More General Ostrowski-Type Inequalities in the Fuzzy Context

Abstract

In this study, Ostrowski-type inequalities in fuzzy settings were investigated. A detailed theory of fuzzy analysis is provided and utilized to establish the Ostrowski-type inequality in the fuzzy number-valued space. The results obtained in this research not only provide a generalization of the results of Dragomir but also give an extended version of the Ostrowski-type inequalities obtained by Anastassiou.

1. Introduction

In 1938, A. Ostrowski [1] proved the following inequality concerning the distance between the integral mean $\frac{1}{\nu -\kappa }{\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho$ and the value $\phi \left(t\right)$, $t\in \left[\kappa ,\nu \right]$.

Theorem 1

([1]). Let $\phi :\left[\kappa ,\nu \right]\to \mathbb{R}$ be continuous on $\left[\kappa ,\nu \right]$ and differentiable on $\left(\kappa ,\nu \right)$ such that ${\phi }^{\prime }:\left(\kappa ,\nu \right)\to \mathbb{R}$ be bounded on $\left(\kappa ,\nu \right)$, i.e., ${∥{\phi }^{\prime }∥}_{\infty }:=\underset{\varrho \in \left(\kappa ,\nu \right)}{\sup }\left|{\phi }^{\prime }\left(\varrho \right)\right|<\infty$. Then

$$\left|\phi \left(t\right)-\frac{1}{\nu -\kappa }{\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho \right|\le \left[\frac{1}{4}+{\left(\frac{t-\frac{\kappa +\nu }{2}}{\nu -\kappa }\right)}^2\right]{∥{\phi }^{\prime }∥}_{\infty }\left(\nu -\kappa \right)$$

(1)

for all $t\in \left[\kappa ,\nu \right]$, and the constant $\frac{1}{4}$ is the best possible.

An improvement to Ostrowski’s inequality was investigated by Dragomir in [2]. Dragomir obtained the following important result:

Theorem 2

([2]). Let $\phi :\left[\kappa ,\nu \right]\to \mathbb{C}$ (where $\mathbb{C}$ is the set of complex numbers) be an absolutely continuous function on $\left[\kappa ,\nu \right]$, whose derivative ${\phi }^{\prime }\in {\mathcal{L}}_{\infty }\left[\kappa ,\nu \right]$. Then

$$\begin{array}{c}\left|\phi \left(t\right)-\frac{1}{\nu -\kappa }{\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho \right|\le \frac{1}{2\left(\nu -\kappa \right)}\left[{∥{\phi }^{\prime }∥}_{\left[\kappa ,t\right],\infty }{\left(t-\kappa \right)}^2+{∥{\phi }^{\prime }∥}_{\left[t,\nu \right],\infty }{\left(\nu -t\right)}^2\right]\hfill \\ \hfill \le \left\{\begin{array}{c}{∥{\phi }^{\prime }∥}_{\left[\kappa ,\nu \right],\infty }\left[\frac{1}{4}+{\left(\frac{t-\frac{\kappa +\nu }{2}}{\nu -\kappa }\right)}^2\right]\left(\nu -\kappa \right);\\ \\ \frac{1}{2}{\left[{∥{\phi }^{\prime }∥}_{\left[\kappa ,t\right],\infty }^{\alpha }+{∥{\phi }^{\prime }∥}_{\left[t,\nu \right],\infty }^{\alpha }\right]}^{\frac{1}{\alpha }}{\left[{\left(\frac{t-\kappa }{\nu -\kappa }\right)}^{2\beta }+{\left(\frac{\nu -t}{\nu -\kappa }\right)}^{2\beta }\right]}^{\frac{1}{\beta }};\\ \\ \mathit{where}\alpha >1\mathit{with}\frac{1}{\alpha }+\frac{1}{\beta }=1;\\ \\ \frac{1}{2}\left[{∥{\phi }^{\prime }∥}_{\left[\kappa ,t\right],\infty }+{∥{\phi }^{\prime }∥}_{\left[t,\nu \right],\infty }\right]{\left[\frac{1}{2}+\left|\frac{t-\frac{\kappa +\nu }{2}}{\nu -\kappa }\right|\right]}^2\left(\nu -\kappa \right)\end{array}\right.\end{array}$$

(2)

for all $t\in \left[\kappa ,\nu \right]$, where ${∥·∥}_{\left[\kappa ,\nu \right],\infty }$ denotes the usual norm on ${\mathcal{L}}_{\infty }\left[\kappa ,\nu \right]$, i.e.,

${∥\phi ∥}_{\left[\kappa ,\nu \right],\infty }=ess\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\left|\phi \left(\varrho \right)\right|<\infty$.

Anastassiou in [3] proved an Ostrowski-type inequality, which we mention in the theorem below:

Theorem 3

([3]). Let $\left(\mathcal{Y},∥·∥\right)$ be a Banach space and $\phi \in C^1\left(\left[\kappa ,\nu \right],\mathcal{Y}\right)$, with $t\in \left[\kappa ,\nu \right]$, then we obtain the following inequality:

$$∥\frac{1}{\nu -\kappa }{\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho -\phi \left(t\right)∥\le \left(\frac{{\left(b-x\right)}^2+{\left(x-a\right)}^2}{2\left(b-a\right)}\right){∥\left|{\phi }^{\prime }\right|∥}_{\infty },$$

(3)

where ${∥\left|{\phi }^{\prime }\right|∥}_{\infty }:=\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }∥{\phi }^{\prime }\left(\varrho \right)∥<\infty$. Inequality (3) is sharp. In particular, the optimal function is ${\phi }^{*}\left(\tau \right):={\left|\tau -t\right|}^{\alpha }·\left(b-a\right)·t_0$, $\alpha >1$, and $t_0$ is a fixed unit vector in $\mathcal{Y}$.

The interested reader can read [3] for more results on Ostrowski-type inequalities for mapping with values in Banach spaces. Suppose that $\mathcal{Y}$ is a Banach space and $-\infty <\kappa <\nu <\infty$. We denote by $\mathcal{L}\left(\mathcal{Y}\right)$ the Banach algebra of all bounded linear operators acting on $\mathcal{Y}$, and the norms of vectors or operators acting on $\mathcal{Y}$ is $∥·∥$.

A function $\phi :\left[\kappa ,\nu \right]\to \mathcal{Y}$ is called measurable if there exists a sequence of simple functions ${\phi }_n:\left[\kappa ,\nu \right]\to \mathcal{Y}$ which converges pointwise almost everywhere on $\left[\kappa ,\nu \right]$ at $\phi$. We recall that a measurable function $\phi :\left[\kappa ,\nu \right]\to \mathcal{Y}$ is Bochner integrable if and only if its norm function $\varrho ⟼∥\phi \left(\varrho \right)∥:\left[\kappa ,\nu \right]\to {\mathbb{R}}_{+}$ is Lebesgue integrable on $\left[\kappa ,\nu \right]$.

The following generalization of an Ostrowski scalar inequality holds [4].

Theorem 4

([4]). Assume that $\mathcal{B}:\left[\kappa ,\nu \right]\to \mathcal{L}\left(\mathcal{Y}\right)$ is Hölder continuous on $\left[\kappa ,\nu \right]$, i.e.,

$$∥\mathcal{B}\left(\varrho \right)-\mathcal{B}\left(\tau \right)∥\le H{\left|\varrho -\tau \right|}^{\alpha },\mathit{for}\mathit{all}\varrho ,\tau \in \left[\kappa ,\nu \right],$$

where $H>0$ and $\alpha \in \left(0,1\right]$. If $\phi :\left[\kappa ,\nu \right]\to \mathcal{Y}$ is Bochner integrable on $\left[\kappa ,\nu \right]$, then we have the inequality

$$\begin{array}{c}\left|{\int }_{\kappa }^{\nu }\mathcal{B}\left(\tau \right)\phi \left(\tau \right)d\tau -\mathcal{B}\left(\varrho \right){\int }_{\kappa }^{\nu }\phi \left(\tau \right)d\tau \right|\le H{\int }_{\kappa }^{\nu }{\left|\varrho -\tau \right|}^{\alpha }∥\phi \left(\tau \right)∥d\tau \hfill \\ \hfill \le H\times \left\{\begin{array}{c}\frac{{\left(\varrho -\kappa \right)}^{\alpha +1}+{\left(\nu -\varrho \right)}^{\alpha +1}}{\alpha +1}\underset{\varrho \in \left[\kappa ,\nu \right]}{\mathit{ess}\sup }∥\phi \left(\varrho \right)∥;\\ \\ {\left[\frac{{\left(\varrho -\kappa \right)}^{q\alpha +1}+{\left(\nu -\varrho \right)}^{q\alpha +1}}{q\alpha +1}\right]}^{\frac{1}{q}}{\left({\int }_{\kappa }^{\nu }{∥\phi \left(\varrho \right)∥}^{p}d\varrho \right)}^{\frac{1}{p}};\\ \\ \mathit{where}p,q>1\mathit{with}\frac{1}{p}+\frac{1}{q}=1;\\ \\ {\left[\frac{1}{2}\left(\nu -\kappa \right)+\left|\varrho -\frac{\kappa +\nu }{2}\right|\right]}^{\alpha }{\int }_{\kappa }^{\nu }∥\phi \left(\varrho \right)∥d\varrho \end{array}\right.\end{array}$$

(4)

for any $\varrho \in \left[\kappa ,\nu \right]$, provided the integrals and $\underset{\varrho \in \left[\kappa ,\nu \right]}{\mathit{ess}\sup }∥\phi \left(\varrho \right)∥$ from the right-hand side are finite.

A weighted version of Ostrowski’s inequality for two functions with values in Banach spaces was established by Dragomir in [5].

Theorem 5

([5]). Assume that $\varkappa :\left[\kappa ,\nu \right]\to \mathbb{C}$ (where $\mathbb{C}$ is the set of complex numbers) and $\phi :\left[\kappa ,\nu \right]\to \mathcal{Y}$ are continuous and φ is strongly differentiable on $\left(\kappa ,\nu \right)$, then for all $\overline{u}\in \left[\kappa ,\nu \right]$, the inequality

$$∥{\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\phi \left(\varrho \right)d\varrho -\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)d\varrho \right)\phi \left(\overline{u}\right)∥\le C\left(\varkappa ,\phi ,\overline{u}\right),$$

holds and

$$\begin{array}{c}\hfill C\left(\varkappa ,\phi ,\overline{u}\right)\le {\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\left|\varkappa \left(\tau \right)\right|d\tau \right)∥{\phi }^{\prime }\left(\varrho \right)∥d\varrho +{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\left|\varkappa \left(\tau \right)\right|d\tau \right)∥{\phi }^{\prime }\left(\varrho \right)∥d\varrho .\end{array}$$

(5)

We also have the bounds

$$C\left(\varkappa ,\phi ,\overline{u}\right)\le \left\{\begin{array}{c}{\int }_{\varrho }^{\nu }\left|\varkappa \left(\tau \right)\right|d\tau {\int }_{\overline{u}}^{\nu }∥{\phi }^{\prime }\left(\varrho \right)∥d\varrho +{\int }_{\kappa }^{\varrho }\left|\varkappa \left(\tau \right)\right|d\tau {\int }_{\kappa }^{\overline{u}}∥{\phi }^{\prime }\left(\varrho \right)∥d\varrho ,\\ \\ {\left[{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }{\left|\varkappa \left(\tau \right)\right|}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\overline{u}}^{\nu }{∥{\phi }^{\prime }\left(\varrho \right)∥}^{q}d\varrho \right)}^{\frac{1}{q}}\\ +{\left[{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }{\left|\varkappa \left(\tau \right)\right|}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\kappa }^{\overline{u}}{∥{\phi }^{\prime }\left(\varrho \right)∥}^{q}d\varrho \right)}^{\frac{1}{q}},\\ \\ \underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }∥{\phi }^{\prime }\left(\varrho \right)∥{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\left|\varkappa \left(\tau \right)\right|d\tau \right)d\varrho \\ +\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }∥{\phi }^{\prime }\left(\varrho \right)∥{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\left|\varkappa \left(\tau \right)\right|d\tau \right)d\varrho ,\end{array}\right.$$

(6)

where $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$.

The next section is devoted to the basic definitions and results of fuzzy numbers and fuzzy number-valued functions.

2. Preliminaries

In this section, we point out some basic definitions and results that will help us in the sequel to this paper. We begin with the following:

Definition 1

([6]). Let us denote by ${\mathbb{R}}_{\mathcal{F}}$ the class of fuzzy subsets of real axis $\mathbb{R}$ (i.e., $u:\mathbb{R}⟶\left[0,1\right]$), i.e., ${\mathbb{R}}_{\mathcal{F}}$ is the set of functions $u:\mathbb{R}⟶\left[0,1\right]$, satisfying the following properties:

(i) 

$\forall u\in {\mathbb{R}}_{\mathcal{F}},$ u is normal, i.e., $u\left(t\right)=1$, $t\in \mathbb{R}$.

(ii) 

$\forall u\in {\mathbb{R}}_{\mathcal{F}},$ u is a convex fuzzy set, i.e.,

$$u(\varrho t+(1-\varrho )s)\ge \min \left\{u\left(t\right),u\left(s\right)\right\},\forall \varrho \in \left[0,1\right],$$

where $s,t\in \mathbb{R}$.

(iii) 

$\forall u\in$${\mathbb{R}}_{\mathcal{F}},$ u is upper semicontinuous on $\mathbb{R}$.

(iv) 

$\overline{\left\{t\in \mathbb{R}:u\left(t\right)>0\right\}}$ is compact.

The set ${\mathbb{R}}_{\mathcal{F}}$ is called the space of fuzzy real numbers.

Remark 1.

It is clear that $\mathbb{R}\subset {\mathbb{R}}_{\mathcal{F}}$, because any real number $t_0\in \mathbb{R}$ can be described as the fuzzy number whose value is 1 for $t=t_0$ and zero otherwise.

We will collect some further definitions and notations needed in the sequel [7].

For $0<r\le 1$ and $u\in {\mathbb{R}}_F$, we define

$${\left[u\right]}^r=\left\{t\in \mathbb{R}:u\left(t\right)\ge r\right\}$$

and

$${\left[u\right]}^0=\overline{\left\{t\in \mathbb{R}:u\left(t\right)>0\right\}}.$$

Now, it is well-known that for each $r\in \left[0,1\right],$${\left[u\right]}^r$ is a bounded closed interval. For $u,v\in {\mathbb{R}}_{\mathcal{F}}$ and $\lambda \in \mathbb{R}$, the sum $u\oplus v$ and the product $\lambda \odot u$ are defined by ${\left[u\oplus v\right]}^r={\left[u\right]}^r+{\left[v\right]}^r$, ${\left[\lambda \odot u\right]}^r=\lambda {\left[u\right]}^r,$$\forall r\in \left[0,1\right]$, where ${\left[u\right]}^r+{\left[v\right]}^r$ means the usual addition of two intervals as subsets of $\mathbb{R}$ and $\lambda {\left[u\right]}^r$ means the usual product between a scalar and a subset of $\mathbb{R}$.

Now we define $\mathcal{D}:{\mathbb{R}}_{\mathcal{F}}\times {\mathbb{R}}_{\mathcal{F}}⟶\mathbb{R}\cup \left\{0\right\}$ by

$$\mathcal{D}(u,v)=\underset{r\in \left[0,1\right]}{\sup }\left(\max \left\{\left|u_{-}^r-v_{-}^r\right|,\left|u_{+}^r-v_{+}^r\right|\right\}\right),$$

where ${\left[u\right]}^r=\left[u_{-}^r,u_{+}^r\right],$${\left[v\right]}^r=\left[v_{-}^r,v_{+}^r\right],$ then $(\mathcal{D},{\mathbb{R}}_{\mathcal{F}})$ is a metric space, and it possesses the following properties:

(i)

$\mathcal{D}(u\oplus w,v\oplus w)=\mathcal{D}(u,v),$$\forall u,v,w\in {\mathbb{R}}_{\mathcal{F}}.$

(ii)

$\mathcal{D}(\lambda \odot$$u,$$\lambda \odot v)=$$\lambda \mathcal{D}(u,v),\forall u,v\in {\mathbb{R}}_{\mathcal{F}},\forall \lambda \in \mathbb{R}.$

(iii)

$\mathcal{D}(u\oplus v,w\oplus e)\le \mathcal{D}(u,w)+\mathcal{D}(v,e),$$\forall u,v,w,e\in {\mathbb{R}}_{\mathcal{F}}$.

Moreover, it is well-known that $({\mathbb{R}}_{\mathcal{F}},\mathcal{D})$ is a complete metric space.

Furthermore, we have the following theorem:

Theorem 6

([8]). We have the following properties of fuzzy real numbers:

(i) 

If we denote $\tilde{o}={\mathcal{X}}_{\left\{0\right\}}$ (${\mathcal{X}}_{\left\{0\right\}}$ is the characteristic function of $\left\{0\right\}$ that only takes values 0 or 1), then $\tilde{o}\in {\mathbb{R}}_{\mathcal{F}}$ is the neutral element with respect to ⊕, i.e., $u\oplus \tilde{o}=\tilde{o}\oplus u$, for all $u\in {\mathbb{R}}_{\mathcal{F}}$.

(ii) 

If $\tilde{o}\in {\mathbb{R}}_{\mathcal{F}}$ is the neutral element with respect to ⊕, then none of $u\in {\mathbb{R}}_{\mathcal{F}}\setminus \mathbb{R}$ has the opposite in ${\mathbb{R}}_{\mathcal{F}}$ with respect to ⊕.

(iii) 

For all $\kappa ,\nu \in \mathbb{R}$ with $\kappa ,\nu \ge 0$ or $\kappa ,\nu \le 0$ and any $u\in {\mathbb{R}}_{\mathcal{F}}$, we have $(\kappa +\nu )\odot u=\kappa \odot u\oplus \nu \odot u$. For all $\kappa ,\nu \in \mathbb{R}$ if $\kappa ,\nu \ngeqq 0$ or $\kappa ,\nu \nleqq 0$, the above property does not hold for any $u\in {\mathbb{R}}_{\mathcal{F}}$.

(iv) 

For any $\lambda \in \mathbb{R}$ and any $u,v\in {\mathbb{R}}_{\mathcal{F}},$ we have $\lambda \odot (u\oplus v)=\lambda \odot u\oplus \lambda \odot v.$

(v) 

For any $\lambda ,\mu \in \mathbb{R}$ and any $u\in \mathbb{R},$ we have $\lambda \odot (\mu \odot v)=(\lambda .\mu )\odot v.$

(vi) 

If we denote ${∥u∥}_{\mathcal{F}}=\mathcal{D}(u,\tilde{o}),$$\forall u\in {\mathbb{R}}_{\mathcal{F}}$, then ${∥.∥}_{\mathcal{F}}$ is a norm on ${\mathbb{R}}_{\mathcal{F}}$, i.e., ${∥u∥}_{\mathcal{F}}=0$ if and only if $u=\tilde{o}$, ${∥\lambda \odot u∥}_{\mathcal{F}}=\left|\lambda \right|.{∥u∥}_{\mathcal{F},}$ and ${∥u\oplus v∥}_{\mathcal{F}}\le {∥u∥}_{\mathcal{F}}+{∥v∥}_{\mathcal{F}},\left|{∥u∥}_{\mathcal{F}}+{∥v∥}_{\mathcal{F}}\right|\le \mathcal{D}(u,v).$

Remark 2.

The propositions (ii) and (iii) in the above theorem show us that $({\mathbb{R}}_{\mathcal{F}},\oplus ,\odot )$ is not a linear space over $\mathbb{R}$, and consequently, $({\mathbb{R}}_{\mathcal{F}},{∥.∥}_{\mathcal{F}})$ cannot be a normed space. However, the properties of $\mathcal{D}$ and those in theorem propositions (iv)–(vi) have as an effect that most of the metric properties of functions defined on $\mathbb{R}$ with values in a Banach space can be extended to functions $\phi :\mathbb{R}⟶{\mathbb{R}}_{\mathcal{F}}$, called fuzzy number-valued functions.

In this paper, for ranking concept, we will use a partial ordering, introduced in [9].

Definition 2

([9]). Let the partial ordering ≼ in ${\mathbb{R}}_{\mathcal{F}}$ by $u\preccurlyeq v$ if and only if $u_{-}^r\le v_{-}^r$ and $u_{+}^r\le v_{+}^r$, $\forall r\in \left[0,1\right]$, and the strict inequality ≺ in ${\mathbb{R}}_{\mathcal{F}}$ is defined by $u\prec v$ if and only if $u_{-}^r<v_{-}^r$ and $u_{+}^r<v_{+}^r$, $\forall r\in \left[0,1\right]$, where ${\left[u\right]}^r=\left[u_{-}^r,u_{+}^r\right],{\left[v\right]}^r=\left[v_{-}^r,v_{+}^r\right]$.

Definition 3

([10]). Let t, $s\in {\mathbb{R}}_{\mathcal{F}}$. If there exists a $z\in {\mathbb{R}}_{\mathcal{F}}$ such that $t=s\oplus z$, then we call z the $\mathcal{H}$-difference of t and s, denoted by $z=t\ominus s.$

Definition 4

([11]). Given two fuzzy numbers $u,v\in {\mathbb{R}}_{\mathcal{F}}$, the generalized Hukuhara difference ($g\mathcal{H}$-difference for short) is the fuzzy number w, if it exists, such that

$$u{\ominus }_{g\mathcal{H}}v=w⟺\left\{\begin{array}{c}\left(i\right)u=v\oplus w,\hfill \\ \\ \mathit{or}\left(\mathit{ii}\right)u=v\oplus \left(-1\right)w.\hfill \end{array}\right.$$

Remark 3.

It is easy to show that (i) and (ii) are both valid if and only if w is a neutral number, also known as crisp number.

In terms of r-levels, we have

$${\left[u{\ominus }_{g\mathcal{H}}v\right]}^r=\left[\min \left\{u_{-}^r-v_{-}^r,u_{+}^r-v_{+}^r\right\},\max \left\{u_{-}^r-v_{-}^r,u_{+}^r-v_{+}^r\right\}\right],$$

and the conditions for the existence of $w=u{\ominus }_{g\mathcal{H}}v\in {\mathbb{R}}_{\mathcal{F}}$ are as follows:

$$\mathrm{Case}\left(\mathrm{i}\right)\left\{\begin{array}{c}{w}_{-}^r=u_{-}^r-v_{-}^r\mathrm{and}{w}_{+}^r=u_{+}^r-v_{+}^r\forall r\in \left[0,1\right],\\ \\ \mathrm{with}{w}_{-}^r\mathrm{increasing},w_{+}^r\mathrm{decreasing},w_{-}^r\le w_{+}^r.\end{array}\right.$$

$$\mathrm{Case}\left(\mathrm{ii}\right)\left\{\begin{array}{c}{w}_{-}^r=u_{+}^r-v_{+}^r\mathrm{and}{w}_{+}^r=u_{-}^r-v_{-}^r\forall r\in \left[0,1\right],\\ \\ \mathrm{with}{w}_{-}^r\mathrm{increasing},w_{+}^r\mathrm{decreasing},w_{-}^r\le w_{+}^r.\end{array}\right.$$

If the $g\mathcal{H}$-difference $u{\ominus }_{g\mathcal{H}}v$ does not define a proper fuzzy number, the nested property can be used for r-levels and can obtain a proper fuzzy number by

$${\left[u{\ominus }_{g}v\right]}^r=\overline{\underset{r_0\ge r}{\cup }\left({\left[u\right]}^{r_0}{\ominus }_{g\mathcal{H}}{\left[v\right]}^{r_0}\right)},r\in \left[0,1\right],$$

where $u{\ominus }_{g}v$ defines the generalized difference of two fuzzy numbers $u,v\in {\mathbb{R}}_{\mathcal{F}}$, defined in [10] and extended and studied in [11].

Remark 4.

Throughout this paper, we assume that if $u,v\in {\mathbb{R}}_{\mathcal{F}}$, then $u{\ominus }_{g\mathcal{H}}v\in {\mathbb{R}}_{\mathcal{F}}$.

Proposition 1

([12]). For u, $v\in {\mathbb{R}}_{\mathcal{F}}$, we have

$$\mathcal{D}\left(u{\ominus }_{g\mathcal{H}}v,\tilde{o}\right)\le \mathcal{D}\left(u,v\right).$$

Proposition 2

([10]). For u, $v\in {\mathbb{R}}_{\mathcal{F}}$. If $u{\ominus }_{g\mathcal{H}}v$ exists, it is unique and has the following properties ($\tilde{o}$ denotes the crisp set $\left\{0\right\}$):

(i) 

$u{\ominus }_{g\mathcal{H}}u$$=\tilde{o}$.

(ii) 

(a) $(u\oplus v){\ominus }_{g\mathcal{H}}v=u$, (b) $u{\ominus }_{g\mathcal{H}}(u\ominus v)=v$.

(iii) 

If $u{\ominus }_{g\mathcal{H}}v$ exists, then also $(-v){\ominus }_{g\mathcal{H}}(-u)$ exists, and $\tilde{o}{\ominus }_{g\mathcal{H}}\left(u{\ominus }_{g\mathcal{H}}v\right)=(-v){\ominus }_{g\mathcal{H}}(-u)$.

(iv) 

$u{\ominus }_{g\mathcal{H}}v=v{\ominus }_{g\mathcal{H}}u=w$ if and only if $w=-w$ (in particular, $w=\tilde{o}$ if and only if $u=v$).

(v) 

If $v{\ominus }_{g\mathcal{H}}u$ exists, then either $u\oplus \left(v{\ominus }_{g\mathcal{H}}u\right)=u$ or $v\ominus \left(v{\ominus }_{g\mathcal{H}}u\right)=u$, and if both equalities hold, then $v{\ominus }_{g\mathcal{H}}u$ is a crisp set $\tilde{o}$.

Definition 5

([10]). Let u, $v\in {\mathbb{R}}_{\mathcal{F}}$ have r-levels ${\left[u\right]}^r=\left[u_{-}^r,u_{+}^r\right],{\left[v\right]}^r=\left[v_{-}^r,v_{+}^r\right]$, with $\tilde{o}\notin {\left[v\right]}^r$$\forall r\in \left[0,1\right]$. The $g\mathcal{H}$-division ${÷}_{g\mathcal{H}}$ is the operation that calculates the fuzzy number $w=u{÷}_{g\mathcal{H}}v\in {\mathbb{R}}_{\mathcal{F}}$, defined by

$$u÷g\mathcal{H}v=w\iff \left\{\begin{array}{c}\left(i\right)u=v\odot w,\hfill \\ \\ \mathit{or}\left(\mathit{ii}\right)v=u\odot w^{-1},\hfill \end{array}\right.$$

provided that w is a proper fuzzy number.

Proposition 3

([10]). Let $u,v\in {\mathbb{R}}_{\mathcal{F}}$ (here 1 is the same as $\left\{1\right\}$). We have the following:

(i) 

If $\tilde{o}\notin {\left[u\right]}^r\forall r$, then $u{÷}_{g\mathcal{H}}u=1$.

(ii) 

If $\tilde{o}\notin {\left[v\right]}^r\forall r$, then $uv{÷}_{g\mathcal{H}}v=u$.

(iii) 

If $\tilde{o}\notin {\left[v\right]}^r\forall r$, then $1{÷}_{g\mathcal{H}}v=v^{-1}$ and $1{÷}_{g\mathcal{H}}{v}^{-1}=v$.

(iv) 

If $v{÷}_{g\mathcal{H}}u$ exists, then either $u\left(v{÷}_{g\mathcal{H}}u\right)=v$ or $v{\left(v{÷}_{g\mathcal{H}}u\right)}^{-1}=u$, and both equalities hold if and only if $v{÷}_{g\mathcal{H}}u$ is a crisp set.

Remark 5.

If $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ is a fuzzy-valued function, then the r-level representation of φ is given by $\phi \left(t;r\right)=\left[\underline{\phi }\left(t;r\right),\overline{\phi }\left(t;r\right)\right]$, $t\in \left[\kappa ,\nu \right],r\in \left[0,1\right]$.

Definition 6

([13]). Let $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ be a fuzzy-valued function and $t_0\in \left[\kappa ,\nu \right]$. If $\forall \epsilon >0$ ∃ $\delta >0$∀t such that

$$0<\left|t-t_0\right|<\delta \Rightarrow \mathcal{D}\left(\phi \left(t\right),\mathcal{L}\right)<\epsilon ,$$

then we say that $\mathcal{L}$∈${\mathbb{R}}_{\mathcal{F}}$ is the limit of φ in $t_0$, which is denoted by $\underset{\varrho \to t_0}{lim}\phi \left(t\right)=\mathcal{L}$.

Definition 7

([13]). A function $\phi :\mathbb{R}⟶{\mathbb{R}}_{\mathcal{F}}$ is said to be continuous at $t_0\in \mathbb{R}$ if for every $\epsilon >0$ we can find $\delta >0$ such that $\mathcal{D}\left(\phi \left(t\right),\phi \left(t_0\right)\right)<\epsilon$, whenever $\left|t-t_0\right|<\delta$. φ is said to be continuous on $\mathbb{R}$ if it is continuous at every $t\in \mathbb{R}$. If φ is continuous at each $t_0\in \left[\kappa ,\nu \right]$, then the continuity is one-sided at end points κ, ν.

Lemma 1

([14]). For any κ, $\nu \in \mathbb{R}$, κ, $\nu \ge 0$ and $u\in {\mathbb{R}}_{\mathcal{F}}$, we have

$$\mathcal{D}\left(\kappa \odot u,\nu \odot u\right)\le \left|\kappa -\nu \right|\mathcal{D}(u,\tilde{o}),$$

where $\tilde{o}\in {\mathbb{R}}_{\mathcal{F}}$ is defined by $\tilde{o}:={\mathcal{X}}_{\left\{0\right\}}$.

Definition 8

([11]). Let $t_0\in \left(\kappa ,\nu \right)$ and h be such that $t_0+h\in \left(\kappa ,\nu \right)$, then the $g\mathcal{H}$-derivative of a function $\phi :\left(\kappa ,\nu \right)\to {\mathbb{R}}_{\mathcal{F}}$ at $t_0$ is defined as

$${\phi }_{g\mathcal{H}}^{\prime }\left(t_0\right)=\underset{h\to 0^{+}}{lim}\frac{\phi \left(t_0+h\right){\ominus }_{g\mathcal{H}}\phi \left(t_0\right)}{h}$$

If ${\phi }_{g\mathcal{H}}^{\prime }\left(t_0\right)\in {\mathbb{R}}_{\mathcal{F}}$, we say that φ is generalized Hukuhara differentiable ($g\mathcal{H}$-differentiable for short) at $t_0$.

Definition 9

([11]). Let $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ and $t_0\in \left(\kappa ,\nu \right)$, with $\underline{\phi }\left(t;r\right)$ and $\overline{\phi }\left(t;r\right)$ both differentiable at $t_0$. Furthermore, we say the following:

(a) 

φ is (i) $-g\mathcal{H}$-differentiable at $t_0$ if

$$\left(i\right){\left(\phi \right)}_{g\mathcal{H}}^{\prime }\left(t_0;r\right)=\left[{\left(\underline{\phi }\right)}^{\prime }\left(t_0;r\right),{\left(\overline{\phi }\right)}^{\prime }\left(t_0;r\right)\right],0\le r\le 1.$$

(7)

(b) 

φ is (ii) $-g\mathcal{H}$-differentiable at $t_0$ if

$$\left(\mathit{ii}\right){\left(\phi \right)}_{g\mathcal{H}}^{\prime }\left(t_0;r\right)=\left[{\left(\overline{\phi }\right)}^{\prime }\left(t_0;r\right),{\left(\underline{\phi }\right)}^{\prime }\left(t_0;r\right)\right],0\le r\le 1.$$

(8)

Definition 10

([12]). We say that a point $t_0\in \left(\kappa ,\nu \right)$ is a switching point for the differentiability of φ if in any neighborhood $\mathcal{V}$ of $t_0$ there exist points $t_1<t_0<t_2$ such that

type (I): at $t_1$ (7) holds while (8) does not hold and at $t_2$ (8) holds and ((7) does not hold, or

type (II): at $t_1$ (8) holds while (7) does not hold and at $t_2$ (7) holds and (8) does not hold.

Definition 11

([13]). Let $\phi :\left(\kappa ,\nu \right)\to {\mathbb{R}}_{\mathcal{F}}$ be $g\mathcal{H}$-differentiable at $c\in \left(\kappa ,\nu \right)$. Then φ is fuzzy continuous at c.

Theorem 7

([13]). Let I be a closed interval in $\mathbb{R}$. Let $\varkappa :I\to \varphi :=\varkappa \left(I\right)\subseteq \mathbb{R}$ be differentiable at t and $\phi :\varphi \to {\mathbb{R}}_{\mathcal{F}}$ be $g\mathcal{H}$-differentiable $\overline{u}=\varkappa \left(t\right)$. Assume that g is strictly increasing on I. Then ${\left(\phi \circ \varkappa \right)}_{g\mathcal{H}}^{\prime }\left(t\right)$ exists and

$${\left(\phi \circ \varkappa \right)}_{g\mathcal{H}}^{\prime }\left(t\right)={\phi }_{g\mathcal{H}}^{\prime }\left(\varkappa \left(t\right)\right)\odot {\varkappa }^{\prime }\left(t\right),\forall t\in I.$$

Definition 12

([7]). Let $\phi :\left[\kappa ,\nu \right]⟶{\mathbb{R}}_{\mathcal{F}}$. We say that φ is fuzzy Riemann integrable to $I\in {\mathbb{R}}_{\mathcal{F}}$ if for every $\epsilon >0$, there exists $\delta >0$ such that for any division $P=\left\{\left[\overline{u},\overline{v}\right];\xi \right\}$ of $\left[\kappa ,\nu \right]$ with the norms $\Delta \left(P\right)<\delta$, and so we have

$$\mathcal{D}\left({\sum }^{*}\left(\overline{v}-\overline{u}\right)\odot \phi \left(\xi \right),I\right)<\epsilon ,$$

where ${\sum }^{*}$ denotes the fuzzy summation. We choose to write

$$I:=\left(FR\right){\int }_{\kappa }^{\nu }\phi \left(t\right)dt.$$

We also call a φ as above $\left(FR\right)$-integrable.

Theorem 8

([15]). Let $\phi :\left[\kappa ,\nu \right]⟶{\mathbb{R}}_{\mathcal{F}}$ be integrable and $c\in \left[\kappa ,\nu \right]$. Then

$${\int }_{\kappa }^{\nu }\phi \left(t\right)dt={\int }_{\kappa }^c\phi \left(t\right)dt\oplus {\int }_c^{\nu }\phi \left(t\right)dt.$$

Corollary 1

([7]). If $\phi \in C\left(\left[\kappa ,\nu \right],{\mathbb{R}}_{\mathcal{F}}\right)$, then φ is $\left(FR\right)$-integrable.

Lemma 2

([16]). If $\phi ,\varkappa :\left[\kappa ,\nu \right]\subseteq \mathbb{R}⟶{\mathbb{R}}_{\mathcal{F}}$ are fuzzy continuous (with respect to the metric $\mathcal{D}$), then the function $F:\left[\kappa ,\nu \right]⟶{\mathbb{R}}_{+}\cup \left\{0\right\}$ defined by $F\left(t\right):=\mathcal{D}\left(\phi \left(t\right),\varkappa \left(t\right)\right)$ is continuous on $\left[\kappa ,\nu \right]$, and

$$\mathcal{D}\left(\left(FR\right){\int }_{\kappa }^{\nu }\phi \left(\overline{u}\right)d\overline{u},\left(FR\right){\int }_{\kappa }^{\nu }\varkappa \left(\overline{u}\right)d\overline{u}\right)\le {\int }_{\kappa }^{\nu }\mathcal{D}\left(\phi \left(t\right),\varkappa \left(t\right)\right)dt.$$

Lemma 3

([16]). Let $\phi :\left[\kappa ,\nu \right]\subseteq \mathbb{R}⟶{\mathbb{R}}_{\mathcal{F}}$ be fuzzy continuous. Then

$$\left(FR\right){\int }_{\kappa }^t\phi \left(\varrho \right)d\varrho$$

is a fuzzy continuous function in $t\in \left[\kappa ,\nu \right]$.

Proposition 4

([17]). Let $F\left(\varrho \right):={\varrho }^n\odot u$, $\varrho \ge 0$, $n\in \mathbb{N}$, and $u\in {\mathbb{R}}_{\mathcal{F}}$ be fixed. Then (the $g\mathcal{H}$-derivative)

$$F^{\prime }\left(\varrho \right)=n{\varrho }^{n-1}\odot u.$$

In particular, when $n=1$ then $F^{\prime }\left(\varrho \right)=u$.

Theorem 9

([18]). Let I be an open interval of $\mathbb{R}$ and let $\phi :I⟶{\mathbb{R}}_{\mathcal{F}}$ be $g\mathcal{H}$-fuzzy differentiable, $c\in \mathbb{R}$. Then ${\left(c\odot \phi \right)}_{g\mathcal{H}}^{\prime }$ exists and ${\left(c\odot \phi \left(t\right)\right)}_{g\mathcal{H}}^{\prime }=c\odot {\phi }_{g\mathcal{H}}^{\prime }\left(t\right)$.

Theorem 10

([17]). Let $\phi :\left[\kappa ,\nu \right]⟶{\mathbb{R}}_{\mathcal{F}}$ be a fuzzy differentiable function on $\left[\kappa ,\nu \right]$ with $g\mathcal{H}$-derivative ${\phi }^{\prime }$, which is assumed to be fuzzy continuous. Then

$$\mathcal{D}\left(\phi \left(d\right),\phi \left(c\right)\right)\le \left(d-c\right)\underset{\varrho \in \left[c,d\right]}{\sup }\mathcal{D}\left({\phi }^{\prime }\left(\varrho \right),\tilde{o}\right),$$

for any c, $d\in \left[\kappa ,\nu \right]$ with $d\ge c$.

Theorem 11

([11]). If φ is $g\mathcal{H}$-differentiable with no switching point in the interval $\left[\kappa ,\nu \right]$, then we have

$${\int }_{\kappa }^{\nu }{\phi }_{g\mathcal{H}}^{\prime }\left(t\right)dt=\phi \left(\nu \right){\ominus }_{g\mathcal{H}}\phi \left(\kappa \right).$$

Theorem 12

([13]). Let $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ be a continuous fuzzy-valued function. Then

$$F\left(\varrho \right)={\int }_{\kappa }^{\varrho }\phi \left(t\right)dt,\varrho \in \left[\kappa ,\nu \right]$$

is $g\mathcal{H}$-differentiable and $F_{g\mathcal{H}}^{\prime }\left(\varrho \right)=\phi \left(\varrho \right)$.

Theorem 13

([18]). Let $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ and $\varkappa :\left[\kappa ,\nu \right]\to {\mathbb{R}}^{+}$ be two differentiable functions (φ is $g\mathcal{H}$-differentiable); then

$${\int }_{\kappa }^{\nu }{\phi }_{g\mathcal{H}}^{\prime }\left(t\right)\odot \varkappa \left(t\right)dt=\left(\phi \left(\nu \right)\odot \varkappa \left(\nu \right)\right){\ominus }_{g\mathcal{H}}\left(\phi \left(\kappa \right)\odot \varkappa \left(\kappa \right)\right){\ominus }_{g\mathcal{H}}{\int }_{\kappa }^{\nu }\phi \left(t\right)\odot {\varkappa }^{\prime }\left(t\right)dt.$$

Theorem 14

([18]). Let $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ and $\varkappa :\left[\kappa ,\nu \right]\to {\mathbb{R}}^{+}$ be two differentiable functions (φ is $g\mathcal{H}$-differentiable); then

$${\int }_{\kappa }^t{\phi }_{g\mathcal{H}}^{\prime }\left(t\right)\odot \varkappa \left(t\right)dt=\left(\phi \left(t\right)\odot \varkappa \left(t\right)\right){\ominus }_{g\mathcal{H}}{\int }_{\kappa }^t\phi \left(t\right)\odot {\varkappa }^{\prime }\left(t\right)dt.$$

Since fuzziness is a natural reality different than randomness and determinism, therefore an attempt was made by Anastassiou [19] in 2003 to extend (1) to the fuzzy setting context. In fact, Anastassiou [19] proved the following important results for fuzzy Hölder and fuzzy differentiable functions, respectively:

Theorem 15

([19]). Let $\phi \in C\left([\kappa ,\nu ],{\mathbb{R}}_{\mathcal{F}}\right)$, the space of fuzzy continuous functions, $t\in \left[\kappa ,\nu \right]$ be fixed. If φ fulfills the Hölder condition

$$\mathcal{D}\left(\phi \left(s\right),\phi \left(z\right)\right)\le {\mathcal{L}}_{\phi }{\left|s-z\right|}^{\alpha },0<\alpha \le 1,\mathit{for}\mathit{all}s,z\in \left[\kappa ,\nu \right],$$

for some ${\mathcal{L}}_{\phi }>0$. Then

$$\mathcal{D}\left(\frac{1}{\nu -\kappa }\odot \left(FR\right){\int }_{\kappa }^{\nu }\phi \left(s\right)ds,\phi \left(t\right)\right)\le {\mathcal{L}}_{\phi }\left(\frac{{\left(t-\kappa \right)}^{\alpha +1}+{\left(\nu -t\right)}^{\alpha +1}}{\left(\alpha +1\right)\left(\nu -\kappa \right)}\right).$$

(9)

Theorem 16

([19]). Let $\phi \in C^1\left([\kappa ,\nu ],{\mathbb{R}}_{\mathcal{F}}\right)$, the space of one-time continuously differentiable functions in the fuzzy sense. Then for $t\in \left[\kappa ,\nu \right]$,

$$\begin{array}{c}\hfill \mathcal{D}\left(\frac{1}{\nu -\kappa }\odot \left(FR\right){\int }_{\kappa }^{\nu }\phi \left(s\right)ds,\phi \left(t\right)\right)\le \left(\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }^{\prime }\left(\varrho \right),\tilde{o}\right)\right)\left(\frac{{\left(t-\kappa \right)}^2+{\left(\nu -t\right)}^2}{2\left(\nu -\kappa \right)}\right).\end{array}$$

(10)

The inequalities in (9) and (10) are sharp, as equalities are attained by the choice of simple fuzzy number-valued functions. For further details on these inequalities, we refer interested readers to [19].

The main purpose of the present paper is to establish new Ostrowski-type inequalities for functions with values in the fuzzy environment that generalize the results of Dragomir from [2,5] and extend the results of Anastassiou proved in [19]. The results of this paper extend the result of Theorem 5 to fuzzy settings and hence also generalize the results of Theorems 2 and 4. To the best of our knowledge, this work has not been carried out before in any studies related to the fuzzy environment, and we hope to discover wide continuations and lots of applications in mathematical sciences and other areas of the sciences related to fuzzy mathematics.

3. Main Results

We begin with the following result, which generalizes Theorem 5.

Theorem 17.

Assume that $\varkappa :\left[\kappa ,\nu \right]\to {\mathbb{R}}^{+}$ and $\phi :\left[\kappa ,\nu \right]\to {\mathbb{R}}_{\mathcal{F}}$ are continuous and φ, ϰ are differentiable on $\left(\kappa ,\nu \right)$ (φ is $g\mathcal{H}$-differentiable), then for all $\overline{u}\in \left[\kappa ,\nu \right]$, the inequality

$$\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\le \mathcal{B}\left(\varkappa ,\phi ,\overline{u}\right),$$

(11)

where

$$\begin{array}{cc}\hfill \mathcal{B}\left(\varkappa ,\phi ,\overline{u}\right)& :={\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \hfill & +{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .\hfill \end{array}$$

We have the following bounds for

$$\begin{array}{c}\hfill \mathcal{B}\left(\varkappa ,\phi ,\overline{u}\right)\hspace{1em}\hspace{1em}\le \hspace{1em}\hspace{1em}\left\{\begin{array}{c}\underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right){\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ +\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right){\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \\ {\left[{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }{\left[\varkappa \left(\tau \right)\right]}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\overline{u}}^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}\hfill \\ +{\left[{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }{\left[\varkappa \left(\tau \right)\right]}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\kappa }^{\overline{u}}{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}},\hfill \\ \\ \underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ +\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho .\hfill \end{array}\right.\end{array}$$

(12)

Proof. 

Let $\overline{u}\in \left[\kappa ,\nu \right]$. Using the integration by parts formula given in Theorem 13, we have

$$\begin{array}{c}{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ =\left(0\odot \phi \left(\nu \right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}{\int }_{\overline{u}}^{\nu }\left(-\varkappa \left(\varrho \right)\right)\odot \phi \left(\varrho \right)d\varrho \hfill \\ =\tilde{o}{\ominus }_{g\mathcal{H}}\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}{\int }_{\overline{u}}^{\nu }\left(-\varkappa \left(\varrho \right)\right)\odot \phi \left(\varrho \right)d\varrho \hfill \\ =\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left(-{\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right)\hfill \\ =-\left(\left(-{\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right)\hfill \\ =\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right)\hfill \end{array}$$

(13)

and

$$\begin{array}{c}{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ =\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left(0\odot \phi \left(\kappa \right)\right){\ominus }_{g\mathcal{H}}{\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \hfill \\ =\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\tilde{o}{\ominus }_{g\mathcal{H}}{\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \hfill \\ =-\tilde{o}{\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}{\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \hfill \\ =-\left(-{\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left(-{\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right)\hfill \\ =\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right).\hfill \end{array}$$

(14)

Hence,

$$\begin{array}{c}{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho {\ominus }_{g\mathcal{H}}{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho \hfill \\ =\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right)\hfill \\ {\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right)\hfill \\ =-\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left(-1\right)\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right)\hfill \\ {\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left(-1\right)\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right)\hfill \\ =-\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right)\hfill \\ =\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right).\hfill \end{array}$$

(15)

The equality (15) implies that

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ =\mathcal{D}\left({\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho {\ominus }_{g\mathcal{H}}{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\hfill \\ =\mathcal{D}\left({\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\odot {\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho \right)\hfill \\ =\left|{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)-{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\right|\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \le {\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ +{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .\hfill \end{array}$$

(16)

Using Hölder’s inequality and properties of supremum, we obtain for $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ that

$$\begin{array}{cc}{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\hfill & \left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ & \hfill \le \left\{\begin{array}{c}\underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right){\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho ,\hfill \\ \\ {\left[{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }{\left[\varkappa \left(\tau \right)\right]}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\overline{u}}^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}},\hfill \\ \\ \underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho ,\hfill \end{array}\right.\end{array}$$

(17)

and

$$\begin{array}{cc}{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\hfill & \left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ & \hfill \le \left\{\begin{array}{c}\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right){\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho ,\hfill \\ \\ {\left[{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }{\left[\varkappa \left(\tau \right)\right]}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\kappa }^{\overline{u}}{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}},\hfill \\ \\ \underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho .\hfill \end{array}\right.\end{array}$$

(18)

Substituting (17) and (18) in (16), we obtain the inequality (11). □

The immediate consequence of Theorem 17 is the following corollary.

Corollary 2.

Suppose that the assumptions of Theorem 17 are satisfied. Then the inequalities

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le {\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau {\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho +{\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau {\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \le \left\{\begin{array}{c}\max \left\{{\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau ,{\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \right\}{\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \\ \max \left\{{\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho ,{\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \right\}{\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \hfill \end{array}\right.\hfill \\ \le {\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau {\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \end{array}$$

(19)

for all $\overline{u}\in \left[\kappa ,\nu \right]$.

Proof. 

Proof follows from the first inequality in (11) and by using the properties of the max function. □

Remark 6.

If $m\in \left(\kappa ,\nu \right)$ is such that

$${\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau ={\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau =\frac{1}{2}{\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau ,$$

then (19) becomes the following inequality:

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(m\right)\right),\tilde{0}\right)\hfill \\ \le {\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau {\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .\hfill \end{array}$$

(20)

Corollary 3.

With the assumptions of Theorem 17, we have

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le \underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ +\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ \le \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\nu }\left|\varrho -\overline{u}\right|\varkappa \left(\tau \right)d\tau \hfill \end{array}$$

(21)

for all $\overline{u}\in \left[\kappa ,\nu \right]$.

Proof. 

From the third part in the bounds (11), we have

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le \underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho +\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\hfill \\ \times {\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho \le \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\hfill \\ \times \left[{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho +{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho \right].\hfill \end{array}$$

(22)

Using integration by parts, we have for $\overline{u}\in \left[\kappa ,\nu \right]$ that

$$\begin{array}{cc}\hfill {\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho & ={\left.\varrho {\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right|}_{\overline{u}}^{\nu }+{\int }_{\overline{u}}^{\nu }\varrho \varkappa \left(\varrho \right)d\varrho \hfill \\ \hfill & =-\overline{u}{\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)d\varrho +{\int }_{\overline{u}}^{\nu }\varrho \varkappa \left(\varrho \right)d\varrho \hfill \\ \hfill & ={\int }_{\overline{u}}^{\nu }\left|\varrho -\overline{u}\right|\varkappa \left(\varrho \right)d\varrho \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho & ={\left.\varrho {\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \right|}_{\kappa }^{\overline{u}}-{\int }_{\kappa }^{\overline{u}}\varrho \varkappa \left(\varrho \right)d\varrho \hfill \\ \hfill & =\overline{u}{\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)d\varrho -{\int }_{\kappa }^{\overline{u}}\varrho \varkappa \left(\varrho \right)d\varrho \hfill \\ \hfill & ={\int }_{\kappa }^{\overline{u}}\left|\varrho -\overline{u}\right|\varkappa \left(\varrho \right)d\varrho .\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \begin{array}{c}{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho +{\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ ={\int }_{\kappa }^{\overline{u}}\left|\varrho -\overline{u}\right|\varkappa \left(\varrho \right)d\varrho +{\int }_{\overline{u}}^{\nu }\left|\varrho -\overline{\overline{u}}\right|\varkappa \left(\varrho \right)d\varrho ={\int }_{\kappa }^{\nu }\left|\varrho -\overline{u}\right|\varkappa \left(\varrho \right)d\varrho .\hfill \end{array}\end{array}$$

(23)

Making use of (23), we derive (19). □

Remark 7.

An application of the Hölder’s inequality, we have for p, $q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ that

$${\int }_{\kappa }^{\nu }\left|\varrho -\overline{u}\right|\varkappa \left(\varrho \right)d\varrho \le \left\{\begin{array}{c}\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\left|\varrho -\overline{u}\right|{\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)d\varrho ,\hfill \\ \\ {\left({\int }_{\kappa }^{\nu }{\left|\varrho -\overline{u}\right|}^{q}d\varrho \right)}^{\frac{1}{q}}{\left({\int }_{\kappa }^{\nu }{\left(\varkappa \left(\varrho \right)\right)}^{p}d\varrho \right)}^{\frac{1}{p}},\hfill \\ \\ \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\varkappa \left(\varrho \right){\int }_{\kappa }^{\nu }\left|\varrho -\overline{u}\right|d\varrho .\hfill \end{array}\right.$$

We know that

$$\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\left|\varrho -\overline{u}\right|=\max \left\{\overline{u}-\kappa ,\nu -\overline{u}\right\}=\frac{1}{2}\left(\nu -\kappa \right)+\left|\overline{u}-\frac{\kappa +\nu }{2}\right|,$$

$${\int }_{\kappa }^{\nu }{\left|\varrho -\overline{u}\right|}^{q}d\varrho =\frac{{\left(\overline{u}-\kappa \right)}^{q+1}+{\left(\nu -\overline{u}\right)}^{q+1}}{q+1}$$

and

$${\int }_{\kappa }^{\nu }\left|\varrho -\overline{u}\right|d\varrho =\frac{{\left(\overline{u}-\kappa \right)}^2+{\left(\nu -\overline{u}\right)}^2}{2}=\frac{1}{2}{\left(\nu -\kappa \right)}^2+{\left(\overline{u}-\frac{\kappa +\nu }{2}\right)}^2.$$

Then by (21), we derive the noncommutative Ostrowski-type inequalities for functions with values in the space of fuzzy real numbers:

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \hfill \le \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\left\{\begin{array}{c}\left[\frac{1}{2}\left(\nu -\kappa \right)+\left|\overline{u}-\frac{\kappa +\nu }{2}\right|\right]{\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)d\varrho ,\hfill \\ \\ {\left[\frac{{\left(\overline{u}-\kappa \right)}^{q+1}+{\left(\nu -\overline{\overline{u}}\right)}^{q+1}}{q+1}\right]}^{\frac{1}{q}}{\left({\int }_{\kappa }^{\nu }{\left(\varkappa \left(\varrho \right)\right)}^{p}d\varrho \right)}^{\frac{1}{p}},\hfill \\ \\ \left[\frac{1}{4}{\left(\nu -\kappa \right)}^2+{\left(\overline{u}-\frac{\kappa +\nu }{2}\right)}^2\right]\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\varkappa \left(\varrho \right)\hfill \end{array}\right.\end{array}$$

(24)

for all $\overline{u}\in \left[\kappa ,\nu \right]$.

One more interesting consequence of Theorem 17 is the following result.

Corollary 4.

Suppose that the assumptions of Theorem 17 are satisfied:

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le {\left[{\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)d\varrho \right)}^p\left(\nu -\overline{u}\right)+{\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)d\varrho \right)}^p\left(\overline{u}-\kappa \right)\right]}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}\hfill \\ \le {\left(\nu -\kappa \right)}^{\frac{1}{p}}{\left[{\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\varrho \right)d\varrho \right)}^p+{\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\varrho \right)d\varrho \right)}^p\right]}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(25)

Proof. 

By using the inequality

$$\left(\kappa \nu +cd\right)\le {\left({\kappa }^p+c^p\right)}^{\frac{1}{p}}{\left({\nu }^q+d^q\right)}^{\frac{1}{q}}$$

for $\kappa ,\nu ,c,d>0$ and $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, we have

$$\begin{array}{c}{\left({\int }_{\kappa }^{\overline{u}}{\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)}^{p}d\varrho \right)}^{\frac{1}{p}}{\left({\int }_{\kappa }^{\overline{u}}{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}\hfill \\ +{\left({\int }_{\overline{u}}^{\nu }{\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)}^{p}d\varrho \right)}^{\frac{1}{p}}{\left({\int }_{\overline{u}}^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}\\ \le {\left[{\int }_{\kappa }^{\overline{u}}{\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)}^{p}d\varrho +{\int }_{\overline{u}}^{\nu }{\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)}^{p}d\varrho \right]}^{\frac{1}{p}}\\ \times {\left[{\int }_{\kappa }^{\overline{u}}{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho ,\tilde{0}\right)\right]}^{q}d\varrho +{\int }_{\overline{u}}^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right]}^{\frac{1}{q}}\\ ={\left[{\int }_{\kappa }^{\overline{u}}{\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)}^{p}d\varrho +{\int }_{\overline{u}}^{\nu }{\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)}^{p}d\varrho \right]}^{\frac{1}{p}}\\ \times {\left[{\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right]}^{\frac{1}{q}}\\ \le {\left[{\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \right)}^p{\int }_{\kappa }^{\overline{u}}d\varrho +{\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \right)}^p{\int }_{\overline{u}}^{\nu }d\varrho \right]}^{\frac{1}{p}}\\ \times {\left[{\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right]}^{\frac{1}{q}}\\ ={\left[{\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \right)}^p\left(\overline{u}-\kappa \right)+{\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \right)}^p\left(\nu -\overline{u}\right)\right]}^{\frac{1}{p}}\\ \times {\left[{\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right]}^{\frac{1}{q}}\\ \le {\left(\nu -\kappa \right)}^{\frac{1}{p}}{\left[{\left({\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau \right)}^p+{\left({\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau \right)}^p\right]}^{\frac{1}{p}}\\ \hfill \times {\left[{\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right]}^{\frac{1}{q}}\end{array}$$

which proves the inequality (25). □

Remark 8.

If $m\in \left(\kappa ,\nu \right)$ is such that

$${\int }_{\kappa }^{\overline{u}}\varkappa \left(\tau \right)d\tau ={\int }_{\overline{u}}^{\nu }\varkappa \left(\tau \right)d\tau =\frac{1}{2}{\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau ,$$

then from (25), we obtain

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(m\right)\right),\tilde{0}\right)\hfill \\ \le \frac{1}{2}{\left(\nu -\kappa \right)}^{\frac{1}{p}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)d\varrho \right){\left({\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(26)

Remark 9.

Suppose that the assumptions of Theorem 17

$$\begin{array}{c}\hfill \mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\frac{\kappa +\nu }{2}\right)\right),\tilde{0}\right)\le M\left(\varkappa ,\phi \right),\end{array}$$

(27)

where

$$\begin{array}{cc}\hfill M\left(\varkappa ,\phi \right)& :={\int }_{\frac{\kappa +\nu }{2}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \hfill & +{\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .\hfill \end{array}$$

We have the following bounds for $M\left(\varkappa ,\phi \right)$:

$$M\left(\varkappa ,\phi \right)\le \left\{\begin{array}{c}\left({\int }_{\frac{\kappa +\nu }{2}}^{\nu }\varkappa \left(\tau \right)d\tau \right){\int }_{\frac{\kappa +\nu }{2}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ +\left({\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\varkappa \left(\tau \right)d\tau \right){\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \\ {\left[{\int }_{\frac{\kappa +\nu }{2}}^{\nu }\left({\int }_{\varrho }^{\nu }{\left[\varkappa \left(\tau \right)\right]}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\frac{\kappa +\nu }{2}}^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}\hfill \\ +{\left[{\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\left({\int }_{\kappa }^{\varrho }{\left[\varkappa \left(\tau \right)\right]}^{p}d\tau \right)d\varrho \right]}^{\frac{1}{p}}{\left({\int }_{\kappa }^{\frac{\kappa +\nu }{2}}{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}},\hfill \\ \\ \underset{\varrho \in \left[\frac{\kappa +\nu }{2},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\frac{\kappa +\nu }{2}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ +\underset{\varrho \in \left[\kappa ,\frac{\kappa +\nu }{2}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho .\hfill \end{array}\right.$$

(28)

From (19), we obtain that

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le {\int }_{\frac{\kappa +\nu }{2}}^{\nu }\varkappa \left(\tau \right)d\tau {\int }_{\frac{\kappa +\nu }{2}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho +{\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\varkappa \left(\tau \right)d\tau {\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \le \left\{\begin{array}{c}\max \left\{{\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\varkappa \left(\tau \right)d\tau ,{\int }_{\frac{\kappa +\nu }{2}}^{\nu }\varkappa \left(\tau \right)d\tau \right\}{\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \\ \max \left\{{\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho ,{\int }_{\frac{\kappa +\nu }{2}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \right\}{\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \hfill \end{array}\right.\hfill \\ \le {\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau {\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .\hfill \end{array}$$

(29)

and by (21), we obtain

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le \underset{\varrho \in \left[\frac{\kappa +\nu }{2},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\frac{\kappa +\nu }{2}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ +\underset{\varrho \in \left[\kappa ,\frac{\kappa +\nu }{2}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)d\varrho \hfill \\ \le \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right){\int }_{\kappa }^{\nu }\left|\varrho -\frac{\kappa +\nu }{2}\right|\varkappa \left(\varrho \right)d\varrho .\hfill \end{array}$$

(30)

From (24), we derive the noncommutative midpoint-type inequalities for functions with values in the space of fuzzy real numbers:

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{\varkappa \mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\frac{\kappa +\nu }{2}\right)\right),\tilde{0}\right)\hfill \\ \le \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\left\{\begin{array}{c}\frac{1}{2}\left(\nu -\kappa \right){\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)d\varrho ,\hfill \\ \\ \frac{{\left(\nu -\kappa \right)}^{1+\frac{1}{q}}}{2{\left(q+1\right)}^{1+\frac{1}{q}}}{\left({\int }_{\kappa }^{\nu }{\left(\varkappa \left(\varrho \right)\right)}^{p}d\varrho \right)}^{\frac{1}{p}},\hfill \\ \\ \frac{1}{4}{\left(\nu -\kappa \right)}^2\underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\varkappa \left(\varrho \right).\hfill \end{array}\right.\hfill \end{array}$$

(31)

From (25), we can obtain

$$\begin{array}{c}\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\frac{\kappa +\nu }{2}\right)\right),\tilde{0}\right)\hfill \\ \le {\left(\frac{\nu -\kappa }{2}\right)}^{\frac{1}{p}}{\left[{\left({\int }_{\frac{\kappa +\nu }{2}}^{\nu }\varkappa \left(\varrho \right)d\varrho \right)}^p+{\left({\int }_{\kappa }^{\frac{\kappa +\nu }{2}}\varkappa \left(\varrho \right)d\varrho \right)}^p\right]}^{\frac{1}{p}}\hfill \\ \times {\left({\int }_{\kappa }^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}.\hfill \end{array}$$

(32)

If we consider the case when $\varkappa \left(\varrho \right)=1$, $\varrho \in \left[\kappa ,\nu \right]$, then by (11), we obtain

$$\mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left(\left(\nu -\kappa \right)\odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\le \mathcal{B}\left(\phi ,\overline{u}\right),$$

(33)

where

$$\mathcal{B}\left(\phi ,\overline{u}\right):={\int }_{\overline{u}}^{\nu }\left(\nu -\varrho \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho +{\int }_{\kappa }^{\overline{u}}\left(\varrho -\kappa \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .$$

The bounds of $\mathcal{B}\left(\phi ,\overline{u}\right)$ are given by

$$\mathcal{B}\left(\phi ,\overline{u}\right)\le \left\{\begin{array}{c}\left(\nu -\overline{u}\right){\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ +\left(\overline{u}-\kappa \right){\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \\ \frac{{\left(\nu -\overline{u}\right)}^{1+\frac{1}{p}}}{{\left(p+1\right)}^{1+\frac{1}{p}}}{\left({\int }_{\overline{u}}^{\nu }{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}}\hfill \\ +\frac{{\left(\overline{u}-\kappa \right)}^{1+\frac{1}{p}}}{{\left(p+1\right)}^{1+\frac{1}{p}}}{\left({\int }_{\kappa }^{\overline{u}}{\left[\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\right]}^{q}d\varrho \right)}^{\frac{1}{q}},\hfill \\ \\ \frac{1}{2}{\left(\nu -\overline{u}\right)}^2\underset{\varrho \in \left[\overline{u},\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\hfill \\ +\frac{1}{2}{\left(\kappa -\overline{u}\right)}^2\underset{\varrho \in \left[\kappa ,\overline{u}\right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\hfill \end{array}\right.$$

(34)

for all $\overline{u}\in \left[\kappa ,\nu \right]$.

From (19), we obtain

$$\begin{array}{c}\mathcal{D}\left({\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho {\ominus }_{g\mathcal{H}}\left(\left(\nu -\kappa \right)\odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \le \left(\nu -\overline{u}\right){\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho +\left(\overline{u}-\kappa \right){\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \le \left\{\begin{array}{c}\left[\frac{1}{2}\left(\nu -\kappa \right)+\left|\overline{u}-\frac{\kappa +\nu }{2}\right|\right]{\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \\ \left(\nu -\kappa \right)\max \left\{{\int }_{\kappa }^{\overline{u}}\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right)d\varrho ,\tilde{0}\right)d\varrho ,{\int }_{\overline{u}}^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \right\}\hfill \end{array}\right.\hfill \\ \le \left(\nu -\kappa \right){\int }_{\kappa }^{\nu }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \end{array}$$

(35)

for all $\overline{u}\in \left[\kappa ,\nu \right]$.

From (24), we also have the following Ostrowski-type inequality:

$$\begin{array}{cc}\mathcal{D}({\int }_{\kappa }^{\nu }\phi \left(\varrho \right)d\varrho {\ominus }_{g\mathcal{H}}\hfill & \left(\left(\nu -\kappa \right)\odot \phi \left(\overline{u}\right)\right),\tilde{0})\hfill \\ & \le \underset{\varrho \in \left[\kappa ,\nu \right]}{\sup }\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)\left\{\begin{array}{c}\left(\nu -\kappa \right)\left[\frac{1}{2}\left(\nu -\kappa \right)+\left|\overline{u}-\frac{\kappa +\nu }{2}\right|\right],\hfill \\ \\ {\left(\nu -\kappa \right)}^{\frac{1}{p}}{\left[\frac{{\left(\overline{u}-\kappa \right)}^{q+1}+{\left(\nu -\overline{u}\right)}^{q+1}}{q+1}\right]}^{\frac{1}{q}},\hfill \\ \\ \left[\frac{1}{4}{\left(\nu -\kappa \right)}^2+{\left(\overline{u}-\frac{\kappa +\nu }{2}\right)}^2\right]\hfill \end{array}\right.\hfill \end{array}$$

(36)

for all $\overline{u}\in \left[\kappa ,\nu \right]$.

Example 1.

Consider the fuzzy number-valued mapping $\phi :\left[2,3\right]\to {\mathbb{R}}_{\mathcal{F}}$ defined by

$$\phi \left(t\right)\left(\theta \right)=\left\{\begin{array}{cc}\frac{\theta -2+t^{\frac{1}{2}}}{1-t^{\frac{1}{2}}},& \theta \in \left[2-t^{\frac{1}{2}},3\right]\\ & \\ \frac{2+t^{\frac{1}{2}}-\theta }{t^{\frac{1}{2}}-1},& \theta \in \left(3,2+t^{\frac{1}{2}}\right]\\ & \\ 0,& \mathit{otherwise}.\end{array}\right.$$

Then for each $r\in \left[0,1\right]$, we have ${\phi }^r\left(t\right)=\left[\left(1-r\right)\left(2-t^{\frac{1}{2}}\right)+3r,\left(1-r\right)\left(2+t^{\frac{1}{2}}\right)+3r\right]=\left[{\phi }_{-}^r\left(t\right),{\phi }_{+}^r\left(t\right)\right]$. We also define a mapping $\varkappa :\left[2,3\right]\to {\mathbb{R}}^{+}$ by $\varkappa \left(t\right)=t^2$. Then according to the metric $\mathcal{D}:{\mathbb{R}}_{\mathcal{F}}\times {\mathbb{R}}_{\mathcal{F}}⟶{\mathbb{R}}^{+}\cup \left\{0\right\}$ as defined in the beginning of Section 2 with $\overline{u}=\frac{5}{2}\in \left[2,3\right]$ and $p=4,q=\frac{4}{3}$, we have the left-hand side of inequality (11):

$$\begin{array}{cc}\hfill & \mathcal{D}\left(\left({\int }_{\kappa }^{\nu }\varkappa \left(\varrho \right)\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_{\kappa }^{\nu }\varkappa \left(\tau \right)d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \hfill & =\mathcal{D}\left(\left({\int }_2^3{\varrho }^2\odot \phi \left(\varrho \right)d\varrho \right){\ominus }_{g\mathcal{H}}\left({\int }_2^3{\tau }^{2}d\tau \odot \phi \left(\overline{u}\right)\right),\tilde{0}\right)\hfill \\ \hfill & =\left({\int }_2^3\left(2+{\varrho }^{\frac{1}{2}}\right){\varrho }^{2}d\varrho \right){\ominus }_{g\mathcal{H}}\left(\left(2+{\left(\frac{5}{2}\right)}^{\frac{1}{2}}\right){\int }_2^3{\tau }^{2}d\tau \right)\hfill \\ \hfill & =\frac{2}{7}\left(27\sqrt{3}-8\sqrt{2}\right)-\frac{19}{3}\left(\sqrt{\frac{5}{2}}+2\right)+\frac{38}{3}\hfill \\ \hfill & =0.115167.\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \mathcal{B}\left(\varkappa ,\phi ,\overline{u}\right)& :={\int }_{\overline{u}}^{\nu }\left({\int }_{\varrho }^{\nu }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho \hfill \\ \hfill & +{\int }_{\kappa }^{\overline{u}}\left({\int }_{\kappa }^{\varrho }\varkappa \left(\tau \right)d\tau \right)\mathcal{D}\left({\phi }_{g\mathcal{H}}^{\prime }\left(\varrho \right),\tilde{0}\right)d\varrho .\hfill \end{array}$$

Now we calculate the bounds for $\mathcal{B}\left(\varkappa ,\phi ,\overline{u}\right)$ as follows:

$$\begin{array}{c}\hfill \mathcal{B}\left(\varkappa ,\phi ,\frac{5}{2}\right)\hspace{1em}\hspace{1em}\le \hspace{1em}\hspace{1em}\left\{\begin{array}{c}\frac{1}{2}\underset{\varrho \in \left[\frac{5}{2},3\right]}{\sup }\left({\int }_{\varrho }^3{\tau }^{2}d\tau \right){\int }_{\frac{5}{2}}^3\frac{1}{{\varrho }^{\frac{1}{2}}}d\varrho \hfill \\ +\frac{1}{2}\underset{\varrho \in \left[2,\frac{5}{2}\right]}{\sup }\left({\int }_2^{\varrho }{\tau }^{2}d\tau \right){\int }_2^{\frac{5}{2}}\frac{1}{{\varrho }^{\frac{1}{2}}}d\varrho =0.568053,\hfill \\ \\ {\left[{\int }_{\frac{5}{2}}^3\left({\int }_{\varrho }^3{\tau }^{8}d\tau \right)d\varrho \right]}^{\frac{1}{4}}{\left({\int }_{\frac{5}{2}}^3\frac{1}{{\varrho }^{\frac{2}{3}}}d\varrho \right)}^{\frac{3}{4}}\hfill \\ +{\left[{\int }_2^{\frac{5}{2}}\left({\int }_2^{\varrho }{\tau }^{8}d\tau \right)d\varrho \right]}^{\frac{1}{4}}{\left({\int }_2^{\frac{5}{2}}\frac{1}{{\varrho }^{\frac{2}{3}}}d\varrho \right)}^{\frac{3}{4}}=2.86555,\hfill \\ \\ \underset{\varrho \in \left[\frac{5}{2},3\right]}{\sup }\left(\frac{1}{{\varrho }^{\frac{1}{2}}}\right){\int }_{\frac{5}{2}}^3\left({\int }_{\varrho }^3{\tau }^{2}d\tau \right)d\varrho \hfill \\ +\underset{\varrho \in \left[2,\frac{5}{2}\right]}{\sup }\left(\frac{1}{{\varrho }^{\frac{1}{2}}}\right){\int }_2^{\frac{5}{2}}\left({\int }_2^{\varrho }{\tau }^{2}d\tau \right)d\varrho =0.263523.\hfill \end{array}\right.\end{array}$$

(37)

Hence, it can be observed that the inequality (11) of Theorem 17 is valid for the above choices of functions over the interval $[2,3]$.

Remark 10.

We suggest the interested reader check the validity of the other results of this study with their own choices of functions and any interval of the real numbers.

4. Concluding Remarks

Over the past forty years, the field of mathematical inequality has expanded significantly, and numerous researchers have published numerous articles on the subject using novel approaches. In the rich literature on mathematical inequalities, one of the important inequalities is Ostrowski’s inequality. The Ostrowski-type inequality can be used to estimate the absolute deviation of a function from its integral mean. Various generalizations of the Ostrowski inequality, such as continuous and discrete versions, have been established by many mathematicians in the recent few decades. Among the most notable studies on the generalizations of Ostrowski-type inequalities include the papers [2,5,19,20,21]. In the present study, we proved a more general result of the Ostrowski type in the fuzzy context, which generalizes results from [5] and extends the results from [19]. In order to obtained our results, we used the novel results from the calculus of fuzzy number-valued functions. We hope that the results can be a good source to obtain more new results for the researchers working in the field of mathematical inequalities in fuzzy number-valued calculus.