Alternating series in terms of Riemann zeta function and Dirichlet beta function

1.   Introduction

In convex function theory, the classical Hermite-Hadamard inequality is one of the most well-known inequalities with geometrical interpretation, and it has a wide range of applications, see ^[1,2].

Let $\mathfrak{S}:K\to {\mathbb{R}}_{}^{+}$ be a convex function on a convex set $K$ and $\rho , \varsigma \in K$ with $\rho \ne \varsigma$. Then,

In ^[3], Fejér looked at the key extensions of HH-inequality which is known as Hermite-Hadamard-Fejér inequality (HH-Fejér inequality).

Let $\mathfrak{S}:K\to {\mathbb{R}}_{}^{+}$ be a convex function on a convex set $K$ and $\rho , \varsigma$ $\in K$ with $\rho \ne \varsigma$. Then,

If $\mathfrak{D}\left(\varpi \right) = 1$, then we obtain (1) from (2). We should remark that Hermite-Hadamard inequality is a refinement of the idea of convexity, and it can be simply deduced from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and there have been a lot of improvements and generalizations examined. Sarikaya ^[4] proved the Hadamard type inequality for coordinated convex functions such that

Let $\mathfrak{G}:\Delta \to {\mathbb{R}}_{}^{+}$ be a coordinate convex function on $\Delta = \left[\varsigma , \rho \right]\times \left[\mu , \nu \right]$. If $\mathfrak{G}$ is double fractional integrable, then following inequalities hold:

If $\alpha = 1$, then we obtain the following Dragomir inequality ^[5] on coordinates:

For more details related to inequalities, see ^[6,7,8,9] and reference therein.

Interval analysis, on the other hand, is a well-known example of set-valued analysis, which is the study of sets in the context of mathematical analysis and general topology. It was created as a way of dealing with the interval uncertainty that can be found in many mathematical or computer models of deterministic real-world phenomena. Archimede's method, which is used to calculate the circumference of a circle, is an old example of an interval enclosure. Moore ^[10], who is credited with being the first user of intervals in computational mathematics, published the first book on interval analysis in 1966. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a helpful technique in a variety of fields that are interested in ambiguous data because of its applicability. Computer graphics, experimental and computational physics, error analysis, robotics, and many more fields have applications.

Furthermore, in recent years, numerous major inequalities (Hermite-Hadamard, Ostrowski and others) have been addressed for interval-valued functions. Chalco-Cano et al. used the Hukuhara derivative for interval-valued functions to construct Ostrowski type inequalities for interval-valued functions in ^{[11,12,13,14]}. For interval-valued functions, Román-Flores et al. developed Minkowski and Beckenbach's inequality in ^[15]. For fuzzy interval-valued function, Khan et al. ^[16,17,18] derived some new versions of Hermite-Hadamard type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. ^[19,20] discussed some novel types of Hermite-Hadamard type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities. Recently, Khan et al. ^[21] introduced the new class of convexity in fuzzy-interval calculus which is known as coordinated convex fuzzy-interval-valued functions and with the support of these classes, some Hermite-Hadamard type inequalities are obtained via newly defined fuzzy-interval double integrals. We encourage readers to ^{[22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54]} for other related results.

The following is an overview of the paper's structure. Section 2 recalls some preliminary notions and definitions. Moreover, some properties of introduced coordinated LR-convex IVF are also discussed. Section 3 presents some Hermite-Hadamard type inequalities for coordinated LR-convex IVF. With the help of this class, some fractional integral inequalities are also derived for the coordinated LR-convex IVF and for the product of two coordinated LR-convex IVFs. The fourth section, Conclusions and Future Work, brings us to a close.

2.   Preliminaries and known results

Let $\mathbb{R}$ be the set of real numbers and ${\mathbb{R}}_{I}$ be the space of all closed and bounded intervals of $\mathbb{R}$, such that $\mathfrak{U}\in {\mathbb{R}}_{I}$ is defined by

If ${\mathfrak{U}}_{\mathfrak{*}} = {\mathfrak{U}}^{\mathfrak{*}}$, then $\mathfrak{U}$ is said to be degenerate. If ${\mathfrak{U}}_{\mathfrak{*}}\ge 0$, then $\left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]$ is called positive interval. The set of all positive interval is denoted by ${\mathbb{R}}_{I}^{+}$ and defined as ${\mathbb{R}}_{I}^{+} = \left\{\left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]:\left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]\in {\mathbb{R}}_{I}\mathrm{a}\mathrm{n}\mathrm{d}{\mathfrak{U}}_{\mathfrak{*}}\ge 0\right\}.$

Let $\varrho \in \mathbb{R}$ and $\varrho \mathfrak{U}$ be defined by

Then, the Minkowski difference $\mathfrak{D}-\mathfrak{U}$, addition $\mathfrak{U}+\mathfrak{D}$ and $\mathfrak{U}\times \mathfrak{D}$ for $\mathfrak{U}, \mathfrak{D}\in {\mathbb{R}}_{I}$ are defined by

and

The inclusion $"\supseteq "$ means that

$\mathfrak{U}\supseteq \mathfrak{D}$ if and only if, $\left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]\supseteq \left[{\mathfrak{D}}_{\mathfrak{*}}, {\mathfrak{D}}^{\mathfrak{*}}\right]$, and if and only if

Remark 1. ^[36] (ⅰ) The relation ${"\le }_{p}"$ is defined on ${\mathbb{R}}_{I}$ by

for all $\left[{\mathfrak{D}}_{\mathfrak{*}}, {\mathfrak{D}}^{\mathfrak{*}}\right], \left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]\in {\mathbb{R}}_{I},$ and it is a pseudo order relation. The relation $\left[{\mathfrak{D}}_{\mathfrak{*}}, {\mathfrak{D}}^{\mathfrak{*}}\right]{\le }_{p}\left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]$ coincident to $\left[{\mathfrak{D}}_{\mathfrak{*}}, {\mathfrak{D}}^{\mathfrak{*}}\right]\le \left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]$ on ${\mathbb{R}}_{I}$ when it is ${"\le }_{p}"$

(ⅱ) It can be easily seen that $"{\le }_{p}"$ looks like "left and right" on the real line $\mathbb{R},$ so we call $"{\le }_{p}"$ is "left and right" (or "LR" order, in short).

For $\left[{\mathfrak{D}}_{\mathfrak{*}}, {\mathfrak{D}}^{\mathfrak{*}}\right], \left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]\in {\mathbb{R}}_{I},$ the Hausdorff-Pompeiu distance between intervals $\left[{\mathfrak{D}}_{\mathfrak{*}}, {\mathfrak{D}}^{\mathfrak{*}}\right]$ and $\left[{\mathfrak{U}}_{\mathfrak{*}}, {\mathfrak{U}}^{\mathfrak{*}}\right]$ is defined by

It is familiar fact that $\left({\mathbb{R}}_{I}, d\right)$ is a complete metric space.

Theorem 1. ^[10] If $\mathfrak{G}:[\mu , \nu ]\subset \mathbb{R}\to {\mathbb{R}}_{I}$ is an I-V-F given by $\left(\mathcal{x}\right)$ $\left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}\right)\right]$, then $\mathfrak{G}$ is Riemann integrable over $[\mu , \nu ]$ if and only if, ${\mathfrak{G}}_{\mathfrak{*}}$ and ${\mathfrak{G}}^{\mathfrak{*}}$ both are Riemann integrable over $\left[\mu , \nu \right]$ such that

The collection of all Riemann integrable real valued functions and Riemann integrable I-V-F is denoted by ${\mathcal{R}}_{[\mu , \nu ]}$ and ${\mathfrak{T}\mathcal{R}}_{[\mu , \nu ]},$ respectively.

Definition 1. ^[31,33] Let $\mathfrak{G}:\left[\mu , \nu \right]\to {\mathbb{R}}_{I}$ be interval-valued function and $\mathfrak{G}\in {\mathfrak{T}\mathcal{R}}_{\left[\mu , \nu \right]}$. Then interval Riemann-Liouville-type integrals of $\mathfrak{G}$ are defined as

where $\alpha > 0$ and $\mathit{\Gamma }$ is the gamma function.

Theorem 2. ^[20] Let $\mathfrak{G}:\left[\varsigma , \rho \right]\to {{\mathbb{R}}_{I}}^{+}$ be a LR-convex I-V.F such that $\mathfrak{G}\left(\mathit{y}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathit{y}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathit{y}\right)\right]$ for all $\mathit{y}\in \left[\varsigma , \rho \right]$. If $\mathfrak{G}\in L\left(\left[\varsigma , \rho \right], {\mathbb{R}}_{I}^{+}\right)$, then

Theorem 3. ^[20] Let $\mathfrak{G}, \mathfrak{S}:\left[\varsigma , \rho \right]\to {\mathbb{R}}_{I}^{+}$ be two LR-convex I-V.Fs such that $\mathfrak{G}\left(\mathcal{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}\right)\right]$ and $\mathfrak{S}\left(\mathcal{x}\right) = \left[{\mathfrak{S}}_{\mathfrak{*}}\left(\mathcal{x}\right), {\mathfrak{S}}^{\mathfrak{*}}\left(\mathcal{x}\right)\right]$ for all $\mathcal{x}\in \left[\varsigma , \rho \right]$. If $\mathfrak{G}\times \mathfrak{S}\in L\left(\left[\varsigma , \rho \right], {\mathbb{R}}_{I}^{+}\right)$ is fuzzy Riemann integrable, then

and

where $\mathcal{M}\left(\varsigma , \rho \right) = \mathfrak{G}\left(\varsigma \right)\times \mathfrak{S}\left(\varsigma \right)+\mathfrak{G}\left(\rho \right)\times \mathfrak{S}\left(\rho \right),$ $\mathcal{N}\left(\varsigma , \rho \right) = \mathfrak{G}\left(\varsigma \right)\times \mathfrak{S}\left(\rho \right)+\mathfrak{G}\left(\rho \right)\times \mathfrak{S}\left(\varsigma \right),$

and $\mathcal{M}\left(\varsigma , \rho \right) = \left[{\mathcal{M}}_{\mathcal{*}}\left(\varsigma , \rho \right), {\mathcal{M}}^{\mathcal{*}}\left(\varsigma , \rho \right)\right]$ and $\mathcal{N}\left(\varsigma , \rho \right) = \left[{\mathcal{N}}_{\mathcal{*}}\left(\varsigma , \rho \right), {\mathcal{N}}^{\mathcal{*}}\left(\varsigma , \rho \right)\right].$

Note that, the Theorem 1 is also true for interval double integrals. The collection of all double integrable I-V-F is denoted ${\mathfrak{T}\mathfrak{O}}_{\Delta },$ respectively.

Theorem 4. ^[35] Let $\Delta = \left[\varsigma , \rho \right]\times \left[\mu , \nu \right]$. If $\mathfrak{G}:\Delta \to {\mathbb{R}}_{I}$ is interval-valued doubl integrable ($ID$-integrable) on $\Delta$. Then, we have

Definition 2. ^[36] Let $\mathfrak{G}:\Delta \to {\mathbb{R}}_{I}^{+}$ and $\mathfrak{G}\in {\mathfrak{T}\mathfrak{O}}_{\Delta }$. The interval Riemann-Liouville-type integrals ${\mathcal{I}}_{{\mu }^{+}, {\varsigma }^{+}}^{\alpha , \beta }, {\mathcal{I}}_{{\mu }^{+}, {\rho }^{-}}^{\alpha , \beta },$ ${\mathcal{I}}_{{\nu }^{-}, {\varsigma }^{+}}^{\alpha , \beta }, {\mathcal{I}}_{{\nu }^{-}, {\rho }^{-}}^{\alpha , \beta }$ of $\mathfrak{G}$ order $\alpha , \beta > 0$ are defined by

Definition 3. ^[38] The I-V.F $\mathfrak{G}:\Delta \to {\mathbb{R}}_{I}^{+}$ is said to be coordinated LR-convex I-V.F on $\Delta$ if

for all $\left(\mu , \nu \right), \left(\varsigma , \rho \right)\in \Delta ,$ and $\tau , s\in \left[0, 1\right].$ If inequality (21) is reversed, then $\mathfrak{G}$ is called coordinate LR-concave I-V.F on $\Delta$.

Lemma 1. ^[38] Let $\mathfrak{G}:\Delta \to {\mathbb{R}}_{I}^{+}$ be an coordinated I-V.F on $\Delta$. Then, $\mathfrak{G}$ is coordinated LR-convex I-V.F on $\Delta ,$ if and only if there exist two coordinated LR-convex I-V.Fs ${\mathfrak{G}}_{\mathcal{x}}:\left[\varsigma , \rho \right]\to {\mathbb{R}}_{I}^{+}$, ${\mathfrak{G}}_{\mathcal{x}}\left(w\right) = \mathfrak{G}\left(\mathcal{x}, w\right)$ and ${\mathfrak{G}}_{\mathit{y}}:\left[\mu , \nu \right]\to {\mathbb{R}}_{I}^{+}$, ${\mathfrak{G}}_{\mathit{y}}\left(z\right) = \mathfrak{G}\left(z, \mathit{y}\right)$.

Theorem 5. ^[38] Let $\mathfrak{G}:\Delta \to {\mathbb{R}}_{I}^{+}$ be a I-V.F on $\Delta$ such that

for all $\left(\mathcal{x}, \varpi \right)\in \Delta$. Then, $\mathfrak{G}$ is coordinated LR-convex I-V.F on $\Delta ,$ if and only if, ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)$ are coordinated convex functions.

Example 1. We consider the I-V.Fs $\mathfrak{G}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{R}}_{I}^{+}$ defined by,

Then, for each $\theta \in \left[0, 1\right],$ we have $\mathfrak{G}\left(\mathcal{x}\right) = \left[2\theta \left(6+{e}^{\mathcal{x}}\right)\left(6+{e}^{\varpi }\right), \left(4+2\theta \right)\left(6+{e}^{\mathcal{x}}\right)\left(6+{e}^{\varpi }\right)\right]$. Since end point functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\left(\mathcal{x}, \varpi \right), \theta \right),$ ${\mathfrak{G}}^{\mathfrak{*}}\left(\left(\mathcal{x}, \varpi \right), \theta \right)$ are coordinate concave functions for each $\theta \in \left[0, 1\right]$. Hence $\mathfrak{S}\left(\mathcal{x}, \varpi \right)$ is coordinate LR-concave I-V.F.

From Lemma 1, we can easily note that each LR-convex I-V.F is coordinated LR-convex I-V.F. But the converse is not true.

Remark 2. If one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \varpi \right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)$, then $\mathfrak{G}$ is known as coordinated function if $\mathfrak{G}$ satisfies the coming inequality

is valid which is defined by Dragomir ^[5]

Let one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)$, where ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)$ is affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \varpi \right)$ is a concave function. If coming inequality,

is valid, then $\mathfrak{G}$ is named as coordinated IVF which is defined by Zhao et al. [37, Definition 2 and Example 2]

3.   Main results

In this section, we shall continue with the following fractional $HH$-inequality for coordinated LR-convex I-V.Fs, and we also give fractional $HH$-Fejér inequality for coordinated LR-convex I-V.F through fuzzy order relation.

Theorem 6. Let $\mathfrak{G}:\Delta \to {\mathbb{R}}_{I}^{+}$ be a coordinate LR-convex I-V.F on $\Delta$ such that $\mathfrak{G}\left(\mathcal{x}, \mathit{y}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\right]$ for all $\left(\mathcal{x}, \mathit{y}\right)\in \Delta$. If ${\mathfrak{G}\in \mathfrak{T}\mathfrak{O}}_{\Delta }$, then following inequalities holds:

If $\mathfrak{G}\left(\mathcal{x}\right)$ coordinated LR-concave I-V.F, then

Proof. Let $\mathfrak{G}:\left[\mu , \nu \right]\to {\mathbb{R}}_{I}^{+}$ be a coordinated LR-convex I-V.F. Then, by hypothesis, we have

By using Theorem 5, we have

By using Lemma 1, we have

and

From (25) and (26), we have

and

It follows that

and

Since $\mathfrak{G}\left(\mathcal{x}, .\right)$ and $\mathfrak{G}\left(., \mathit{y}\right)$, both are coordinated LR-convex-IVFs, then from inequality (14), inequalities (27) and (28) we have

and

Since ${\mathfrak{G}}_{\mathcal{x}}\left(w\right) = \mathfrak{G}\left(\mathcal{x}, w\right)$, the inequality (29) can be written as

That is

Multiplying double inequality (31) by $\frac{\alpha {\left(\nu -\mathcal{x}\right)}^{\alpha -1}}{{2\left(\nu -\mu \right)}^{\alpha }}$ and integrating with respect to $\mathcal{x}$ over $\left[\mu , \nu \right],$ we have

Again multiplying double inequality (31) by $\frac{\alpha {\left(\mathcal{x}-\mu \right)}^{\alpha -1}}{{2\left(\nu -\mu \right)}^{\alpha }}$ and integrating with respect to $\mathcal{x}$ over $\left[\mu , \nu \right],$ we have

From (32), we have

From (33), we have

Similarly, since ${\mathfrak{G}}_{\mathit{y}}\left(z\right) = \mathfrak{G}\left(z, \mathit{y}\right)$ then, from (34) and (35), (30) we have

and

After adding the inequalities (46), (35), (36) and (37), we will obtain as resultant second, third and fourth inequalities of (23).

Now, from left part of inequality (14), we have

and

Summing the inequalities (38) and (39), we obtain the following inequality:

this is the first inequality of (23).

Now, from right part of inequality (14), we have

Summing inequalities (41), (42), (43) and (44), and then taking multiplication of the resultant with $\frac{1}{4}$, we have

This is last inequality of (23) and the result has been proven.

Remark 3. If one to take $\alpha = 1$ and $\beta = 1$, then from (23), we achieve the coming inequality, see ^[38]:

Let one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is an affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is concave function. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then from Remark 2 and (24), we acquire the coming inequality, see ^[31]:

Let one takes $\alpha = 1$ and $\beta = 1$, ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is an affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is concave function. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then Remark 2 and from (24), we acquire the coming inequality, see ^[37]:

Example 2. We consider the I-V-Fs $\mathfrak{G}:\left[0, \; 1\right]\times \left[0, \; 1\right]\to {\mathbb{R}}_{I}^{+}$ defined by,

Since end point functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right),$ ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ are convex functions on coordinate, then $\mathfrak{G}\left(\mathcal{x}, \mathit{y}\right)$ is convex I-V-F on coordinate. Then for $\alpha = 1$ and $\beta = 1$, we have

That is

Hence, Theorem 3.1 has been verified

Next both results obtain Hermite-Hadamard type inequalities for the product of two coordinate LR-convex I-V.Fs

Theorem 7. Let $\mathfrak{G}, \mathfrak{S}:\Delta \to {\mathbb{R}}_{I}^{+}$ be a coordinate LR-convex I-V.Fs on $\Delta$ such that $\mathfrak{G}\left(\mathcal{x}, \mathit{y}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\right]$ and $\mathfrak{S}\left(\mathcal{x}, \mathit{y}\right) = \left[{\mathfrak{S}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right), {\mathfrak{S}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\right]$ for all $\left(\mathcal{x}, \mathit{y}\right)\in \Delta$. If ${\mathfrak{G}\times \mathfrak{S}\in \mathfrak{T}\mathfrak{O}}_{\Delta }$, then following inequalities holds:

If $\mathfrak{G}$ and $\mathfrak{S}$ both are coordinate LR-concave I-V.Fs on $\Delta$, then above inequality can be written as

Where

and $K\left(\mu , \nu , \varsigma , \rho \right)$, $\widetilde {L}\left(\mu , \nu , \varsigma , \rho \right)$, $\mathcal{M}\left(\mu , \nu , \varsigma , \rho \right)$ and $\mathcal{N}\left(\mu , \nu , \varsigma , \rho \right)$ are defined as follows:

Proof. Let $\mathfrak{G}$ and $\mathfrak{S}$ both are coordinated LR-convex I-V.Fs on $\left[\mu , \nu \right]\times \left[\varsigma , \rho \right]$. Then

and

Since $\mathfrak{G}$ and $\mathfrak{S}$ both are coordinated LR-convex I-V.Fs, then by Lemma 1, there exist

Since ${\mathfrak{G}}_{\mathcal{x}}$, and ${\mathfrak{S}}_{\mathcal{x}}$ are I-V.Fs, then by inequality (15), we have

That is

Multiplying double inequality (51) by $\frac{\alpha {\left(\nu -\mathcal{x}\right)}^{\alpha -1}}{{2\left(\nu -\mu \right)}^{\alpha }}$ and integrating with respect to $\mathcal{x}$ over $\left[\mu , \nu \right],$ we get

Again, multiplying double inequality (51) by $\frac{\alpha {\left(\mathcal{x}-\mu \right)}^{\alpha -1}}{{2\left(\nu -\mu \right)}^{\alpha }}$ and integrating with respect to $\mathcal{x}$ over $\left[\mu , \nu \right],$ we gain

Summing (52) and (53), we have

Now, again with the help of integral inequality (15) for first two integrals on the right-hand side of (54), we have the following relation

And

From (55)–(58), inequality (54) we have

Hence, the result has been proven.

Remark 4. If one to take $\alpha = 1$ and $\beta = 1$, then from (49), we achieve the coming inequality, see ^[38]:

Let one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is an affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is concave function. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then by Remark 2 and (50), we acquire the coming inequality, see ^[36]:

Let one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is an affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is concave function. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then by Remark 2 and (50), we acquire the coming inequality, see ^[37]:

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ and ${\mathfrak{S}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right) = {\mathfrak{S}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then from (49), we acquire the coming inequality, see ^[39]:

Theorem 8. Let $\mathfrak{G}, \mathfrak{S}:\Delta \to {\mathbb{R}}_{I}^{+}$ be a coordinate LR-convex I-V.F on $\Delta$ such that $\mathfrak{G}\left(\mathcal{x}, \mathit{y}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\right]$ and $\mathfrak{S}\left(\mathcal{x}, \mathit{y}\right) = \left[{\mathfrak{S}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right), {\mathfrak{S}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\right]$ for all $\left(\mathcal{x}, \mathit{y}\right)\in \Delta$. If ${\mathfrak{G}\times \mathfrak{S}\in \mathfrak{T}\mathfrak{O}}_{\Delta }$, then following inequalities holds:

If $\mathfrak{G}$ and $\mathfrak{S}$ both are coordinate LR-concave I-V.Fs on $\Delta$, then above inequality can be written as

Where $K\left(\mu , \nu , \varsigma , \rho \right)$, $L\left(\mu , \nu , \varsigma , \rho \right)$, $\mathcal{M}\left(\mu , \nu , \varsigma , \rho \right)$ and $\mathcal{N}\left(\mu , \nu , \varsigma , \rho \right)$ are given in Theorem 7.

Proof. Since $\mathfrak{G}, \mathfrak{S}:\Delta \to {\mathbb{R}}_{I}^{+}$ be two LR-convex I-V.Fs, then from inequality (16)$,$ we have

and

Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain

Again, with the help of integral inequality (16) and Lemma 1 for each integral on the right-hand side of (67), we have

And

and

From inequalities (68) to (79), inequality (67) we have

Again, with the help of integral inequality (15) and Lemma 1, for each integral on the right-hand side of (80), we have

From (77) to (84), (80) we have

This concludes the proof of Theorem 8 result has been proven.

Remark 5. If we take $\alpha = 1$ and $\beta = 1$, then from (63), we achieve the coming inequality, see ^[38]:

Let one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is an affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is convex function. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then from Remark 2 and (64), we acquire the coming inequality, see ^[37]:

Let one takes ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is an affine function and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ is convex function. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)\ne {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then from Remark 2 and (64) we acquire the coming inequality, see ^[36]:

If we take ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$ and ${\mathfrak{S}}_{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right) = {\mathfrak{S}}^{\mathfrak{*}}\left(\mathcal{x}, \mathit{y}\right)$, then from (63), we acquire the coming inequality, see ^[39]:

4.   Conclusions

In this study, with the help of coordinated LR-convexity for interval-valued functions, several novel Hermite-Hadamard type inequalities are presented. It is also demonstrated that the conclusions reached in this study represent a possible extension of previously published equivalent results. Similar inequalities may be discovered in the future using various forms of convexities. This is a novel and intriguing topic, and future study will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.

Acknowledgments

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. All authors read and approved the final manuscript. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.