Some new versions of Jensen, Schur and Hermite-Hadamard type inequalities for $\left({p}, \mathfrak{J}\right)$-convex fuzzy-interval-valued functions

1.   Introduction

Integral inequality is well recognized to be important in both pure and practical mathematics; for examples, see ^{[1,2,3,4,5,6,7,8,9,10]}. The behavior of inequalities makes it clear that mathematical approaches are useless in the absence of inequalities. For this reason, exact inequalities are now required in order to demonstrate the validity and uniqueness of the mathematical procedures. Convexity also plays a big part in the subject of inequalities, owing to the behavior of its definition.

Let $K$ be a convex set. Then, a real valued function $\mathfrak{G}:K\to \mathbb{R}$ is named as convex on $K$ if the inequality

holds for all $\mathfrak{x}, y\in K, \mathfrak{s}\in \left[0, 1\right].$ If $\mathfrak{G}$ is concave, then $-\mathfrak{G}$ is convex. Over the years, convex sets and convex functions have been modified to a remarkable variety of convexities, such as harmonic convexity ^[11], quasi convexity ^[12], Schur convexity ^[13,14], strong convexity ^[15,16], p-convexity ^[17], generalized convexity ^[18] and so on. For more information, see ^{[19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]} and the references therein.

The Jensen inequality ^[35,36] is one of these inequalities for convex functions, and it can be stated as follows.

Let ${\mathfrak{w}}_{j}\in [0, 1]$, ${\mathfrak{a}}_{j}\in \left[\mathfrak{a}, \mathfrak{b}\right],$ $\left(j = 1, 2, 3, \dots \kappa, \kappa \ge 2\right)$ and $\mathfrak{G}$ be a convex function. Then,

with $\sum _{j = 1}^{\kappa }{\mathfrak{w}}_{j} = 1.$ If $\mathfrak{G}$ is concave, then inequality (2) is reversed.

Research on the idea of convexity with integral problems is fascinating. As a result, several writers have proved numerous equalities or inequalities as applications of convex functions. Among the notable outcomes are the Gagliardo-Nirenberg inequality ^[37], the Hardy inequality ^[38], the Ostrowski inequality ^[39], the Olsen inequality ^[40] and the Hermite-Hadamard inequality ($HH$-inequality, in short) ^[41]. The $HH$-inequality is an interesting outcome in convex analysis which is formulated for convex function $\mathfrak{G}:K\to {\mathbb{R}}^{+}$ on an interval $K = \left[\mathfrak{a}, \mathfrak{b}\right]$ by

for all $\mathfrak{a}, \mathfrak{b}\in K.$ If $\mathfrak{G}$ is concave, then inequality (3) is reversed.

Fejér created the Hermite-Hadamard-Fejér inequality (𝐻𝐻-Fejér inequality), which is the most significant weighted extension of the 𝐻𝐻-inequality, in ^[42].

Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{R}}^{+}$ be a convex function on a convex set $K$ and $\mathfrak{a}, \mathfrak{b}$ $\in K$ with $\mathfrak{a}\le \mathfrak{b}$. Then,

If $\mathit{\Omega }\left(\mathfrak{x}\right) = 1$, then we obtain (3) from (4). With the assistance of inequality (4), many inequalities can be obtained through special symmetric function $\mathit{\Omega }\left(\mathfrak{x}\right)$ for convex functions.

Meanwhile, to increase the accuracy of measurement findings and to carry out error analysis automatically, interval analysis has been proposed and researched by Moore ^[43], Kulish and Miranker ^[44], and they have substituted interval operations with real operations. In this field, an interval of real numbers is used to represent an uncertain variable. Following their research, numerous authors concentrated on the literature and employed this idea in various contexts. An h-convex interval-valued function (h-convex 𝘐𝘝𝘍) was first proposed in 2018 by Zhao et al. ^[45], who demonstrated that convex 𝘐𝘝𝘍 is a specific example of the 𝐻𝐻-inequality.

Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ be a convex 𝘐𝘝𝘍 given by $\mathfrak{G}\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$, where ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}\right)$ is a convex function, and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}\right)$ is a concave function. If $\mathfrak{G}$ is Riemann integrable, then

We refer readers to ^{[46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]} and the references therein for more examination of the literature on the uses and characteristics of generalized convex functions and 𝐻𝐻-integral inequalities.

Operation research, computer science, management sciences, artificial intelligence, control engineering and decision sciences are just a few of the applied sciences and pure mathematics problems that are studied in ^[64], where a large amount of research on fuzzy sets and systems has been devoted to the development of various fields. Similar to this, the concepts of convexity and non-convexity are crucial in optimization in the fuzzy domain because they allow us to distinguish the optimality condition of convexity and produce fuzzy variational inequalities. As a result, the theories of variational inequality and fuzzy complementary problems have powerful mechanisms of mathematical problems and a cordial relationship. This field is fascinating and has produced many writers. Additionally, the concepts of convex fuzzy mapping and finding its optimality condition with the aid of fuzzy variational inequality were studied by Nanda and Kar ^[65] and Chang ^[66]. Fuzzy convexity's generalization and extension are crucial to its application in a variety of contexts. Let's remark that preinvex fuzzy mapping is one of the most often discussed kinds of nonconvex fuzzy mapping. This concept was first proposed by Noor ^[67], who also demonstrated some findings that show how fuzzy variational-like inequality distinguishes the fuzzy optimality condition of differentiable fuzzy preinvex mappings. For a more in-depth review of the literature on the uses and characterization of generalized convex fuzzy mappings and variational-like inequalities, see ^{[68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85]} and the references therein.

Fuzzy-interval-valued functions are the fuzzy mappings. There are certain integrals that deal with 𝘍𝘐𝘝𝘍s and have 𝘍𝘐𝘝𝘍s as their integrands. For instance, Oseuna-Gomez et al. ^[86] and Costa et al. ^[87] created the Kulisch-Miranker order relation to establish Jensen's integral inequality for 𝘍𝘐𝘝𝘍s. Costa and Floures provided Minkowski and Beckenbach's inequalities, where the integrands are 𝘍𝘐𝘝𝘍s, using the same methodology. Because Costa et al. ^[88] established a relationship between the components of fuzzy-interval space and interval space and introduced a level-wise fuzzy order relation on fuzzy-interval space through the Kulisch-Miranker order relation defined on interval space, these authors were particularly inspired by their work. By creating a fuzzy-interval integral inequality for extended convex 𝘍𝘐𝘝𝘍, where the integrands are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍, we generalize the integral inequalities (2)–(4). For more information related to fuzzy sets, fuzzy functions and inequalities, see ^{[89,90,91,92,93,94,95,96,97,98,99,100,101]} and the references therein.

The structure of this study is as follows: Preliminary ideas and findings in interval space, the space of fuzzy intervals and convex analysis are presented in Section 2. Additionally, the brand-new idea of $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s is also presented. For $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s, Section 3 derives discrete Jensen and Schur type inequalities. Using fuzzy Riemann integrals, Section 4 derives fuzzy-interval 𝐻𝐻-inequalities for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s. To support our findings, some compelling instances are also provided. Section 5 gives conclusions and future plans.

2.   Definitions and basic results

In this section, we first provide a few definitions, rough notations and findings that will be beneficial for future research. Then, we give new $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s definitions and properties.

Let ${\mathcal{K}}_{C}$ be the space of all closed and bounded intervals of $\mathbb{R}$ and $\mathfrak{o}\in {\mathcal{K}}_{C}$ be defined by

If ${\mathfrak{o}}_{\mathfrak{*}} = {\mathfrak{o}}^{\mathfrak{*}}$, then $\mathfrak{o}$ is named as degenerate. In this article, all intervals will be non-degenerate intervals. If ${\mathfrak{o}}_{\mathfrak{*}}\ge 0$, then $\left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]$ is named as a positive interval. The set of all positive intervals is denoted by ${\mathcal{K}}_{C}^{+}$ and defined as ${\mathcal{K}}_{C}^{+} = \left\{\left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]:\left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]\in {\mathbb{R}}_{I}\mathrm{a}\mathrm{n}\mathrm{d}{\mathfrak{o}}_{\mathfrak{*}}\ge 0\right\}.$

Let $\rho \in \mathbb{R}$ and $\rho \mathfrak{o}$ be defined by

Then, the Minkowski difference $\mathfrak{q}-\mathfrak{o}$, addition $\mathfrak{o}+\mathfrak{q}$ and $\mathfrak{o}\times \mathfrak{q}$ for $\mathfrak{o}, \mathfrak{q}\in {\mathcal{K}}_{C}$ are defined by

and $\left[{\mathfrak{q}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}\right]\times \left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right] = \left[min\left\{{\mathfrak{q}}_{\mathfrak{*}}{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{q}}_{\mathfrak{*}}{\mathfrak{o}}^{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}{\mathfrak{o}}^{\mathfrak{*}}\right\}, max\left\{{\mathfrak{q}}_{\mathfrak{*}}{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{q}}_{\mathfrak{*}}{\mathfrak{o}}^{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}{\mathfrak{o}}^{\mathfrak{*}}\right\}\right].$ The inclusion "$\subseteq$" means that $\mathfrak{q}\subseteq \mathfrak{o}$ If and only if, $\left[{\mathfrak{q}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}\right]\subseteq \left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]$, if and only if ${\mathfrak{o}}_{\mathfrak{*}}\le {\mathfrak{q}}_{\mathfrak{*}}$, ${\mathfrak{q}}^{\mathfrak{*}}\le {\mathfrak{o}}^{\mathfrak{*}}.$

Remark 2.1. ^[44] The relation ${"\le }_{I}"$ is defined on ${\mathcal{K}}_{C}$ by

for all $\left[{\mathfrak{q}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}\right], \left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]\in {\mathcal{K}}_{C},$ and it is an order relation. For given $\left[{\mathfrak{q}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}\right], \left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]\in {\mathcal{K}}_{C},$ we say that $\left[{\mathfrak{q}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}\right]{\le }_{I}\left[{\mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{o}}^{\mathfrak{*}}\right]$ if and only if ${\mathfrak{q}}_{\mathfrak{*}}{\le \mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}{\le \mathfrak{o}}^{\mathfrak{*}}$ or ${\mathfrak{q}}_{\mathfrak{*}}{\le \mathfrak{o}}_{\mathfrak{*}}, {\mathfrak{q}}^{\mathfrak{*}}{ < \mathfrak{o}}^{\mathfrak{*}}$.

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

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

The concept of a Riemann integral for 𝘐𝘝𝘍 first introduced by Moore ^[43] is defined as follows:

Theorem 2.2. ^[43] If $\mathfrak{G}:[\mathfrak{a}, \mathfrak{b}]\subset \mathbb{R}\to {\mathbb{R}}_{I}$ is an 𝘐𝘝𝘍 such that $\mathfrak{G}\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}\right)\right]$, then $\mathfrak{G}$ is Riemann integrable over $[\mathfrak{a}, \mathfrak{b}]$ if and only if ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}\right)$ both are Riemann integrable over $\left[\mathfrak{a}, \mathfrak{b}\right]$ such that

The collections of all Riemann integrable real valued functions and Riemann integrable 𝘐𝘝𝘍s are denoted by ${\mathcal{R}}_{[\mathfrak{a}, \mathfrak{b}]}$ and ${\mathcal{I}\mathcal{R}}_{[\mathfrak{a}, \mathfrak{b}]},$ respectively.

Let $\mathbb{R}$ be the set of real numbers. A fuzzy subset set $\mathcal{A}$ of $\mathbb{R}$ is distinguished by a function $\mathfrak{g}:\mathbb{R}\to \left[\mathrm{0, 1}\right]$ called the membership function. In this study, this depiction is approved. Moreover, the collection of all fuzzy subsets of $\mathbb{R}$ is denoted by $\mathbb{F}\left(\mathbb{R}\right).$

A real fuzzy-interval $\mathfrak{g}$ is a fuzzy set in $\mathbb{R}$ with the following properties:

(1) $\mathfrak{g}$ is normal, i.e., there exists $\mathfrak{x}\in \mathbb{R}$ such that $\mathfrak{g}\left(\mathfrak{x}\right) = 1$.

(2) $\mathfrak{g}$ is upper semi continuous, i.e., for given $\mathfrak{x}\in \mathbb{R},$ and for every $\mathfrak{x}\in \mathbb{R}$ there exist $\epsilon > 0$ there exist $\delta > 0$ such that $\mathfrak{g}\left(\mathfrak{x}\right)-\mathfrak{g}\left(y\right) < \epsilon$ for all $y\in \mathbb{R}$ with $\left|\mathfrak{x}-y\right| < \delta.$

(3) $\mathfrak{g}$ is fuzzy convex, i.e., $\mathfrak{g}\left(\left(1-\mathfrak{s}\right)\mathfrak{x}+\mathfrak{s}y\right)\ge min\left(\mathfrak{g}\left(\mathfrak{x}\right), \mathfrak{g}\left(y\right)\right),$ $\forall$ $\mathfrak{x}, y\in \mathbb{R}$ and $\mathfrak{s}\in [0, 1]$.

(4) $\mathfrak{g}$ is compactly supported, i.e., $cl\left\{\mathfrak{x}\in \mathbb{R}|\mathfrak{g}\left(\mathfrak{x}\right) > 0\right\}$ is compact.

The collection of all real fuzzy-intervals is denoted by ${\mathbb{F}}_{C}\left(\mathbb{R}\right)$.

Since ${\mathbb{F}}_{C}\left(\mathbb{R}\right)$ denotes the set of all real fuzzy-intervals and let $\mathfrak{g}\in {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be real fuzzy-interval, if and only if, $𝒿$-levels ${\left[\mathfrak{g}\right]}^𝒿$ is a nonempty compact convex set of $\mathbb{R}$. This is represented by

and from these definitions, we have

where

Proposition 2.3. ^[88] Let $\mathfrak{g}, \mathfrak{d}\in {\mathbb{F}}_{C}\left(\mathbb{R}\right)$. Then, relation "$\preccurlyeq$" given on ${\mathbb{F}}_{C}\left(\mathbb{R}\right)$ by

is a partial order relation.

We now discuss some properties of real fuzzy-intervals under addition, scalar multiplication, multiplication and division. If $\mathfrak{g}, \mathfrak{d}\in {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ and $\rho \in \mathbb{R}$, then arithmetic operations are defined by

Remark 2.4. Obviously, ${\mathbb{F}}_{C}\left(\mathbb{R}\right)$ is closed under addition and nonnegative scalar multiplication, and the above defined properties on ${\mathbb{F}}_{C}\left(\mathbb{R}\right)$ are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number $\rho \in \mathbb{R},$

Theorem 2.5. ^[71,74] The space ${\mathbb{F}}_{C}\left(\mathbb{R}\right)$ dealing with a supremum metric, i.e., for $\psi, \mathfrak{d}\in {\mathbb{F}}_{C}\left(\mathbb{R}\right)$

is a complete metric space, where $H$ denotes the well-known Hausdorff metric on the space of intervals.

Definition 2.6. ^[88] A mapping $\mathfrak{G}:K\subset \mathbb{R}\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ is named as 𝘍𝘐𝘝𝘍. For each $𝒿\in \left[0, 1\right],$ whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:K\subset \mathbb{R}\to {\mathcal{K}}_{C}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in K.$ Here, for each $𝒿\in \left[0, 1\right],$ the end point real functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right):K\to \mathbb{R}$ are called lower and upper functions of $\mathfrak{G}$.

Remark 2.7. Let $\mathfrak{G}:K\subset \mathbb{R}\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a 𝘍𝘐𝘝𝘍. Then, $\mathfrak{G}\left(\mathfrak{x}\right)$ is named as continuous at $\mathfrak{x}\in K$ if for each $𝒿\in \left[0, 1\right],$ both end point functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ are continuous at $\mathfrak{x}\in K.$

From the above literature review, the following results can be concluded (see ^{[88,44,43,86]}):

Definition 2.8. Let $\mathfrak{G}:[c, d]\subset \mathbb{R}\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a 𝘍𝘐𝘝𝘍. The fuzzy Riemann integral of $\mathfrak{G}$ over $\left[c, d\right],$ denoted by $\left(FR\right){\int }_{c}^{d}\mathfrak{G}\left(\mathfrak{x}\right)d\mathfrak{x}$, is defined level-wise by

for all $𝒿\in \left[0, 1\right],$ where ${\mathcal{R}}_{[c, d]}$ is the collection of end point functions of 𝘐𝘝𝘍s. $\mathfrak{G}$ is $\left(FR\right)$-integrable over $[c, d]$ if $\left(FR\right){\int }_{c}^{d}\mathfrak{G}\left(\mathfrak{x}\right)d\mathfrak{x}\in {\mathbb{F}}_{C}\left(\mathbb{R}\right).$ Note that if both end point functions are Lebesgue-integrable, then $\mathfrak{G}$ is fuzzy Aumann-integrable, see ^[81].

Theorem 2.9. Let $\mathfrak{G}:[c, d]\subset \mathbb{R}\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a 𝘍𝘐𝘝𝘍, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:[c, d]\subset \mathbb{R}\to {\mathcal{K}}_{C}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in [c, d]$ and for all $𝒿\in \left[0, 1\right].$ Then, $\mathfrak{G}$ is $\left(FR\right)$-integrable over $[c, d]$ if and only if, ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ both are $R$-integrable over $[c, d]$. Moreover, if $\mathfrak{G}$ is $\left(FR\right)$-integrable over $\left[c, d\right],$ then

for all $𝒿\in \left[0, 1\right].$

The families of all $\left(FR\right)$-integrable 𝘍𝘐𝘝𝘍s and $R$-integrable functions over $[c, d]$ are denoted by ${\mathcal{F}\mathcal{R}}_{\left(\left[c, d\right], 𝒿\right)}$ and ${\mathcal{R}}_{\left(\left[c, d\right], 𝒿\right)},$ for all $𝒿\in \left[0, 1\right].$

Definition 2.10. ^[51] A function $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{R}}^{+}$ is named as a $P$-convex function if

for all $\mathfrak{x}, y\in \left[\mathfrak{a}, \mathfrak{b}\right], \mathfrak{s}\in \left[0, 1\right].$ If (13) is reversed, then $\mathfrak{G}$ is named as $P$-concave.

Definition 2.11. ^[47] A function $\mathfrak{G}:K\to {\mathbb{R}}^{+}$ is named as an $s$-convex function in the second sense if

for all $\mathfrak{x}, y\in \left[\mathfrak{a}, \mathfrak{b}\right], \mathfrak{s}\in \left[0, 1\right],$ where $s\in (0, 1)$. If (14) is reversed, then $\mathfrak{G}$ is named as $s$-concave in the second sense.

Definition 2.12. ^[55] A function $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{R}}^{+}$ is named as a $\mathfrak{J}$-convex function if for all $\mathfrak{x}, y\in \left[\mathfrak{a}, \mathfrak{b}\right], \mathfrak{s}\in \left[0, 1\right],$ we have

where $\mathfrak{J}:\mathcal{L}\to {\mathbb{R}}^{+}$ such that $\mathfrak{J}\not\equiv 0$, $\left[0, 1\right]\subseteq \mathcal{L}.$ If (15) is reversed, then $\mathfrak{G}$ is named as $\mathfrak{J}$-concave in the second sense.

A function $\mathfrak{J}:\mathcal{L}\to {\mathbb{R}}^{+}$ is named as super multiplicative if for all $\mathfrak{x}, y\in \mathcal{L}$, we have

If (16) is reversed, then $\mathfrak{J}$ is known as sub multiplicative. If the equality holds in (16), then $\mathfrak{J}$ is named as multiplicative.

Definition 2.13. ^[17] Let $\mathfrak{p}\in \mathbb{R}$ with $\mathfrak{p}\ne 0$. Then, the interval ${K}_{\mathfrak{p}}$ is named as $\mathfrak{p}$-convex if

for all $\mathfrak{x}, y\in {K}_{\mathfrak{p}}, \mathfrak{s}\in \left[0, 1\right],$ where $\mathfrak{p} = 2n+1$ and $n\in N.$

Definition 2.14. ^[17] Let $\mathfrak{p}\in \mathbb{R}$ with $\mathfrak{p}\ne 0$ and ${K}_{\mathfrak{p}} = \left[\mathfrak{a}, \mathfrak{b}\right]\subseteq \mathbb{R}$. Then, the function $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{R}}^{+}$ is named as a $\mathfrak{p}$-convex function if

for all $\mathfrak{x}, y\in \left[\mathfrak{a}, \mathfrak{b}\right], \mathfrak{s}\in \left[0, 1\right].$ If the inequality (18) is reversed, then $\mathfrak{G}$ is named as a $\mathfrak{p}$-concave function.

Definition 2.15. ^[52] Let ${K}_{\mathfrak{p}}$ be a $\mathfrak{p}$-convex set and $\mathfrak{J}:[0, 1]\subseteq \mathcal{L}\to {\mathbb{R}}^{+}$ be a nonnegative real-valued function such that $\mathfrak{J}\not\equiv 0$, where $\mathcal{L}\subseteq \mathbb{R}$. Then, function $\mathfrak{G}:{K}_{\mathfrak{p}}\to \mathbb{R}$ is named as $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex on ${K}_{\mathfrak{p}}$ such that

for all $\mathfrak{x}, y\in {K}_{\mathfrak{p}} = \left[\mathfrak{a}, \mathfrak{b}\right], \mathfrak{s}\in \left[0, 1\right],$ where $\mathfrak{G}\left(\mathfrak{x}\right)\ge 0$ and $\mathfrak{J}:\mathcal{L}\to {\mathbb{R}}^{+}$ such that $\mathfrak{J}\not\equiv 0$ and $\left[0, 1\right]\subseteq \mathcal{L}.$ If (19) is reversed, then $\mathfrak{G}$ is named as $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave on $\left[\mathfrak{a}, \mathfrak{b}\right]$. $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-affine if and only if it is both a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex and $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave function.

Definition 2.16. ^[65,66] Let $K$ be a convex set. Then, 𝘍𝘐𝘝𝘍 $\mathfrak{G}:K\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ is named as convex on $K$ if

for all $\mathfrak{x}, y\in K, \mathfrak{s}\in \left[0, 1\right],$ where $\mathfrak{G}\left(\mathfrak{x}\right)\succcurlyeq \widetilde{0}.$ If (20) is reversed, then $\mathfrak{G}$ is named as concave on $\left[\mathfrak{a}, \mathfrak{b}\right]$. $\mathfrak{G}$ is affine if and only if it is both a convex and concave function.

Definition 2.17. Let ${K}_{\mathfrak{p}}$ be a $\mathfrak{p}$-convex set and $\mathfrak{J}:[0, 1]\subseteq \mathcal{L}\to {\mathbb{R}}^{+}$ be a nonnegative real-valued function such that $\mathfrak{J}\not\equiv 0$, where $\mathcal{L}\subseteq \mathbb{R}$. Then, 𝘍𝘐𝘝𝘍 $\mathfrak{G}:{K}_{\mathfrak{p}}\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ is named as $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex on ${K}_{\mathfrak{p}}$ such that

for all $\mathfrak{x}, y\in {K}_{\mathfrak{p}}, \mathfrak{s}\in \left[0, 1\right],$ where $\mathfrak{G}\left(\mathfrak{x}\right)\succcurlyeq \widetilde{0}$ and $\mathfrak{J}:\mathcal{L}\to {\mathbb{R}}^{+}$ such that $\mathfrak{J}\not\equiv 0$ and $\left[0, 1\right]\subseteq \mathcal{L}.$ If (21) is reversed, then $\mathfrak{G}$ is named as $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave on $\left[\mathfrak{a}, \mathfrak{b}\right]$. $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-affine if and only if it is both $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex and $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave 𝘍𝘐𝘝𝘍.

Remark 2.18. The $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s have some very nice properties similar to convex 𝘍𝘐𝘝𝘍:

If $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍, then ${\rm Y}\mathfrak{G}$ is also $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex for ${\rm Y}\ge 0$.

If $\mathcal{F}$ and $\mathfrak{G}$ both are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s, then $\mathrm{m}\mathrm{a}\mathrm{x}\left(\mathcal{F}\left(\mathfrak{x}\right), \mathfrak{G}\left(\mathfrak{x}\right)\right)$ is also $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍.

We now discuss some new and known special cases of $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s:

If $\mathfrak{J}\left(\mathfrak{s}\right) = {\mathfrak{s}}^{s}$ with $s\in \left(\mathrm{0, 1}\right)$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 becomes $\left(\mathfrak{p}, s\right)$-convex 𝘍𝘐𝘝𝘍, that is,

If $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s},$ then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 becomes $\mathfrak{p}$-convex 𝘍𝘐𝘝𝘍, that is,

If $\mathfrak{p}\equiv 1$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 becomes $\mathfrak{J}$-convex 𝘍𝘐𝘝𝘍, that is,

If $\mathfrak{J}\left(\mathfrak{s}\right) = {\mathfrak{s}}^{s}$ with $s\in \left(\mathrm{0, 1}\right)$ and $\mathfrak{p}\equiv 1$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 becomes $s$-convex 𝘍𝘐𝘝𝘍, that is,

If $\mathfrak{p}\equiv 1$ and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 becomes convex 𝘍𝘐𝘝𝘍, see ^[65,66], that is,

Theorem 2.19. Let ${K}_{\mathfrak{p}}$ be a $\mathfrak{p}$-convex set, non-negative real valued function $\mathfrak{J}:[0, 1]\subseteq {K}_{\mathfrak{p}}\to \mathbb{R}$ such that $\mathfrak{J}\not\equiv 0$, and let $\mathfrak{G}:{K}_{\mathfrak{p}}\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a 𝘍𝘐𝘝𝘍, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:{K}_{\mathfrak{p}}\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}\subset {\mathcal{K}}_{C}$ are given by

for all $\mathfrak{x}\in {K}_{\mathfrak{p}}$ and for all $𝒿\in \left[0, 1\right]$. Then, $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex on ${K}_{\mathfrak{p}}$ if and only if, for all $𝒿\in \left[0, 1\right],$ ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ both are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex functions $.$

Proof. Assume that for each $𝒿\in \left[0, 1\right],$ ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex on ${K}_{\mathfrak{p}}.$ Then, from (19), we have

and

Then, by (27), (6) and (8), we obtain

that is,

Hence, $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 on ${K}_{\mathfrak{p}}$.

Conversely, let $\mathfrak{G}$ be $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 on ${K}_{\mathfrak{p}}.$ Then, for all $\mathfrak{x}, y\in {K}_{\mathfrak{p}}$ and $\mathfrak{s}\in \left[0, 1\right],$ we have $\mathfrak{G}\left({\left[\mathfrak{s}{\mathfrak{x}}^{\mathfrak{p}}+\left(1-\mathfrak{s}\right){y}^{\mathfrak{p}}\right]}^{\frac{1}{\mathfrak{p}}}\right)\preccurlyeq \mathfrak{J}\left(\mathfrak{s}\right)\mathfrak{G}\left(\mathfrak{x}\right){\widetilde + }\mathfrak{J}\left(1-\mathfrak{s}\right)\mathfrak{G}\left(y\right).$ Therefore, from (27), we have

Again, from (27), (6) and (8), we obtain

Then by $\left(\mathfrak{p}, \mathfrak{J}\right)$-convexity of $\mathfrak{G}$, we have

and

for each $𝒿\in \left[0, 1\right].$ Hence, the result follows.

Remark 2.20. If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ with $𝒿 = 1$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 reduces to the classical $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex function (see ^[52]).

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ with $𝒿 = 1$ and $\mathfrak{J}\left(\mathfrak{s}\right) = {\mathfrak{s}}^{s}$ with $s\in (0, 1)$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 reduces to the classical $\left(\mathfrak{p}, s\right)$-convex function (see ^[47]).

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ with $𝒿 = 1$ and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 reduces to the classical $\mathfrak{p}$-convex function (see ^[17]).

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ with $𝒿 = 1$ and $\mathfrak{p} = 1$, then $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 reduces to the classical $\mathfrak{J}$-convex function (see ^[55]).

Example 2.21. We consider $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s},$ for $\mathfrak{s}\in \left[0, 1\right],$ and the 𝘍𝘐𝘝𝘍 $\mathfrak{G}:[0, 1]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ defined by

Then, for each $𝒿\in \left[0, 1\right],$ we have ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[2𝒿{\mathfrak{x}}^{\mathfrak{p}}, (4-2𝒿){\mathfrak{x}}^{\mathfrak{p}}\right]$. Since end point functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ both are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex functions for each $𝒿\in [0, 1]$. Hence, $\mathfrak{G}\left(\mathfrak{x}\right)$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍.

3.   Discrete Jensen and Schur type inequalities

We introduce the idea of discrete Jensen and Schur type inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s in this section. The discrete Jensen type inequality is further refined in several ways. The discrete Jensen type inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 is shown first in the following result.

Theorem 3.1. (Discrete Jensen type inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍) Let ${\mathfrak{w}}_{j}\in {\mathbb{R}}^{+}$, ${\mathfrak{x}}_{j}\in \left[\mathfrak{a}, \mathfrak{b}\right],$ $\left(j = 1, 2, 3, \dots \kappa, \kappa \ge 2\right)$ and $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex fuzzy 𝘍𝘐𝘝𝘍 with non-negative real valued function $\mathfrak{J}:[0, 1]\to \mathbb{R}$, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{J}$ is a super-multiplicative function on $\mathcal{L}$, then

where ${W}_{\kappa } = \sum _{j = 1}^{\kappa }{\mathfrak{w}}_{j}.$ If function $\mathfrak{J}$ is sub-multiplicative and $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave, then inequality (29) is reversed.

Proof. When $\kappa = 2$, inequality (29) is true. Consider that inequality (19) is true for $\kappa = n-1,$ and then

Now, let us prove that inequality (29) holds for $\kappa = n.$

Therefore, for each $𝒿\in \left[0, 1\right],$ we have

From that, we have

that is,

and the result follows.

If ${\mathfrak{w}}_{1} = {\mathfrak{w}}_{2} = {\mathfrak{w}}_{3} = \dots = {\mathfrak{w}}_{\kappa } = 1,$ then Theorem 3.1 reduces to the following result:

Corollary 3.2. Let ${\mathfrak{x}}_{j}\in \left[\mathfrak{a}, \mathfrak{b}\right],$ $\left(j = 1, 2, 3, \dots \kappa, \kappa \ge 2\right)$ and $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex fuzzy 𝘍𝘐𝘝𝘍 with non-negative real valued function $\mathfrak{J}:[0, 1]\to \mathbb{R}$, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{J}$ is a super-multiplicative function, then

If function $\mathfrak{J}$ is sub-multiplicative, and $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave, then inequality (30) is reversed.

Next, Theorem 3.3 gives the Schur-type inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s.

Theorem 3.3. (Discrete Schur-type inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍) Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 with non-negative real valued function $\mathfrak{J}:[0, 1]\to \mathbb{R}$, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{J}:\mathcal{L}\to {\mathbb{R}}^{+}$ is a nonnegative super-multiplicative function, then for ${\mathfrak{x}}_{1}, {\mathfrak{x}}_{2}, {\mathfrak{x}}_{3}\in \left[\mathfrak{a}, \mathfrak{b}\right]$, such that ${\mathfrak{x}}_{1} < {\mathfrak{x}}_{2} < {\mathfrak{x}}_{3}$ and ${{\mathfrak{x}}_{3}}^{\mathfrak{p}}-{{\mathfrak{x}}_{1}}^{\mathfrak{p}}$, ${{\mathfrak{x}}_{3}}^{\mathfrak{p}}-{{\mathfrak{x}}_{2}}^{\mathfrak{p}},$ ${{\mathfrak{x}}_{2}}^{\mathfrak{p}}-{{\mathfrak{x}}_{1}}^{\mathfrak{p}}\in \mathcal{L}$, we have

If the function $\mathfrak{J}$ is a nonnegative sub-multiplicative function, and $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave, then inequality (31) is reversed.

Proof. Let ${\mathfrak{x}}_{1}, {\mathfrak{x}}_{2}, {\mathfrak{x}}_{3}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and $\mathfrak{J}\left({{\mathfrak{x}}_{3}}^{\mathfrak{p}}-{{\mathfrak{x}}_{1}}^{\mathfrak{p}}\right) > 0.$ Then, by hypothesis, we have

Consider $\mathfrak{s} = \frac{{{\mathfrak{x}}_{3}}^{\mathfrak{p}}-{{\mathfrak{x}}_{2}}^{\mathfrak{p}}}{{{\mathfrak{x}}_{3}}^{\mathfrak{p}}-{{\mathfrak{x}}_{1}}^{\mathfrak{p}}}$, and then ${{\mathfrak{x}}_{2}}^{\mathfrak{p}} = \mathfrak{s}{{\mathfrak{x}}_{1}}^{\mathfrak{p}}+\left(1-\mathfrak{s}\right){{\mathfrak{x}}_{3}}^{\mathfrak{p}}.$ Since $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍, by hypothesis, we have

Therefore, for each $𝒿\in \left[0, 1\right],$ we have

From (33), we have

that is,

Hence,

A refinement of a Jensen type inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 is given in the following theorem.

Theorem 3.4. Let ${\mathfrak{w}}_{j}\in {\mathbb{R}}^{+}$, ${\mathfrak{x}}_{j}\in \left[\mathfrak{a}, \mathfrak{b}\right],$ $\left(j = 1, 2, 3, \dots \kappa, \kappa \ge 2\right)$ and $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 with non-negative real valued function $\mathfrak{J}:[0, 1]\to \mathbb{R}$, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\left(L, U\right)\subseteq [\mathfrak{a}, \mathfrak{b}]$ and $\mathfrak{J}$ is a nonnegative super-multiplicative function, then

where ${W}_{\kappa } = \sum _{j = 1}^{\kappa }{\mathfrak{w}}_{j}.$ If $\mathfrak{J}$ is a sub-multiplicative function and $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave, then inequality (34) is reversed.

Proof. Consider $L = {\mathfrak{x}}_{1}, {\mathfrak{x}}_{j} = {\mathfrak{x}}_{2},$ $\left(j = 1, 2, 3, \dots \kappa \right)$, $U = {\mathfrak{x}}_{3}$. Then, by hypothesis and inequality (32), we have

Therefore, for each $𝒿\in \left[0, 1\right]$, we have

The above inequality can be written as

Taking the sum of all inequalities (35) for $j = 1, 2, 3, \dots \kappa,$ we have

That is,

Thus,

and this completes the proof.

We now consider some special cases of Theorems 3.1 and 3.4.

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = {\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$, then Theorems 3.1 and 3.4 reduce to the following results:

Corollary 3.5. ^[52] (Jensen inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex function) Let ${\mathfrak{w}}_{j}\in {\mathbb{R}}^{+}$, ${\mathfrak{x}}_{j}\in \left[\mathfrak{a}, \mathfrak{b}\right],$ $\left(j = 1, 2, 3, \dots \kappa, \kappa \ge 2\right)$ and let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{R}}^{+}$ be a non-negative real-valued function. If $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex function, and $\mathfrak{J}$ is a nonnegative super-multiplicative function on $\mathcal{L}$, then

where ${W}_{\kappa } = \sum _{j = 1}^{\kappa }{\mathfrak{w}}_{j}.$ If $\mathfrak{J}$ is a sub-multiplicative function, and $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave function, then inequality (36) is reversed.

Corollary 3.6. Let ${\mathfrak{w}}_{j}\in {\mathbb{R}}^{+}$, ${\mathfrak{x}}_{j}\in \left[\mathfrak{a}, \mathfrak{b}\right],$ $\left(j = 1, 2, 3, \dots \kappa, \kappa \ge 2\right),$ $\mathfrak{J}$ be a nonnegative super-multiplicative function on $\mathcal{L}$ and $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{R}}^{+}$ be a non-negative real-valued function. If $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex function, and ${\mathfrak{x}}_{1}, {\mathfrak{x}}_{2}, \dots, {\mathfrak{x}}_{j}\in \left(L, U\right)\subseteq [\mathfrak{a}, \mathfrak{b}]$, then

where ${W}_{\kappa } = \sum _{j = 1}^{\kappa }{\mathfrak{w}}_{j}.$ If $\mathfrak{J}$ is a sub-multiplicative function, and $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave function, then inequality (37) is reversed.

4.   Hermite-Hadamard type inequalities

In view of the 𝐻𝐻-inequality in Eq (3) and 𝐻𝐻-inequality in Eq (4), we can deduce the following version of the 𝐻𝐻-inequalities for $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave 𝘍𝘐𝘝𝘍s.

Theorem 4.1. Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 with non-negative real valued function $\mathfrak{J}:[0, 1]\to {\mathbb{R}}^{+}$ and $\mathfrak{J}\left(\frac{1}{2}\right)\ne 0$, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{G}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathfrak{a}, \mathfrak{b}\right], 𝒿\right)}$, then

If $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave 𝘍𝘐𝘝𝘍, then

Proof. Let $\mathfrak{G}$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍. Then, by hypothesis, we have

Therefore, for each

$𝒿\in \left[0, 1\right]$, we have

Then,

It follows that

That is,

Thus,

In a similar way as above, we have

Combining (40) and (41), we have

Hence, we have the required result.

Remark 4.2. If $\mathfrak{J}\left(\mathfrak{s}\right) = {\mathfrak{s}}^{s}$, then Theorem 4.1 reduces to the result for $\left(\mathfrak{p}, s\right)$-convex 𝘍𝘐𝘝𝘍 (see ^[85]):

If $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then Theorem 4.1 reduces to the result for $\mathfrak{p}$-convex 𝘍𝘐𝘝𝘍 (see ^[85]):

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{b}, 𝒿\right)$ with $𝒿 = 1$, then Theorem 4.1 reduces to the result for classical $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex function (see ^[52]):

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{b}, 𝒿\right)$ with $𝒿 = 1$ and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then Theorem 4.1 reduces to the result for classical $\mathfrak{p}$-convex function (see ^[17]):

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{b}, 𝒿\right)$ with $𝒿 = 1,$ $\mathfrak{p} = 1$ and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then Theorem 4.1 reduces to the result for classical convex function (see ^[41]):

Example 4.3. Let $\mathfrak{p}$ be an odd number and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$ for $\mathfrak{s}\in \left[0, 1\right]$, and the 𝘍𝘐𝘝𝘍 $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right] = [2, 3]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ is defined by

Then, for each $𝒿\in \left[0, 1\right],$ we have ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[𝒿\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right), \left(2-𝒿\right)\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right)\right]$. Since end point functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = 𝒿\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right),$ ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = (2-𝒿)\left(2-{\mathfrak{x}}^{\frac{\begin{array}{c} p\end{array}}{2}}\right)$ are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex functions for each $𝒿\in [0, 1]$. Then, $\mathfrak{G}\left(\mathfrak{x}\right)$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍. We now compute the following:

for all $𝒿\in \left[0, 1\right].$ That means

$\left[\frac{4-\sqrt{10}}{2}𝒿, \frac{4-\sqrt{10}}{2}\left(2-𝒿\right)\right]{\le }_{I}\left[\frac{21}{50}𝒿, \frac{21}{50}(2-𝒿)\right]{\le }_{I}\left[\frac{4-\sqrt{2}-\sqrt{3}}{2}𝒿, \frac{4-\sqrt{2}-\sqrt{3}}{2}(2-𝒿)\right],$ for all $𝒿\in \left[0, 1\right],$ and the Theorem has been demonstrated.

Theorem 4.4. Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 with non-negative real valued function $\mathfrak{J}:[0, 1]\to {\mathbb{R}}^{+}$ and $\mathfrak{J}\left(\frac{1}{2}\right)\ne 0,$ whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{G}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathfrak{a}, \mathfrak{b}\right], 𝒿\right)}$, then

where

and

Proof. Take $\left[{\mathfrak{a}}^{\mathfrak{p}}, \frac{{\mathfrak{a}}^{\mathfrak{p}}+{\mathfrak{b}}^{\mathfrak{p}}}{2}\right],$ and we have

Therefore, for each $𝒿\in \left[0, 1\right]$, we have

In consequence, we obtain

That is,

It follows that

In a similar way as above, we have

Combining (49) and (50), we have

By using Theorem 4.1, we have

Therefore, for each $𝒿\in \left[0, 1\right]$, we have

that is,

and hence, the result follows.

Example 4.5. Let $\mathfrak{p}$ be an odd number and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s},$ for $\mathfrak{s}\in \left[0, 1\right]$, and the 𝘍𝘐𝘝𝘍 $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right] = [2, 3]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ defined by, ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[𝒿\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right), \left(2-𝒿\right)\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right)\right],$ as in Example 4.3, then $\mathfrak{G}\left(\mathfrak{x}\right)$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 and satisfying (38). We have ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = 𝒿\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right)$ and ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = \left(2-𝒿\right)\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right)$. We now compute the following

Then, we obtain that

Hence, Theorem 4.4 is verified.

Next, Theorems 4.6 and 4.7 obtain the fuzzy interval integral inequalities for the product of $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s

Theorem 4.6. Let $\mathfrak{G}, \mathcal{J}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be two $\left(\mathfrak{p}, {\mathfrak{J}}_{1}\right)$-convex and $\left(\mathfrak{p}, {\mathfrak{J}}_{2}\right)$-convex 𝘍𝘐𝘝𝘍s with non-negative real valued functions ${\mathfrak{J}}_{1}, {\mathfrak{J}}_{2}:[0, 1]\to \mathbb{R},$ respectively, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿, {\mathcal{J}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are, respectively, given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ and ${\mathcal{J}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathcal{J}}_{\mathcal{*}}\left(\mathfrak{x}, 𝒿\right), {\mathcal{J}}^{\mathcal{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{G}\widetilde{\times }\mathcal{J}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathfrak{a}, \mathfrak{b}\right], 𝒿\right)}$, then

where $\mathcal{M}\left(\mathfrak{a}, \mathfrak{b}\right) = \mathfrak{G}\left(\mathfrak{a}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{a}\right){\widetilde + }\mathfrak{G}\left(\mathfrak{b}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{b}\right),$ $\mathcal{N}\left(\mathfrak{a}, \mathfrak{b}\right) = \mathfrak{G}\left(\mathfrak{a}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{b}\right){\widetilde + }\mathfrak{G}\left(\mathfrak{b}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{a}\right),$ and $\mathcal{M}\left(\mathfrak{a}, \mathfrak{b}\right) = \left[{\mathcal{M}}_{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right), {\mathcal{M}}^{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right)\right]$ and $\mathcal{N}\left(\mathfrak{a}, \mathfrak{b}\right) = \left[{\mathcal{N}}_{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right), {\mathcal{N}}^{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right)\right].$

Proof. Since $\mathfrak{G}$ is a $\left(\mathfrak{p}, {\mathfrak{J}}_{1}\right)$-convex 𝘍𝘐𝘝𝘍 and $\mathcal{J}$ is a $\left(\mathfrak{p}, {\mathfrak{J}}_{2}\right)$-convex 𝘍𝘐𝘝𝘍 then, for each $𝒿\in \left[0, 1\right]$, we have

and

From the definition of $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s it follows that $\mathfrak{G}\left(\mathfrak{x}\right)\succcurlyeq \widetilde{0}$ and $\mathcal{J}\left(\mathfrak{x}\right)\succcurlyeq \widetilde{0}$, so

Integrating both sides of above inequality over [0, 1] we get

It follows that

that is,

Thus,

and the theorem has been established.

Theorem 4.7. Let $\mathfrak{G}, \mathcal{J}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be two $\left(\mathfrak{p}, {\mathfrak{J}}_{1}\right)$-convex and $\left(\mathfrak{p}, {\mathfrak{J}}_{2}\right)$-convex 𝘍𝘐𝘝𝘍s with non-negative real valued functions ${\mathfrak{J}}_{1}, {\mathfrak{J}}_{2}:[0, 1]\to {\mathbb{R}}^{+}$ such that ${\mathfrak{J}}_{1}, {\mathfrak{J}}_{2}\not\equiv 0$ and ${\mathfrak{J}}_{1}\left(\frac{1}{2}\right){\mathfrak{J}}_{2}\left(\frac{1}{2}\right)\ne 0$, respectively, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿, {\mathcal{J}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are, respectively, given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ and ${\mathcal{J}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathcal{J}}_{\mathcal{*}}\left(\mathfrak{x}, 𝒿\right), {\mathcal{J}}^{\mathcal{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{G}\widetilde{\times }\mathcal{J}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathfrak{a}, \mathfrak{b}\right], 𝒿\right)}$, then

where $\mathcal{M}\left(\mathfrak{a}, \mathfrak{b}\right) = \mathfrak{G}\left(\mathfrak{a}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{a}\right){\widetilde + }\mathfrak{G}\left(\mathfrak{b}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{b}\right),$ $\mathcal{N}\left(\mathfrak{a}, \mathfrak{b}\right) = \mathfrak{G}\left(\mathfrak{a}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{b}\right){\widetilde + }\mathfrak{G}\left(\mathfrak{b}\right)\widetilde{\times }\mathcal{J}\left(\mathfrak{a}\right),$ and $\mathcal{M}\left(\mathfrak{a}, \mathfrak{b}\right) = \left[{\mathcal{M}}_{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right), {\mathcal{M}}^{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right)\right]$ and $\mathcal{N}\left(\mathfrak{a}, \mathfrak{b}\right) = \left[{\mathcal{N}}_{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right), {\mathcal{N}}^{\mathcal{*}}\left(\left(\mathfrak{a}, \mathfrak{b}\right), 𝒿\right)\right].$

Proof. By hypothesis, for each $𝒿\in \left[0, 1\right]$, we have

Integrating over $\left[0, 1\right],$ we have

that is,

Hence, we have the required result.

Example 4.8. Let $\mathfrak{p}$ be an odd number and ${\mathfrak{J}}_{1}\left(\mathfrak{s}\right) = \mathfrak{s},$ ${\mathfrak{J}}_{2}\left(\mathfrak{s}\right) = 1,$ for $\mathfrak{s}\in \left[0, 1\right]$, and the $\left(\mathfrak{p}, {\mathfrak{J}}_{1}\right)$-convex and $\left(\mathfrak{p}, {\mathfrak{J}}_{2}\right)$-convex 𝘍𝘐𝘝𝘍s $\mathfrak{G}, \mathcal{J}:\left[\mathfrak{a}, \mathfrak{b}\right] = [2, 3]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ are, respectively, defined by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[𝒿\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right), \left(2-𝒿\right)\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right)\right],$ as in Example 4.3, and ${\mathcal{J}}_𝒿\left(\mathfrak{x}\right) = \left[𝒿{\mathfrak{x}}^{\mathfrak{p}}, (2-𝒿){\mathfrak{x}}^{\mathfrak{p}}\right]$. Since $\mathfrak{G}\left(x\right)$ and $\mathcal{J}\left(x\right)$ both are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s, ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = 𝒿\left(2-{\mathfrak{x}}^{\frac{\begin{array}{c} p\end{array}}{2}}\right)$, ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right) = \left(2-𝒿\right)\left(2-{\mathfrak{x}}^{\frac{\mathfrak{p}}{2}}\right)$, and ${\mathcal{J}}_{\mathcal{*}}\left(\mathfrak{x}, 𝒿\right) = 𝒿{\mathfrak{x}}^{\mathfrak{p}}$, ${\mathcal{J}}^{\mathcal{*}}\left(\mathfrak{x}, 𝒿\right) = (2-𝒿){\mathfrak{x}}^{\mathfrak{p}}$, we compute the following.

for each $𝒿\in \left[0, 1\right],$ which means

Hence, Theorem 4.6 is demonstrated.

For Theorem 4.7, we have

for each $𝒿\in \left[0, 1\right],$ which means

Hence, Theorem 4.7 is verified.

Next, Theorems 4.9 and 4.10 give the second 𝐻𝐻-Fejér inequality and the first 𝐻𝐻-Fejér inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍, respectively.

Theorem 4.9. (Second 𝐻𝐻-Fejér inequality for $\mathfrak{J}$-convex 𝘍𝘐𝘝𝘍) Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 with $\mathfrak{a} < \mathfrak{b}$ and $\mathfrak{J}:[0, 1]\to {\mathbb{R}}^{+}$, whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{G}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathfrak{a}, \mathfrak{b}\right], 𝒿\right)}$ and $\mathit{\Omega }:\left[\mathfrak{a}, \mathfrak{b}\right]\to \mathbb{R}, \mathit{\Omega }\left(\mathfrak{x}\right)\ge 0,$ $\mathfrak{p}$-symmetric with respect to ${\left[\frac{{\mathfrak{a}}^{\mathfrak{p}}+{\mathfrak{b}}^{\mathfrak{p}}}{2}\right]}^{\frac{1}{\mathfrak{p}}},$ then

If $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave 𝘍𝘐𝘝𝘍, then inequality (53) is reversed.

Proof. Let $\mathfrak{G}$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍. Then, for each $𝒿\in \left[0, 1\right]$, we have

Also,

After adding (54) and (55) and integrating over $\left[0, 1\right],$ we get

Since $\mathit{\Omega }$ is symmetric,

Since

From (57) and integrating with respect to $\mathfrak{s}$ over $[0, 1]$, we have

that is,

and hence,

Theorem 4.10. (First 𝐻𝐻-Fejér inequality for $\mathfrak{J}$-convex 𝘍𝘐𝘝𝘍) Let $\mathfrak{G}:\left[\mathfrak{a}, \mathfrak{b}\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ be a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍 with $\mathfrak{a} < \mathfrak{b}$ and $\mathfrak{J}:[0, 1]\to {\mathbb{R}}^{+}$ such that $\mathfrak{J}\left(\frac{1}{2}\right)\not\equiv 0,$ whose $𝒿$-levels define the family of 𝘐𝘝𝘍s ${\mathfrak{G}}_𝒿:\left[\mathfrak{a}, \mathfrak{b}\right]\subset \mathbb{R}\to {{\mathcal{K}}_{C}}^{+}$ are given by ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[{\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right), {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)\right]$ for all $\mathfrak{x}\in \left[\mathfrak{a}, \mathfrak{b}\right]$ and for all $𝒿\in \left[0, 1\right]$. If $\mathfrak{G}\in {\mathcal{F}\mathcal{R}}_{\left(\left[\mathfrak{a}, \mathfrak{b}\right], 𝒿\right)}$ and $\mathit{\Omega }:\left[\mathfrak{a}, \mathfrak{b}\right]\to \mathbb{R}, \mathit{\Omega }\left(\mathfrak{x}\right)\ge 0,$ $\mathfrak{p}$-symmetric with respect to ${\left[\frac{{\mathfrak{a}}^{\mathfrak{p}}+{\mathfrak{b}}^{\mathfrak{p}}}{2}\right]}^{\frac{1}{\mathfrak{p}}},$ and ${\int }_{\mathfrak{a}}^{\mathfrak{b}}\mathit{\Omega }\left(\mathfrak{x}\right)d\mathfrak{x} > 0$, then

If $\mathfrak{G}$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-concave 𝘍𝘐𝘝𝘍, then inequality (58) is reversed.

Proof. Since $\mathfrak{G}$ is a $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍, for each $𝒿\in \left[0, 1\right]$ we have

By multiplying (59) by $\mathit{\Omega }\left({\left[\mathfrak{s}{\mathfrak{a}}^{\mathfrak{p}}+\left(1-\mathfrak{s}\right){\mathfrak{b}}^{\mathfrak{p}}\right]}^{\frac{1}{\mathfrak{p}}}\right) = \mathit{\Omega }\left({\left[\left(1-\mathfrak{s}\right){\mathfrak{a}}^{\mathfrak{p}}+\mathfrak{s}{\mathfrak{b}}^{\mathfrak{p}}\right]}^{\frac{1}{\mathfrak{p}}}\right)$ and integrating it by $\mathfrak{s}$ over $\left[0, 1\right],$ we obtain

Since

from (61), (60) we have

From that, we have

that is,

This completes the proof.

Remark 4.11. If in the Theorems 4.9 and 4.10 $\mathfrak{J}\left(\mathfrak{s}\right) = {\mathfrak{s}}^{s}$, then we obtain the appropriate theorems for $\left(\mathfrak{p}, s\right)$-convex 𝘍𝘐𝘝𝘍s on the second sense (see ^[85]):

If in the Theorems 4.9 and 10 $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then we obtain the appropriate theorems for $\mathfrak{p}$-convex 𝘍𝘐𝘝𝘍s (see ^[85]).

If ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right)$ with $𝒿 = 1$, then Theorems 4.9 and 4.10 reduce to classical first and second 𝐻𝐻-Fejér inequality for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex function.

If in the Theorems 4.9 and 4.10, ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right)$ with $𝒿 = 1$ and $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$, then we obtain the appropriate theorems for $\mathfrak{p}$-convex function (see ^[53]).

If in the Theorems 4.9 and 4.10, ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right) = {\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{a}, 𝒿\right)$ with $𝒿 = 1$, $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s}$ and $\mathfrak{p} = 1$, then we obtain the appropriate theorems for convex function, ^[42].

If $\mathit{\Omega }\left(\mathfrak{x}\right) = 1,$ then combining Theorems 4.9 and 4.10, we get Theorem 4.1.

Example 4.12. We consider $\mathfrak{J}\left(\mathfrak{s}\right) = \mathfrak{s},$ for $\mathfrak{s}\in \left[0, 1\right],$ and the 𝘍𝘐𝘝𝘍 $\mathfrak{G}:\left[1, 4\right]\to {\mathbb{F}}_{C}\left(\mathbb{R}\right)$ defined by,

Then, for each $𝒿\in \left[0, 1\right],$ we have ${\mathfrak{G}}_𝒿\left(\mathfrak{x}\right) = \left[(1+𝒿){e}^{{\mathfrak{x}}^{\mathfrak{p}}}, 2(2-𝒿){e}^{{\mathfrak{x}}^{\mathfrak{p}}}\right]$. Since end point functions ${\mathfrak{G}}_{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right),$ ${\mathfrak{G}}^{\mathfrak{*}}\left(\mathfrak{x}, 𝒿\right)$ are $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex functions, for each $𝒿\in [0, 1]$, $\mathfrak{G}\left(\mathfrak{x}\right)$ is $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍. If

where $\mathfrak{p} = 1$, then we have

and

From (64) and (65), we have

Hence, Theorem 4.9 is verified.

For Theorem 4.10, we have

From (66) and (67), we have

Hence, Theorem 4.10 is demonstrated.

5.   Conclusions

The $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex (concave, affine) class for 𝘍𝘐𝘝𝘍s was established in this paper. For $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍, we created some brand-new discrete Jensen and Schur type inequalities. Additionally, using fuzzy Riemann integrals, we discovered several $HH$-inequalities for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s. Examples were used to demonstrate how our findings apply to a broad class of previously undiscovered and well-known inequalities for $\left(\mathfrak{p}, \mathfrak{J}\right)$-convex 𝘍𝘐𝘝𝘍s and their variant forms. We will try to analyze Jensen and $HH$-inequalities for 𝘐𝘝𝘍 and 𝘍𝘐𝘝𝘍s on a temporal scale in the future as we explore these ideas. We hope that the theories and methods presented in this paper can serve as a springboard for additional study in this field.

Acknowledgments

The Rector of COMSATS University Islamabad in Islamabad, Pakistan, is acknowledged by the writers for offering top-notch research and academic settings.

Conflict of interest

The authors declare that they have no competing interests.