrow \mathcal{K}_c$, we say that $F$ is bounded on $[a, b]_{\mathbb{K}}$ if there exists $\mathcal{M} > 0$ such that

Remark 3.2. For any $F :[a, b] \rightarrow \mathcal{K}_c$, $F$ is bounded on $[a, b]_{\mathbb{K}}$ doesn't imply that $F$ is bounded on the whole $[a, b]$. For example, consider $\mathbb{K} = \mathbb{Q}$ and $F :[1, 2] \rightarrow \mathcal{K}_c$ given by

One can easily check that $F$ is bounded on $[1, 2]_{\mathbb{Q}}$ but not bounded on $[1, 2]$.

Definition 3.3. We say that $S \in \mathcal{K}_c$ is the supremum interval of $F :[a, b] \rightarrow \mathcal{K}_c$ on $[a, b]_{\mathbb{K}}$ if the following conditions hold.

(1) For any $x \in [a, b]_{\mathbb{K}}$, we have $F(x) \preceq S$.

(2) For any $\varepsilon > 0$, there is $x_0 \in [a, b]_{\mathbb{K}}$ such that $S \ominus_g \varepsilon \prec F(x_0)$.

Moreover,

Definition 3.4. We say that $T \in \mathcal{K}_c$ is the infimum interval of $F :[a, b] \rightarrow \mathcal{K}_c$ on $[a, b]_{\mathbb{K}}$ if the following conditions hold.

(1) For any $x \in [a, b]_{\mathbb{K}}$, we have $T \preceq F(x)$.

(2) For any $\varepsilon > 0$, there is $x_0 \in [a, b]_{\mathbb{K}}$ such that $F(x_0) \prec T + \varepsilon$.

Moreover,

Example 3.5. Consider $F :\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathcal{K}_c$ defined by $F(x) = \left[-\frac{1}{x+2}, \sin x\right]$. Let $\mathbb{K} = \mathbb{Q}$, then we have

Then

and

For a given $I = [a, b]$, let $I_1 \preceq I_2 \preceq \ldots \preceq I_n$ and $I_i = [a_{i}, b_i]$, where $a_{i+1} = b_i$, $1 \leq i \leq n, i \in \mathbb{N}$. The finite set of intervals $\tau = \{(I_{1}, I_{2}, \ldots, I_{n}), \; \bigcup_{i = 1}^n I_i = I$, $1 \leq i \leq n, i \in \mathbb{N}\}$ is called a partition of $I$. Let $\mathcal{P}_{[a, b]}$ denote by the set of all partitions of $I$. Through this, we can give the definition of $\mathbb{K}$-partition of $I$.

Consider any $\tau = (I_{1}, I_{2}, \ldots, I_{n}) \in \mathcal{P}_{[a, b]}^{\mathbb{K}}$, let

Note that

If $F :[a, b] \rightarrow \mathcal{K}_c$ is bounded on $[a, b]_{\mathbb{K}}$, these supremum and infimum intervals are finite well-defined intervals.

The upper and lower $\mathbb{K}$-Darboux sum associated to $F$ and partition $\tau$ can be defined by

and

Note that $U_{\mathbb{K}}(F, \tau), L_{\mathbb{K}}(F, \tau) \in \mathcal{K}_c$ and

Then the upper and lower $\mathbb{K}$-Darboux integral of $F$ on $[a, b]$ can be defined by

and

Since $F$ is bounded on $[a, b]_\mathbb{K}$, the above integrals exist.

Definition 3.6. Let $F :[a, b] \rightarrow \mathcal{K}_c$ be bounded on $[a, b]_{\mathbb{K}}$. If we have

then $F$ is said to be $\mathbb{K}$-Darboux integrable on $[a, b]$. Let $\int_{a}^{b} F(x) d_{\mathbb{K}} x \in \mathcal{K}_c$ refer to the $\mathbb{K}$-Darboux integral of $F$ on $[a, b]$. We denote $D_\mathbb{K}([a, b], \mathcal{K}_c)$ by the set of all $\mathbb{K}$-Darboux integrable interval-valued functions on $[a, b]$.

For the special case $\mathbb{K} = \mathbb{R}$, we use $\int_{a}^{b} F(x) d x$ instead of $\int_{a}^{b} F(x) d_{\mathbb{K}} x$ and $D([a, b], \mathcal{K}_c)$ instead of $D_\mathbb{K}([a, b], \mathcal{K}_c)$. The next statement shows that $\mathbb{K}$-Darboux integral is well-defined.

Theorem 3.7. Let $F :[a, b] \rightarrow \mathcal{K}_c$. Then $F \in D_\mathbb{K}([a, b], \mathcal{K}_c)$ if and only if for any $\varepsilon > 0$, there exists a partition $\tau \in \mathcal{P}_{[a, b]}^{\mathbb{K}}$ such that

Proof. First, consider $F \in D_\mathbb{K}([a, b], \mathcal{K}_c)$. For any $\varepsilon > 0$, there exist two partitions $\tau_1$, $\tau_2 \in \mathcal{P}_{[a, b]}^{\mathbb{K}}$ such that

Let $\tau = \tau_1 \cup \tau_2$, then we can get

Conversely, let $\varepsilon > 0$ and choose $\tau \in \mathcal{P}_{[a, b]}^{\mathbb{K}}$ which satisfies (3.5). Since

we have

Which finishes the proof.

By Theorem 3.5 and the properties of classical Darboux integral, we can obtain the following statement.

Corollary 3.8. Let $F :[a, b] \rightarrow \mathcal{K}_c$. Then we say that $F \in D_\mathbb{K}([a, b], \mathcal{K}_c)$ if and only if for any sequence $\{\tau_n\}_{n\in\mathbb{N}} \subseteq \mathcal{P}_{[a, b]}^{\mathbb{K}}$, where $\tau_n = (I_1^{(n)}, I_2^{(n)}, \ldots, I_{k_n}^{(n)})$ such that

and for any point $\xi^{(n)}_j \in [a_{j}^{(n)}, b_{j}^{(n)}]_\mathbb{K}$ we have

Now, we can give the definition of interval-valued $\mathbb{K}$-Riemann integral.

Definition 3.9. We say that $F :[a, b] \rightarrow \mathcal{K}_c$ is $\mathbb{K}$-Riemann integrable on $[a, b]$, if there exists an $H \in \mathcal{K}_c$ for any $\tau \in \mathcal{P}_{[a, b]}^{\mathbb{K}}$ such that

and $\xi_i \in [a_i, b_{i}]_\mathbb{K}$ for all $1 \leq i \leq n, i \in \mathbb{N}$, we have

We say that interval $H$ is the $\mathbb{K}$-Riemann integral of $F$ and we denote $R_\mathbb{K}([a, b], \mathcal{K}_c)$ by the set of all $\mathbb{K}$-Riemann integrable interval-valued functions on $[a, b]$.

Remark 3.10. From Corollary 3.8, it's not hard to see that $\mathbb{K}$-Darboux integral is equivalent to $\mathbb{K}$-Riemann integral of interval-valued functions. We use $\mathbb{K}$-Riemann integral and the symbol $\int^b_a F(x)d_{\mathbb{K}} x$ instead of $\mathbb{K}$-Darboux integral in the rest of this paper.

Theorem 3.11. Let $\mathbb{Q}\subseteq \mathbb{K}_1 \subseteq \mathbb{K}_2 \subseteq \mathbb{R}$ and let $F :[a, b] \rightarrow \mathcal{K}_c$, if $F \in R_{\mathbb{K}_2}([a, b], \mathcal{K}_c)$, then $F \in R_{\mathbb{K}_1}([a, b], \mathcal{K}_c)$ and

Proof. Consider $\tau_n = (I_1^{(n)}, I_2^{(n)}, \ldots, I_{k_n}^{(n)})\in \mathcal{P}_{[a, b]}^{\mathbb{K}_1}$ such that

Since $F \in R_{\mathbb{K}_2}([a, b], \mathcal{K}_c)$, for any point $\xi^{(n)}_j \in [a_{j}^{(n)}, b_{j}^{(n)}]_{\mathbb{K}_1}$, we have

By the arbitrariness of $\tau_n \in \mathcal{P}_{[a, b]}^{\mathbb{K}_1}$, we have

Corollary 3.12. Let $F :[a, b] \rightarrow \mathcal{K}_c$. If $F \in R([a, b], \mathcal{K}_c)$ then for any $\mathbb{K} \subseteq \mathbb{R}$, we have $F \in R_\mathbb{K}([a, b], \mathcal{K}_c)$ and

The next example shows that the reverse of the above result is not true.

Example 3.13. Let $\mathbb{K}_1 \subseteq \mathbb{K}_2 \subseteq \mathbb{R}$, and $\mathbb{K}_1 \neq \mathbb{K}_2$. Consider $F :[a, b] \rightarrow \mathcal{K}_c$ given by

It's obviously that $F \in R_{\mathbb{K}_1}([a, b], \mathcal{K}_c)$ and

Now, we consider any partition $\tau \in \mathcal{P}_{[a, b]}^{\mathbb{K}_2}\setminus \mathcal{P}_{[a, b]}^{\mathbb{K}_1}$. Since $[a_{i}, b_i]_{\mathbb{K}_{1}} \subset [a_{i}, b_i]_{\mathbb{K}_{2}}\subset [a_{i}, b_i]$, we obtain

It implies

Thus

which illustrates that $F \notin R_{\mathbb{K}_2}([a, b], \mathcal{K}_c)$. Moreover, if we pick another subfield $\mathbb{K}_3$ with $\mathbb{Q}\subseteq \mathbb{K}_3 \subset \mathbb{K}_2$, one can shows that $F \in R_{\mathbb{K}_3}([a, b], \mathcal{K}_c)$.

The following results can be obtained easily by the case of real-valued functions(see ^[3]).

Theorem 3.14. Let $F :[a, b] \rightarrow \mathcal{K}_c$. If $F \in CX_{\mathbb{K}}([a, b], \mathcal{K}_c)$, then there always have a unique $\Psi :[a, b] \rightarrow \mathcal{K}_c$ and

Moreover, $\Psi$ is convex and $\Psi \in C([a, b], \mathcal{K}_c)$.

Proposition 3.15. Let $F, G :[a, b] \rightarrow \mathcal{K}_c$ and $F, G \in R_{\mathbb{K}}([a, b], \mathcal{K}_c)$.

(i) For any $\alpha, \beta \in \mathbb{R}$, if $\alpha F + \beta G \in IR^{\mathbb{K}}_{[a, b]}$, then

(ii) If $F \subseteq G$, then

(iii) If $\left\|F(x)\right\|: [a, b] \rightarrow[0, \infty)$ is $\mathbb{K}$-Riemann integrable and

Theorem 3.16. Let $F, G: [a, b] \rightarrow \mathcal{K}_c$. If $F, G \in R_{\mathbb{K}}([a, b], \mathcal{K}_c)$, then

Moreover, if $\ell(F)-\ell(G)$ has a constant sign on $[a, b]$, then

Proof. First, let $\Phi^-: = f^- - g^-$, $\Phi^+: = f^+ - g^+$, then we have

It implies that

Moreover, if $\ell(F)-\ell(G)$ has constant sign on $[a, b]$, then $F \ominus_{g} G = [\phi^-, \phi^+]$, if $\ell(F)\geq \ell(G)$, or $F \ominus_{g} G = [\phi^+, \phi^-]$, if $\ell(F) < \ell(G)$. We now assume that $\ell(F)\geq \ell(G)$ on $[a, b]$, and $F \ominus_{g} G = [\phi^-, \phi^+]$. Thus, we have $\int^b_a \phi^- d_{\mathbb{K}} x \leq \int^b_a \phi^+ d_{\mathbb{K}} x$, which shows that

The other case $\ell(F) < \ell(G)$ can be proved in the same way.

Theorem 3.17. Let $c \in [a, b]_\mathbb{K}$. For any $F :[a, b] \rightarrow \mathcal{K}_c$, if $F \in R_{\mathbb{K}}([a, b], \mathcal{K}_c)$, then $F \in R_{\mathbb{K}}([a, c], \mathcal{K}_c)$ and $F \in R_{\mathbb{K}}([c, b], \mathcal{K}_c)$. Moreover

Proof. Suppose $F \in R_{\mathbb{K}}([a, b], \mathcal{K}_c)$. For any $\varepsilon > 0$ there exists $\tau \in \mathcal{P}_{[a, b]}^{\mathbb{K}}$ such that

By adding point $c$ to $\tau$ write as $\tau_c$ such that $\tau_c = (I_{1}, \ldots, I_{k}, I_c, I_{k+1}, \ldots, I_n)$, where $I_k = [a_k, c]$ and $I_c = [c, b_k]$. Consider $\tau_c = \tau_1 \cup \tau_2$ and

Since

It seems that

Similarly, we obtain

This illustrates $F \in R_{\mathbb{K}}([a, c], \mathcal{K}_c)$ and $F \in R_{\mathbb{K}}([c, b], \mathcal{K}_c)$.

Finally, throughout above result, we obtain

Similarly, we have

It seems (3.16) holds.

Next, we give a calculation of the integral for $\mathbb{K}$-linear interval-valued functions. Note that the $\mathbb{K}$-linear interval-valued functions may not continuous at all point and the set of discontinuous points may not countable, so the classical Riemann integrability has failed here.

Proposition 3.18. Let $F:[a, b] \rightarrow \mathcal{K}_c$. If $F$ is $\mathbb{K}$-linear, then $F \in R_{\mathbb{K}}([a, b], \mathcal{K}_c)$. Moreover,

Proof. It's obviously that if $F$ is $\mathbb{K}$-linear then $F \in R_{\mathbb{K}}([a, b], \mathcal{K}_c)$. Consider partitions $\tau_n = \left(I_1^{(n)}, I_2^{(n)}, \ldots, I_{k_n}^{(n)}\right) \in \mathcal{P}_{[a, b]}^{\mathbb{K}_1}$ and $\xi_{j}^{(n)} = a+\frac{j}{n}(b-a)$. By Corollary 3.8,

Example 3.19. Let $\mathbb{K} = \mathbb{Q}$. Consider $F:[0, 1] \rightarrow \mathcal{K}_c$ such that $F(x) = [-x, x].$ Note that $F$ is $\mathbb{K}$-linear.

By Proposition 3.18, we have

On the other hand, for a given partition $\tau = (I_{1}, I_{2}, \ldots, I_{n})\in \mathcal{P}_{[0, 1]}^{\mathbb{K}}$ such that $\ell(I_i) = \frac{1}{n}$, and $a_i, b_i \in [0, 1]_{\mathbb{K}}$, $1 \leq i \leq n, i \in \mathbb{N}$. Then

and

Thus

Consequently, Proposition 3.18 is verified.

The next example shows that if $c\in(a, b)$, Theorem 3.17 may not hold.

Example 3.20. Consider $F:[a, b] \rightarrow \mathcal{K}_c$ is $\mathbb{K}$-linear, let $c = \alpha a + (1-\alpha)b$, where $\alpha \in (0, 1)$. If $\mathbb{K} \neq \mathbb{R}$, then

Obviously, if we have $\alpha \in [0, 1]_\mathbb{K}$, (3.18) is equal to zero. But for some $\alpha \in (0, 1) \setminus \mathbb{K}$, i.e. $c \in [a, b] \setminus [a, b]_\mathbb{K}$, (3.18) is different from zero.

4.   Radial g$H$-$\mathbb{K}$-derivative for interval-valued functions

In this section, we introduce the definition of interval-valued radial g$H$-$\mathbb{K}$-derivative.

Definition 4.1. Let $F :[a, b] \rightarrow \mathcal{K}_c$, we say that $F(x)$ is radial g$H$-$\mathbb{K}$ differential at $x_0$ in the direction $\lambda$, if there exists $\mathcal{I} \in \mathcal{K}_c$ for any $h \in \mathbb{K}_+$ such that

$\mathcal{I}$ defined in this fashion is denoted by the symbol $D_\mathbb{K}^\lambda F(x_0)$ and $D_\mathbb{K}^\lambda F(x_0)$ is said to be the radial g$H$-$\mathbb{K}$-derivative of $F(x)$ at $x_0$ in the direction $\lambda$.

Check that if $F :[a, b] \rightarrow \mathcal{K}_c$ is $\mathbb{K}$-linear, then we have

If $F :[a, b] \rightarrow \mathcal{K}_c$ is g$H$-differential at $x_0 \in [a, b]$, then $F$ is radial g$H$-$\mathbb{K}$-differential at $x_0$ in the direction $\lambda$ and

The next statement shows the relationship between interval-valued $\mathbb{K}$-Riemann integral and radial g$H$-$\mathbb{K}$-derivative.

Theorem 4.2. Let $F :[a, b] \rightarrow \mathcal{K}_c$, and $F \in R_{\mathbb{K}}([a, x], \mathcal{K}_c)$, for any $x \in (a, b]$. We define $\Phi :[a, b] \rightarrow \mathcal{K}_c$ by

If $F \in C_\mathbb{K}([a, b], \mathcal{K}_c)$, then $\Phi$ is radial g$H$-$\mathbb{K}$-differential at $x$ in the direction $x-a$ and

Proof. Fix $x \in (a, b]$, for any $\varepsilon > 0$ and $h \in \mathbb{K}_+$, we have

Let $h \in \mathbb{K}_+$ be small enough, then

Thus

Now, we can give the next characterization of $\mathbb{K}$-convex interval-valued functions.

Theorem 4.3. Let $F :[a, b] \rightarrow \mathcal{K}_c$. If $F \in CX_{\mathbb{K}}([a, b], \mathcal{K}_c)$ and $F$ is $\ell$-monotone on $[a, b]$, then for any $x \in [a, b]$, we have

Proof. Since $F \in CX_{\mathbb{K}}([a, b], \mathcal{K}_c)$, by Theorem 3.14, there exist a unique continuous $\Psi :[a, b] \rightarrow \mathcal{K}_c$ and

Then we obtain

where $h \in \mathbb{K}_+$. Thus,

The following example illustrates that if F is not $\ell$-monotone, then (4.6) is false.

Example 4.4. Consider $F:[0, 2] \rightarrow \mathcal{K}_c$ such that

We see that $F \in CX_{\mathbb{K}}([0, 2], \mathcal{K}_c)$ for any subfield $\mathbb{K} \subseteq \mathbb{R}$. It's obviously that $F(x)$ is radial g$H$-$\mathbb{K}$-differential on $[0, 2]$, and

Since $D_\mathbb{K}^2 F(x) \in R([0, 2], \mathcal{K}_c)$, then we have $D_\mathbb{K}^2 F(x) \in R_\mathbb{K}([0, 2], \mathcal{K}_c)$. Check that $F(x)$ is $\ell$-increasing on $[0, 1]$ and $\ell$-decreasing on $[1, 2]$, then we obtain

and

Consequently,

But if we only consider $F(x)$ on $[0, 1]$, then

and

Thus, Theorem 4.3 is verified.

5.   Hermite-Hadamard type inequalities

Above all, let's study the standard Hermite-Hadamard type inequality in the case of $\mathbb{K}$-Riemann integral.

Theorem 5.1. Let $F :[a, b] \rightarrow \mathcal{K}_c$. If $F \in CX_{\mathbb{K}}([a, b], \mathcal{K}_c)$ then

Proof. Since $F \in CX_{\mathbb{K}}([a, b], \mathcal{K}_c)$, by Theorem 3.14, there exist a unique continuous $\Psi :[a, b] \rightarrow \mathcal{K}_c$ and

Note that $\Psi(x)$ is convex and satisfied the classical Hermite-Hadamard inequalities. Thus

By Corollary 3.12, we obtain

which finishes the proof.

Example 5.2. Consider $F :[0, 1] \rightarrow \mathcal{K}_c$ defined by $F(x) = [x^2, \sqrt{x}+1]$. Let $\mathbb{K} = \mathbb{Q}(\sqrt{3}) = \{a+b\sqrt{3}\; |\; a, b\in \mathbb{Q}\}$. Then we have

Since $F \in R([0, 1], \mathcal{K}_c)$, by Corollary 3.12, we have $F \in R_\mathbb{K}([0, 1], \mathcal{K}_c)$, and

It seems that

which illustrates

Consequently, Theorem 5.1 is verified.

Example 5.3. Let $\mathbb{K} \subset \mathbb{R}$, and $\mathbb{K} \neq \mathbb{Q}$. Consider $F :[0, 1] \rightarrow \mathcal{K}_c$ given by

Then

and

Apparently, $F \notin R([0, 1], \mathcal{K}_c)$ but $F \in R_\mathbb{K}([0, 1], \mathcal{K}_c)$, by Theorem 4.3 we have

Finally, we have

Consequently, Theorem 5.1 is verified.

Definition 5.4. Let $F:[a, b] \rightarrow \mathcal{K}_c^+$. We say that $F$ is $log$-$\mathbb{K}$-convex if for any $\alpha \in \mathbb{K} \cap (0, 1)$,

We denote $CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$ by the set of all $log$-$\mathbb{K}$-convex interval-valued function on $[a, b]$.

Remark 5.5. For any interval $A = [a^-, a^+] \in \mathcal{K}_c^+$, and $\alpha > 0$, we have

Proposition 5.6. Let $F:[a, b] \rightarrow \mathcal{K}_c^+$. If $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$, then we have

After that, we can give the following results about $log$-$\mathbb{K}$-convex interval-valued functions.

Theorem 5.7. Let $F:[a, b] \rightarrow \mathcal{K}_c^+$. If $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$, then for any $\alpha \in \mathbb{K} \cap (0, 1) \setminus \{\frac{1}{2}\}$,

Proof. The cases $\alpha = 0, 1$ are trivial. Since $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$, then

By integrating the above inequality, we obtain

We define $t = (1-2\alpha)x+\alpha(a+b)$, and $d_{\mathbb{K}} t = (1-2\alpha)d_{\mathbb{K}} x$.

If $x = a$, we have $t = (1 - \alpha) a+\alpha b$ and if $x = b$, we have $t = \alpha a+(1 - \alpha) b$. Thus,

Remark 5.8. If $\alpha = \frac{1}{2}$, it's obvious that

Theorem 5.9. Let $F:[a, b] \rightarrow \mathcal{K}_c^+$, and $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$. If $\varphi:[a, b]\rightarrow[0, +\infty)$ is $\mathbb{K}$-Riemann integrable on $[a, b]$ with $\int^b_a \varphi(x)d_{\mathbb{K}} x > 0$, then for any $p > 0$ we have

Proof. For any $p > 0$, if $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$, then we have $F^{2p} \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$. By Proposition 5.6,

Multiply the above inequality with $\varphi(x)$ and then integrate, we have

Thus,

Theorem 5.10. Let $F:[a, b] \rightarrow \mathcal{K}_c^+$, and $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$. If $\varphi:[a, b]\rightarrow[0, +\infty)$ is $\mathbb{K}$-Riemann integrable on $[a, b]$ with $\int^b_a \varphi(x)d_{\mathbb{K}} x > 0$, then we have

Proof. Firstly, since $F \in CX^{log}_\mathbb{K}({[a, b]}, \mathcal{K}_c)$, consider Jensen's inequality for interval-valued functions(see ^[20,21]), we obtain

By taking the exponential in the above inequality, we get the first inequality.

Eventually, we have

It seems that

By taking the exponential in the above inequality, we get the second inequality.

Example 5.11. Let $\mathbb{K} = \mathbb{Q}$. Consider $F:[1, 2] \rightarrow \mathcal{K}_c^+$ given by $F(x) = [\frac{1}{x}, x]$. Check that $F \in CX^{log}_\mathbb{K}({[1, 2], \mathcal{K}_c})$. Since $F \in R_\mathbb{K}({[1, 2], \mathcal{K}_c})$, if $\alpha = \frac{1}{3}$, we have

where

and

Also,

Thus, we obtain

Consequently, Theorem 5.7 is verified.

Now, consider $p = 1$, given $\varphi(x) = x$ with $\int_{1}^{2} \varphi(x) d_{\mathbb{K}} x = \frac{3}{2}$, then

Also we have

Thus, we obtain

Consequently, Theorem 5.9 is verified.

Next, compute that

and

Thus, we obtain

Consequently, Theorem 5.10 is verified.

6.   Conclusion

In this work, the concepts of interval-valued $\mathbb{K}$-Riemann integral and radial $\mathbb{K}$-g$H$-derivative are introduced and some basic properties are discussed. Furthermore, some new Hermite-Hadamard type inequalities for $\mathbb{K}$-convex and $log$-$\mathbb{K}$-convex interval-valued functions are established. The main result of this paper give a new type of integral for interval-valued functions, which may be used in the further study of fuzzy-valued functions, interval optimization and interval-valued differential equations. In the future, we intend to study some applications in interval optimizations and interval-valued differential equations by using $\mathbb{K}$-Riemann integral and $\mathbb{K}$-g$H$-derivative.

Acknowledgments

This work was supported in part by the Fundamental Research Funds for Central Universities (2019B44914), Special Soft Science Research Projects of Technological Innovation in Hubei Province (2019ADC146), Key Projects of Educational Commission of Hubei Province of China (D20192501), the Natural Science Foundation of Jiangsu Province (BK20180500) and the National Key Research and Development Program of China (2018YFC1508100).

Conflict of interest

The authors declare that they have no competing interests.