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Post-quantum Hermite–Hadamard type inequalities for interval-valued convex functions

Abstract

In this research, we introduce the notions of \((p,q)\)-derivative and integral for interval-valued functions and discuss their fundamental properties. After that, we prove some new inequalities of Hermite–Hadamard type for interval-valued convex functions employing the newly defined integral and derivative. Moreover, we find the estimates for the newly proved inequalities of Hermite–Hadamard type. It is also shown that the results proved in this study are the generalization of some already proved research in the field of Hermite–Hadamard inequalities.

Introduction

Many studies have recently been carried out in the field of q-analysis, starting with Euler due to a high demand for mathematics that models quantum computing q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions, and other sciences, quantum theory, mechanics, and the theory of relativity [15]. Apparently, Euler was the founder of this branch of mathematics by using the parameter q in Newton’s work of infinite series. Later, Jackson was the first to develop q-calculus known as without limits calculus in a systematic way [2]. In 1908–1909, Jackson defined the general q-integral and q-difference operator [4]. In 1969, Agarwal described the q-fractional derivative for the first time [6]. In 1966–1967 Al-Salam introduced q-analogues of the Riemann–Liouville fractional integral operator and q-fractional integral operator [7]. In 2004, Rajkovic gave a definition of the Riemann-type q-integral which was a generalization of Jackson q-integral. In 2013, Tariboon introduced \({}_{a}D_{q}\)-difference operator [8].

Many integral inequalities well known in classical analysis, such as Hölder’s inequality, Simpson’s inequality, Newton’s inequality, Hermite–Hadamard inequality, Ostrowski inequality, Cauchy–Bunyakovsky–Schwarz, Gruss, Gruss–Cebysev, and other integral inequalities, have been proved and applied for q-calculus using classical convexity. Many mathematicians have done studies in q-calculus analysis, the interested reader can check [925].

Inspired by these ongoing studies, we give the idea about the post-quantum derivative and integral in the setting of interval-valued calculus. We also prove some new inequalities of Hermite–Hadamard type and find their estimates.

Interval calculus

We give notation and preliminary information about the interval analysis in this section. Let the space of all closed intervals of \(\mathbb{R} \) denoted by \(I_{c}\) and K be a bounded element of \(I_{c}\), we have the representation

$$ K= [ \underline{k},\overline{k} ] = \{ t\in \mathbb{R} :\underline{k} \leq t\leq \overline{k} \} , $$

where \(\underline{k},\overline{k}\in \mathbb{R} \) and \(\underline{k}\leq \overline{k}\). The length of the interval \(K= [ \underline{k},\overline{k} ] \) can be stated as \(L ( K ) =\overline{k}-\underline{k}\). The numbers \(\underline{k}\) and are called the left and the right endpoints of interval K, respectively. When \(\overline{k}=\underline{k}\), the interval K is said to be degenerate, and we use the form \(K=k=[k,k]\). Also, we can say that K is positive if \(\underline{k}>0\), or we can say that K is negative if \(\overline{k}<0\). The sets of all closed positive intervals of \(\mathbb{R} \) and closed negative intervals of \(\mathbb{R} \) are denoted by \(I_{c}^{+}\) and \(I_{c}^{-}\), respectively. The Pompeiu–Hausdorff distance between the intervals K and M is defined by

$$ d_{H} ( K,M ) =d_{H} \bigl( [ \underline{k}, \overline{k} ] , [ \underline{m},\overline{m} ] \bigr) =\max \bigl\{ \vert \underline{k}-\underline{m} \vert , \vert \overline{k}-\overline{m} \vert \bigr\} . $$
(2.1)

\(( I_{c},d ) \) is known to be a complete metric space (see [26]).

The absolute value of K, denoted by \(|K|\), is the maximum of the absolute values of its endpoints:

$$ \vert K \vert =\max \bigl\{ \vert \underline{k} \vert , \vert \overline{k} \vert \bigr\} . $$

Now, we mention the definitions of fundamental interval arithmetic operations for the intervals K and M as follows:

$$\begin{aligned}& K+M = [ \underline{k}+\underline{m},\overline{k}+\overline{m} ] , \\& K-M = [ \underline{k}-\overline{m},\overline{k}-\underline{m} ] , \\& K\cdot M = [ \min U,\max U ], \quad \text{where }U= \{ \underline{k} \underline{m},\underline{k} \overline{m,} \overline{k}\underline{m}, \overline{k} \overline{m} \} , \\& K/M = [ \min V,\max V ],\quad \text{where }V= \{ \underline{k}/\underline{m},\underline{k}/\overline{m},\overline{k}/\underline{m},\overline{k}/\overline{m} \} \text{ and }0\notin M. \end{aligned}$$

Scalar multiplication of the interval K is defined by

$$ \mu K=\mu [ \underline{k},\overline{k} ] =\textstyle\begin{cases} [ \mu \underline{k},\mu \overline{k} ] , & \mu >0; \\ \{ 0 \} , & \mu =0; \\ [ \mu \overline{k},\mu \underline{k} ] , & \mu < 0,\end{cases} $$

where \(\mu \in \mathbb{R} \).

The opposite of the interval K is

$$ -K:=(-1)K=[-\overline{k},-\underline{k}], $$

where \(\mu =-1\).

The subtraction is given by

$$ K-M=K+(-M)=[\underline{k}-\overline{m},\overline{k}-\underline{m}]. $$

In general, −K is not additive inverse for K, i.e., \(K-K\neq 0\).

The definitions of operations cause a great many algebraic features which allows \(I_{c}\) to be a quasilinear space (see [27]). These properties can be listed as follows (see [2630]):

  1. (1)

    (Associativity of addition) \((K+M)+N=K+(M+N)\) for all \(K,M,N\in I_{c} \),

  2. (2)

    (Additivity element) \(K+0=0+K=K\) for all \(K\in I_{c}\),

  3. (3)

    (Commutativity of addition) \(K+M=M+K\) for all \(K,M\in I_{c}\),

  4. (4)

    (Cancellation law) \(K+N=M+N\Longrightarrow K=M\) for all \(K,M,N\in I_{c}\),

  5. (5)

    (Associativity of multiplication) \((K\cdot M)\cdot N=K\cdot (M\cdot N)\) for all \(K,M,N\in I_{c}\),

  6. (6)

    (Commutativity of multiplication) \(K\cdot M=M\cdot K\) for all \(K,M\in I_{c}\),

  7. (7)

    (Unity element) \(K\cdot 1=1\cdot K\) for all \(K\in I_{c}\),

  8. (8)

    (Associativity law) \(\lambda (\mu K)= ( \lambda \mu ) K\) for all \(K\in I_{c}\) and all \(\lambda ,\mu \in \mathbb{R} \),

  9. (9)

    (First distributivity law) \(\lambda (K+M)=\lambda K+\lambda M\) for all \(K,M\in I_{c}\) and all \(\lambda \in \mathbb{R} \),

  10. (10)

    (Second distributivity law) \((\lambda +\mu )K=\lambda K+\mu K\) for all \(K\in I_{c}\) and all \(\lambda ,\mu \in \mathbb{R} \).

In addition to all these features, the distributive law is not always true for intervals. As an example, \(K=[1,2]\), \(M=[2,3]\), and \(N=[-2,-1]\).

$$ K\cdot (M+N)=[0,4], $$

whereas

$$ K\cdot M+K\cdot N=[-2,5]. $$

Definition 1

([31])

For the intervals K and M, we state that the \(g\mathcal{H}\)-difference of K and M is the interval T such that

$$ K\ominus _{g}M=T\Leftrightarrow \textstyle\begin{cases} K=M+T, \\ \text{or} \\ T=K+ ( -M ) .\end{cases} $$

It looks beyond dispute that

$$ K\ominus _{g}M=\textstyle\begin{cases} [ \underline{k}-\underline{m},\overline{k}-\overline{m} ] ,& \text{if }L ( K ) \geq L ( M ) , \\ [ \overline{k}-\overline{m},\underline{k}-\underline{m} ] ,& \text{if }L ( K ) < L ( M ) .\end{cases} $$

Particularly, if \(M=m\in \mathbb{R} \) is a constant, we have

$$ K\ominus _{g}M= [ \underline{k}-m,\overline{k}-m ] . $$

Moreover, another set feature is the inclusion that is defined by

$$ K\subseteq M\quad \Longleftrightarrow \quad \underline{k}\leq \underline{m} \quad \text{and}\quad \overline{k}\leq \overline{m}. $$

Throughout this paper, \(0< q<1\) and a function \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\) is called L-increasing (or L-decreasing) if \(L ( f ) : [ a,b ] \rightarrow [ 0, \infty ) \) is increasing (or decreasing) on \([ a,b ] \). Also, \(F= [ \underline{F},\overline{F} ] \) is said to be L-monotone on \([ a,b ] \) if \(L ( f ) \) is monotone on \([ a,b ] \). For condensation, interval-valued quantum calculus and interval-valued post-quantum calculus are denoted by Iq-calculus and \(I ( p,q ) \)-calculus, respectively.

Preliminaries of Iq-calculus and inequalities

In this section, we recollect some formerly regarded concepts about the q-calculus and Iq-calculus. Moreover, here and further we use the following notation (see [5]):

$$ [ n ] _{q}=\frac{1-q^{n}}{1-q}=1+q+q^{2}+ \cdots+q^{n-1},\quad q\in ( 0,1 ) . $$

In [4], Jackson gave the q-Jackson integral from 0 to b for \(0< q<1\) as follows:

$$ \int _{0}^{b}f ( x ) \,d_{q}x= ( 1-q ) b\sum_{n=0}^{\infty }q^{n}f \bigl( bq^{n} \bigr) $$
(3.1)

provided the sum converges absolutely.

Definition 2

([32])

For a function \(f: [ a,b ] \rightarrow \mathbb{R} \), the \(q_{a}\)-derivative of f at \(x\in [ a,b ] \) is characterized by the expression

$$ {}_{a}D_{q}f ( x )= \frac{f ( x ) -f ( qx+ ( 1-q ) a ) }{ ( 1-q ) ( x-a ) }, \quad x\neq a. $$
(3.2)

Moreover, we have \({}_{a}D_{q}f ( a )=\lim_{x\rightarrow a}{}_{a}D_{q}f ( x )\). The function f is said to be q- differentiable on \([ a,b ] \) if \({}_{a}D_{q}f ( x )\) exists for all \(x\in [ a,b ] \). If \(a=0\) in (3.2), then \({}_{0}D_{q}f ( x ) =D_{q}f ( x )\), where \(D_{q}f ( x )\) is familiar q-derivative of f at \(x\in [ 0,b ] \) defined by the expression (see [5])

$$ D_{q}f ( x )= \frac{f ( x ) -f ( qx ) }{ ( 1-q ) x},\quad x\neq 0. $$

Definition 3

([33])

For a function \(f: [ a,b ] \rightarrow \mathbb{R} \), the \(q^{b}\)-derivative of f at \(x\in [ a,b ] \) is characterized by the expression

$$ {}^{b}D_{q}f ( x )= \frac{f ( qx+ ( 1-q ) b ) -f ( x ) }{ ( 1-q ) ( b-x ) },\quad x\neq b. $$

Definition 4

([32])

Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a function. Then the \(q_{a}\)-definite integral on \([ a,b ] \) is defined as

$$\begin{aligned} \int _{a}^{b}f ( x ) \,{}_{a}d_{q}x =& ( 1-q ) ( b-a ) \sum_{n=0}^{\infty }q^{n}f \bigl( q^{n}b+ \bigl( 1-q^{n} \bigr) a \bigr) \\ = &( b-a ) \int _{0}^{1}f \bigl( ( 1-t ) a+tb \bigr) \,d_{q}t. \end{aligned}$$

Definition 5

([33])

Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a function. Then the \(q^{b}\)-definite integral on \([ a,b ] \) is defined as

$$\begin{aligned} \int _{a}^{b}f ( x ) \,{}^{b}d_{q}x =& ( 1-q ) ( b-a ) \sum_{n=0}^{\infty }q^{n}f \bigl( q^{n}a+ \bigl( 1-q^{n} \bigr) b \bigr) \\ = &( b-a ) \int _{0}^{1}f \bigl( ta+ ( 1-t ) b \bigr) \,d_{q}t. \end{aligned}$$

On the other hand, recently, Lou et al. introduced the notions of \(I(q)\)-calculus. They gave the following definitions of \(I(q)\)-derivative and integral, and proved some inequalities of \(I(q)\)-Hermite–Hadamard type for interval-valued convex functions.

Definition 6

([31])

For an interval-valued function \(F= [ \underline{F}, \overline{F} ] : [ a,b ] \rightarrow I_{c}\), the \(Iq_{a}\)-derivative of F at \(x\in [ a,b ] \) is defined by

$$ {}_{a}D_{q}F ( x ) = \frac{F ( x ) \ominus _{g}F ( qx+ ( 1-q ) a ) }{ ( 1-q ) ( x-a ) },\quad x\neq a. $$
(3.3)

Since \(F= [ \underline{F}, \overline{F} ] : [ a,b ] \rightarrow I_{c}\) is a continuous function, we can state

$$ {}_{a}D_{q}F ( a ) =\lim_{x\rightarrow a} {}_{a}D_{q}F ( x ) . $$

The function F is said to be Iq-differentiable on \([ a,b ] \) if \({}_{a}D_{q}F ( x ) \) exist for all \(x\in [ a,b ] \). If we set \(a=0\) in (3.3), then \(_{0}D_{q}F ( a ) =D_{q}F ( a ) \), where \(D_{q}F ( a ) \) is called Iq-Jackson derivative of F at \(x\in [ a,b ] \) defined by the expression

$$ D_{q}F ( x ) = \frac{F ( x ) \ominus _{g}F ( qx ) }{ ( 1-q ) x}. $$

Definition 7

([31])

For an interval-valued function \(F= [ \underline{F}, \overline{F} ] : [ a,b ] \rightarrow I_{c}\), the \(Iq_{a}\)-definite integral is defined by

$$ \int _{a}^{x}F ( s ) \,{}_{a}d_{q}^{I}s= ( 1-q ) ( x-a ) \sum_{n=0}^{\infty }q^{n}F \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) a \bigr) $$
(3.4)

for all \(x\in [ a,b ] \).

Remark 1

If we set \(a=0\) in (3.4), then we have Iq-Jackson integral defined by the following equation:

$$ \int _{0}^{x}F ( s ) \,{}_{0}d_{q}^{I}s= ( 1-q ) x \sum_{n=0}^{\infty }q^{n}F \bigl( q^{n}x \bigr) $$

for all \(x\in [ 0,\infty ) \).

Theorem 1

([31])

Let \(F= [ \underline{F}, \overline{F} ] : [ a,b ] \rightarrow I_{c}\) be \(Iq_{a}\)-differentiable and convex on \([ a,b ] \). Then the \(Iq_{a}\)-Hermite–Hadamard inequality is expressed as follows:

$$ F \biggl( \frac{qa+b}{ [ 2 ] _{q}} \biggr) \supseteq \frac{1}{b-a}\int _{a}^{b}F ( x ) \,{}_{a}d_{q}^{I}x \supseteq \frac{qF ( a ) +F ( b ) }{ [ 2 ] _{q}}. $$

In [34], Alp et al. gave the definition of \(Iq^{b}\)-integral and proved inequalities of Hermite–Hadamard type for interval-valued convex functions by using \(Iq^{b}\)-integral.

Definition 8

For an interval-valued function \(F= [ \underline{F}, \overline{F} ] : [ a,b ] \rightarrow I_{c}\), the \(Iq^{b}\)-definite integral is defined by

$$ \int _{x}^{b}F ( s ) \,{}^{b}d_{q}^{I}s= ( 1-q ) ( b-x ) \sum_{n=0}^{\infty }q^{n}F \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) b \bigr) $$
(3.5)

for all \(x\in [ a,b ] \).

Theorem 2

Let \(F= [ \underline{F}, \overline{F} ] : [ a,b ] \rightarrow I_{c}\) be interval-valued convex on \([ a,b ] \). Then the \(Iq^{b}\)-Hermite–Hadamard inequality is expressed as follows:

$$ F \biggl( \frac{a+qb}{ [ 2 ] _{q}} \biggr) \supseteq \frac{1}{b-a}\int _{a}^{b}F ( x ) \,{}^{b}d_{q}^{I}x \supseteq \frac{F ( a ) +qF ( b ) }{ [ 2 ] _{q}}. $$

\(I ( p,q ) \)-calculus

In this section, the notions and results about the \((p,q)\)-calculus are reviewed, and we are interested in introducing the concepts of \(I ( p,q ) \)-calculus.

The \([ n ] _{p,q}\) is said to be \((p,q)\)-integers and expressed as

$$ [ n ] _{p,q}=\frac{p^{n}-q^{n}}{p-q} $$

with \(0< q< p\leq 1\). The \([ n ] _{p,q}!\) and [ n k ] ! are called \((p,q)\)-factorial and \((p,q)\)-binomial, respectively, and expressed as

[ n ] p , q ! = k = 1 n [ k ] p , q , n 1 , [ 0 ] p , q ! = 1 , [ n k ] ! = [ n ] p , q ! [ n k ] p , q ! [ k ] p , q ! .

Definition 9

([35])

For a function \(f: [ a,b ] \rightarrow \mathbb{R} \), the \(( p,q ) \)-derivative of f at \(x\in [ a,b ] \) is given by

$$ D_{p,q}f ( x ) = \frac{f ( px ) -f ( qx ) }{ ( p-q ) x},\quad x\neq 0 $$
(4.1)

with \(0< q< p\leq 1\).

On the other hand, Tunç and Göv gave the following new definitions of \((p,q)\)-derivative and integrals.

Definition 10

([36])

For a function \(f: [ a,b ] \rightarrow \mathbb{R} \), the \(( p,q ) _{a}\)-derivative of f at \(x\in [ a,b ] \) is given by

$$ {}_{a}D_{p,q}f ( x ) = \frac{f ( px+ ( 1-p ) a ) -f ( qx+ ( 1-q ) a ) }{ ( p-q ) ( x-a ) }, \quad x\neq a, $$
(4.2)

with \(0< q< p\leq 1\).

For \(x=a\), we state \({}_{a}D_{p,q}f ( a ) =\lim_{x\rightarrow a} {}_{a}D_{p,q}f ( x ) \) if it exists and is finite.

Definition 11

([36])

For a function \(f: [ a,b ] \rightarrow \mathbb{R} \), the definite \((p,q)_{a}\)-integral of f on \([ a,b ] \) is stated as

$$ \int _{a}^{x}f ( t )\, {}_{a}d_{p,q}t= ( p-q ) ( x-a ) \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}f \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{p^{n+1}} \biggr) a \biggr) $$
(4.3)

with \(0< q< p\leq 1\).

On the other hand, Ali et al. gave the following new definition of \(( p,q ) \)-derivative and integral, and proved some related inequalities.

Definition 12

([37])

For a continuous function \(f: [ a,b ] \rightarrow \mathbb{R} \), the definite \((p,q)^{b}\)-integral of f on \([ a,b ] \) is stated as

$$ \int _{x}^{b}f ( t ) \,{}^{b}d_{p,q}t= ( p-q ) ( b-x ) \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}f \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{p^{n+1}} \biggr) b \biggr) $$
(4.4)

with \(0< q< p\leq 1\).

Definition 13

([37])

For a continuous function \(f: [ a,b ] \rightarrow \mathbb{R} \), the \(( p,q ) ^{b}\)-derivative of f at \(x\in [ a,b ] \) is given as follows:

$$ {}^{b}D_{p,q}f ( x ) = \frac{f ( qx+ ( 1-q ) b ) -f ( px+ ( 1-p ) b ) }{ ( p-q ) ( b-x ) },\quad x\neq b. $$
(4.5)

For \(x=b\), we state \({}^{b}D_{p,q}f ( b ) =\lim_{x\rightarrow b} {}^{b}D_{p,q}f ( x ) \) if it exists and is finite.

Theorem 3

([37])

Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a convex differentiable function on \([ a,b ] \). Then the following inequalities hold for \(( p,q ) ^{b}\)-integrals:

$$ f \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \leq \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}f ( x ) \,{}^{b}d_{p,q}x \leq \frac{pf ( a ) +qf ( b ) }{ [ 2 ] _{p,q}}, $$
(4.6)

where \(0< q< p\leq 1\).

Theorem 4

([37])

Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a convex differentiable function on \([ a,b ] \). Then the following inequalities hold for \(( p,q ) ^{b}\)-integrals:

$$\begin{aligned} f \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) + \frac{ ( p-q ) ( b-a ) }{ [ 2 ] _{p,q}}f^{\prime } \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) \leq &\frac{1}{p ( b-a ) }\int _{pa+ ( 1-p ) b}^{b}f ( x )\, {}^{b}d_{p,q}x \\ \leq &\frac{pf ( a ) +qf ( b ) }{ [ 2 ] _{p,q}}, \end{aligned}$$
(4.7)

where \(0< q< p\leq 1\).

Theorem 5

([37])

Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a differentiable function on \(( a,b ) \) and \({}^{b}D_{p,q}f\) be continuous and integrable on \([ a,b ] \). If \(\vert {}^{b}D_{p,q}f \vert \) is a convex function over \([ a,b ] \), then we have the following \(( p,q ) \)-midpoint inequality:

$$\begin{aligned}& \biggl\vert \int _{ap+ ( 1-p ) b}^{b}f ( x ) \,{}^{b}d_{p,q}x-f \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert \\& \quad \leq ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q}f ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}f ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}f ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q}f ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr], \end{aligned}$$
(4.8)

where

$$\begin{aligned}& A_{1} ( p,q ) =\frac{qp^{3}}{ ( [ 2 ] _{p,q} ) ^{3} [ 3 ] _{p,q}}, \\& A_{2} ( p,q ) =\frac{q ( p^{3} ( p^{2}+q^{2}-p ) +p^{2} [ 3 ] _{p,q} ) }{ ( [ 2 ] _{p,q} ) ^{4} [ 3 ] _{p,q}}, \\& A_{3} ( p,q ) =\frac{q ( q+2p ) }{ [ 2 ] _{p,q}}- \frac{q^{2} ( q^{2}+3p^{2}+3pq ) }{ ( [ 2 ] _{p,q} ) ^{3} [ 3 ] _{p,q}}, \\& A_{4} ( p,q ) =\frac{q}{ [ 2 ] _{p,q}}- \frac{q^{2} ( q+2p ) }{ ( [ 2 ] _{p,q} ) ^{4}}-A_{3} ( p,q ), \end{aligned}$$

and \(0< q< p\leq 1\).

Theorem 6

([37])

Let \(f: [ a,b ] \rightarrow \mathbb{R} \) be a differentiable function on \(( a,b ) \) and \({}^{b}D_{p,q}f\) be integrable on \([ a,b ] \). If \(\vert {}^{b}D_{p,q}f \vert \) is a convex function over \([ a,b ] \), then we have the following new \(( p,q ) \)-trapezoidal inequality:

$$\begin{aligned}& \biggl\vert \frac{pf ( a ) +qf ( b ) }{ [ 2 ] _{p,q}}-\frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}f ( x ) \,{}^{b}d_{p,q}x \biggr\vert \\& \quad \leq \frac{q ( b-a ) }{ [ 2 ] _{p,q}} \bigl[ \bigl\vert {}^{b}D_{p,q}f ( a ) \bigr\vert A_{5} ( p,q ) + \bigl\vert {}^{b}D_{p,q}f ( b ) \bigr\vert A_{6} ( p,q ) \bigr], \end{aligned}$$
(4.9)

where

$$\begin{aligned}& A_{5} ( p,q ) =\frac{2 ( [ 3 ] _{p,q}- [ 2 ] _{p,q} ) }{ [ 2 ] _{p,q}^{3} [ 3 ] _{p,q}}+\frac{ [ 2 ] _{p,q}^{2}- [ 3 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}}, \\& A_{6} ( p,q ) =\frac{2 ( [ 2 ] _{p,q}-1 ) }{ [ 2 ] _{p,q}^{2}}-A_{5} ( p,q ) . \end{aligned}$$

Now, we are able to introduce the concepts of \(I ( p,q ) ^{b}\)- derivative and integrals.

\(I ( p,q ) ^{b}\)-derivative

Definition 14

For a continuous interval-valued function \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\), the \(I ( p,q ) ^{b}\)-derivative of F at \(x\in [ a,b ] \) is given as follows:

$$ {}^{b}D_{p,q}F ( x ) = \frac{F ( qx+ ( 1-q ) b ) \ominus _{g}F ( px+ ( 1-p ) b ) }{ ( p-q ) ( b-x ) },\quad x\neq b, $$
(4.10)

with \(0< q< p\leq 1\). For \(x=b\), we state \({}^{b}D_{p,q}F ( b ) =\lim_{x\rightarrow b} {}^{b}D_{p,q}F ( x ) \) if it exists and is finite.

Remark 2

If we choose \(p=1\) in (4.10), then we have \(Iq^{b}\)-derivative defined as follows:

$$ {}^{b}D_{q}F ( x ) = \frac{F ( qx+ ( 1-q ) b ) \ominus _{g}F ( x ) }{ ( 1-q ) ( b-x ) },\quad x\neq b. $$

Theorem 7

An interval-valued function \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\) is said to be \(I ( p,q ) ^{b}\)-differentiable at \(x\in [ a,b ] \) if and only if \(\underline{F}\) and are \((p,q)^{b}\)-differentiable at \(x\in [ a,b ] \). Moreover,

$$ {}^{b}D_{p,q}F ( x ) = \bigl[ \min \bigl\{ {}^{b}D_{p,q} \underline{F} ( x ) , {}^{b}D_{p,q}\overline{F} ( x ) \bigr\} ,\max \bigl\{ {}^{b}D_{p,q}\underline{F} ( x ) , {}^{b}D_{p,q} \overline{F} ( x ) \bigr\} \bigr] . $$
(4.11)

Proof

Let F be an \(I ( p,q ) ^{b}\)-differentiable function at \(x\in [ a,b ] \), there exist \(\underline{G}\) and such that \({}^{b}D_{p,q}F ( x ) = [ \underline{G},\overline{G} ] \). From Definition 14, we have that

$$ \underline{G} ( x ) =\min \biggl\{ \frac{\underline{F} ( qx+ ( 1-q ) b ) -\underline{F} ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) }, \frac{\overline{F} ( qx+ ( 1-q ) b ) -\overline{F} ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) } \biggr\} $$

and

$$ \overline{G} ( x ) =\max \biggl\{ \frac{\underline{F} ( qx+ ( 1-q ) b ) -\underline{F} ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) }, \frac{\overline{F} ( qx+ ( 1-q ) b ) -\overline{F} ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) } \biggr\} $$

exist. So, it is clear that the \(\underline{F}\) and are \((p,q)^{b}\)-differentiable at \(x\in [ a,b ] \).

To prove conversely, we suppose that \(\underline{F}\) and are \((p,q)^{b}\)-differentiable at \(x\in [ a,b ] \). Then we have two possibilities \({}^{b}D_{p,q}\underline{F} ( x ) \leq {}^{b}D_{p,q}\overline{F} ( x ) \) or \({}^{b}D_{p,q}\underline{F} ( x ) \geq {}^{b}D_{p,q} \overline{F} ( x ) \) for all \(x\in [ a,b ] \).

If \({}^{b}D_{p,q}\underline{F} ( x ) \leq {}^{b}D_{p,q} \overline{F} ( x ) \), then we have following relation:

$$\begin{aligned}& \bigl[ {}^{b}D_{p,q}\underline{F} ( x ) , {}^{b}D_{p,q} \overline{F} ( x ) \bigr] \\& \quad = \biggl[ \frac{\underline{F} ( qx+ ( 1-q ) b ) -\underline{F} ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) }, \frac{\overline{F} ( qx+ ( 1-q ) b ) -\overline{F} ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) } \biggr] \\& \quad =\frac{F ( qx+ ( 1-q ) b ) \ominus _{g}F ( px+ ( 1-p ) a ) }{ ( p-q ) ( b-x ) } \\& \quad = {}^{b}D_{p,q}F ( x ) . \end{aligned}$$

Thus, \(F ( x ) \) is \(I ( p,q ) ^{b}\)-differentiable at \(x\in [ a,b ] \). Now, if \({}^{b}D_{p,q}\underline{F} ( x ) \geq {}^{b}D_{p,q} \overline{F} ( x ) \), then

$$ {}^{b}D_{p,q}F ( x ) = \bigl[ {}^{b}D_{p,q} \overline{F} ( x ) ,{}^{b}D_{p,q}\underline{F} ( x ) \bigr], $$

and by applying the similar concepts, we can prove \(F ( x ) \) is \(I ( p,q ) ^{b}\)-differentiable at \(x\in [ a,b ] \). □

Theorem 8

Let \(F= [ \underline{F},\overline{F} ] \rightarrow I_{c}\) be an \(I ( p,q ) ^{b}\)-differentiable function on \([ a,b ] \). Then the following equalities hold for all \(x\in [ a,b ] \):

  1. 1.

    \({}^{b}D_{p,q}F ( x ) = [ {}^{b}D_{p,q} \underline{F} ( x ) , {}^{b}D_{p,q}\overline{F} ( x ) ] \), if F is L-increasing;

  2. 2.

    \({}^{b}D_{p,q}F ( x ) = [ {}^{b}D_{p,q} \overline{F} ( x ) , {}^{b}D_{p,q}\underline{F} ( x ) ] \), if F is L-decreasing.

Proof

To prove the first equality, we suppose that F is \(I ( p,q ) ^{b}\)-differentiable and L-decreasing on \([ a,b ] \). So, we have

$$ F \bigl( qx+ ( 1-q ) b \bigr) >F \bigl( px+ ( 1-p ) a \bigr) $$

for any \(x\in [ a,b ] \). Since \(L ( f ) \) is increasing, then we have

$$\begin{aligned}& \bigl[ \overline{F} \bigl( qx+ ( 1-q ) b \bigr) - \underline{F} \bigl( qx+ ( 1-q ) b \bigr) \bigr] \\& \quad - \bigl[ \overline{F} \bigl( px+ ( 1-p ) b \bigr) - \underline{F} \bigl( px+ ( 1-p ) b \bigr) \bigr] >0, \\& \overline{F} \bigl( qx+ ( 1-q ) b \bigr) -\overline{F} \bigl( px+ ( 1-p ) b \bigr) \\& \quad >\underline{F} \bigl( qx+ ( 1-q ) b \bigr) -\underline{F} \bigl( px+ ( 1-p ) b \bigr) . \end{aligned}$$

Therefore,

$$\begin{aligned}& {}^{b}D_{p,q}F ( x ) \\& \quad =\frac{ [ \underline{F} ( qx+ ( 1-q ) b ) ,\overline{F} ( qx+ ( 1-q ) b ) ] \ominus _{g} [ \underline{F} ( px+ ( 1-p ) b ) ,\overline{F} ( px+ ( 1-p ) b ) ] }{ ( p-q ) ( b-x ) } \\& \quad = \biggl[ \frac{\underline{F} ( qx+ ( 1-q ) b ) -\underline{F} ( px+ ( 1-p ) b ) }{ ( p-q ) ( b-x ) }, \frac{\overline{F} ( qx+ ( 1-q ) b ) -\overline{F} ( px+ ( 1-p ) b ) }{ ( p-q ) ( b-x ) } \biggr] \\& \quad = \bigl[ {}^{b}D_{p,q}\underline{F} ( x ) , {}^{b}D_{p,q} \overline{F} ( x ) \bigr] . \end{aligned}$$

With the similar steps, the second equality can be done. □

Theorem 9

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\) be a \(I ( p,q ) ^{b}\)-differentiable function on \([ a,b ] \). Then, for all \(C= [ \underline{C},\overline{C} ] \in I_{c}\) and \(\alpha \in \mathbb{R} \), the functions \(F+C\) and αF are \(I ( p,q ) ^{b}\)-differentiable functions on \([ a,b ] \). Moreover,

$$ {}^{b}D_{p,q} ( F+C ) = {}^{b}D_{p,q}F ( x ) $$

and

$$ {}^{b}D_{p,q}\alpha F ( x ) =\alpha {}^{b}D_{p,q}F ( x ) . $$

Proof

The proof can be easily done using Definition 14, hence we leave the proof for the readers. □

Theorem 10

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\) be an \(I ( p,q ) ^{b}\)-differentiable function on \([ a,b ] \). Then, for all \(C= [ \underline{C},\overline{C} ] \in I_{c}\), if \(L ( F ) -L ( C ) \) has a constant sign on \([ a,b ] \), then \(F\ominus _{g}C\) is an \(I ( p,q ) ^{b}\)-differentiable function and \({}^{b}D_{p,q} ( F\ominus _{g}C ) = {}^{b}D_{p,q}F ( x ) \).

Proof

The proof can be easily done using Definition 14, hence we leave the proof for the readers. □

\(I ( p,q ) ^{b}\)-integral

Definition 15

For a continuous interval-valued function \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\), the definite \(I(p,q)^{b}\)-integral of F on \([ a,b ] \) is stated as

$$ \int _{x}^{b}F ( t ) \,{}^{b}d_{p,q}^{I}t= ( p-q ) ( b-x ) \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}F \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{p^{n+1}} \biggr) b \biggr) $$
(4.12)

with \(0< q< p\leq 1\).

Remark 3

If we set \(p=1\) in (4.12), then we have the definition of \(Iq^{b}\)-integral that we reviewed in the last section.

The following theorem provides us a relation between \(I ( p,q ) ^{b}\)-integral and \(( p,q ) ^{b}\)-integral.

Theorem 11

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\) be a continuous function on \([ a,b ] \), the function F is \(I ( p,q ) ^{b}\)-integrable on \([ a,b ] \) if and only if \(\underline{F}\) and are \(( p,q ) ^{b} \)-integrable functions on \([ a,b ] \). Furthermore,

$$ \int _{a}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x= \biggl[ \int _{a}^{b} \underline{F} ( x ) \,{}^{b}d_{p,q}x, \int _{a}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x \biggr] . $$
(4.13)

Example 1

Let \(F= [ \underline{F},\overline{F} ] : [ 0,1 ] \rightarrow I_{c}\), defined by \(F= [ x^{2},x ] \). For \(0< q< p\leq 1\), we obtain that

$$ \int _{a}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x= \biggl[ \frac{1}{ [ 3 ] _{p,q}},\frac{1}{ [ 2 ] _{p,q}} \biggr] . $$

Theorem 12

Let \(F,G: [ a,b ] \rightarrow I_{c}\) be two continuous and \(I ( p,q ) ^{b}\)-integrable functions on \([ a,b ] \) such that \(F= [ \underline{F},\overline{F} ] \) and \(G= [ \underline{G},\overline{G} ] \). Then, for \(\alpha \in \mathbb{R} \), the following properties hold:

  1. 1.

    \(\int _{a}^{b} [ F ( x ) +G ( x ) ] \,{}^{b}d_{p,q}^{I}x= \int _{a}^{b}F ( x ) {}^{b}d_{p,q}^{I}x+\int _{a}^{b}G ( x ) \,{}^{b}d_{p,q}^{I}x\);

  2. 2.

    \(\int _{a}^{b}\alpha F ( x ) \,{}^{b}d_{p,q}^{I}x=\alpha \int _{a}^{b}F ( x ) \, {}^{b}d_{p,q}^{I}x\);

  3. 3.

    \(\int _{a}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x\ominus _{g}\int _{a}^{b}G ( x ) \,{}^{b}d_{p,q}^{I}x\subseteq \int _{a}^{b}F ( x ) \ominus _{g}G ( x ) \,{}^{b}d_{p,q}^{I}x\).

Moreover, if \(L ( F ) -L ( G ) \) has a constant sign, then we have

$$ \int _{a}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x \ominus _{g} \int _{a}^{b}G ( x ) \,{}^{b}d_{p,q}^{I}x= \int _{a}^{b}F ( x ) \ominus _{g}G ( x ) \,{}^{b}d_{p,q}^{I}x. $$

Theorem 13

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}\) be a continuous function on \([ a,b ] \), if F is an \(I ( p,q ) ^{b}\)-differentiable function on \([ a,b ] \), then \({}^{b}D_{p,q}F\) is \(I ( p,q ) ^{b}\)-integrable. Furthermore, if F is L-increasing on \([ a,b ] \), then the following equality holds for \(c\in [ a,b ] \):

$$ F ( c ) \ominus _{g}F ( x ) = \int _{x}^{c} {}^{b}D_{p,q}F ( s ) \,{}^{b}d_{p,q}^{I}s. $$

Proof

The proof of Theorem 13 can be easily done by using Theorems 15 and 7. □

Hermite–Hadamard inequalities for \(I ( p,q ) ^{b}\)-integral

In this section, we review the concept of interval-valued convex functions and prove inequalities of Hermite–Hadamard type for an interval-valued convex function by using the newly defined \(I ( p,q ) \)-integral.

Definition 16

([31])

A function \(F= [ \underline{F},\overline{F} ] :[a,b]\rightarrow I_{c}^{+}\) is said to be interval-valued convex if, for all \(x,y\in {}[ a,b]\) and \(t\in (0,1)\), we have

$$ tF(x)+(1-t)F(y)\subseteq F\bigl(tx+(1-t)y\bigr). $$

Theorem 14

A function \(F= [ \underline{F},\overline{F} ] :[a,b]\rightarrow I_{c}^{+}\) is said to be interval-valued convex if and only if \(\underline{F}\) is a convex function \([ a,b ] \) and is a concave function on \([ a,b ] \).

Theorem 15

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) be a differentiable interval-valued convex function, then the following inequalities hold for the \(I ( p,q ) ^{b}\)-integral:

$$ F \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \supseteq \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x \supseteq \frac{pF ( a ) +qF ( b ) }{ [ 2 ] _{p,q}}. $$
(5.1)

Proof

Since \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) is an interval-valued convex function, therefore \(\underline{F}\) is a convex function and is a concave function. So, from \(\underline{F}\) and inequality (4.6), we have

$$ \underline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \leq \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b} \underline{F} ( x ) \,{}^{b}d_{p,q}x\leq \frac{p\underline{F} ( a ) +q\underline{F} ( b ) }{ [ 2 ] _{p,q}} , $$
(5.2)

and from the concavity of and (4.6), we have

$$ \frac{p\overline{F} ( a ) +q\overline{F} ( b ) }{ [ 2 ] _{p,q}}\leq \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x\leq \overline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) . $$
(5.3)

From (5.2) and (5.3), we obtain

$$\begin{aligned} \underline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \leq &\frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b} \underline{F} ( x ) \,{}^{b}d_{p,q}x \\ \leq &\frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x\leq \overline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr), \end{aligned}$$

and hence, we have

$$ F \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \supseteq \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x. $$
(5.4)

Also, from (5.2) and (5.3), we obtain

$$\begin{aligned} \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b} \underline{F} ( x ) \,{}^{b}d_{p,q}x \leq &\frac{p\underline{F} ( a ) +q\underline{F} ( b ) }{ [ 2 ] _{p,q}} \\ \leq &\frac{p\overline{F} ( a ) +q\overline{F} ( b ) }{ [ 2 ] _{p,q}}\leq \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x, \end{aligned}$$

and hence, we have

$$ \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x )\, {}^{b}d_{p,q}^{I}x \supseteq \frac{p\underline{F} ( a ) +q\underline{F} ( b ) }{ [ 2 ] _{p,q}}. $$
(5.5)

By combining (5.4) and (5.5), we obtain the required inequality, which accomplishes the proof. □

Theorem 16

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) be a differentiable interval-valued convex function on \([ a,b ] \), then the following inequalities hold for the \(I ( p,q ) ^{b}\)-integral:

$$\begin{aligned} F \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) + \frac{ ( p-q ) ( b-a ) }{ [ 2 ] _{p,q}} F^{\prime } \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) \supseteq & \frac{1}{p ( b-a ) }\int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x \\ \supseteq &\frac{pF ( a ) +qF ( b ) }{ [ 2 ] _{p,q}}. \end{aligned}$$
(5.6)

Proof

Since \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) is an interval-valued convex function, therefore \(\underline{F}\) is a convex function and is a concave function. Because of the convexity of \(\underline{F}\), from inequalities (4.7), we obtain that

$$\begin{aligned} \underline{F} \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) + \frac{ ( p-q ) ( b-a ) }{ [ 2 ] _{p,q}} \underline{F}^{\prime } \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) \leq &\frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b} \underline{F} ( x ) \,{}^{b}d_{p,q}x \\ \leq &\frac{p\underline{F} ( a ) +q\underline{F} ( b ) }{ [ 2 ] _{p,q}}. \end{aligned}$$
(5.7)

Now, using the fact that is a concave function, and from inequality (4.7), we obtain that

$$\begin{aligned} \overline{F} \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) + \frac{ ( p-q ) ( b-a ) }{ [ 2 ] _{p,q}} \overline{F}^{\prime } \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) \geq &\frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b} \overline{F} ( x ) \,{}^{b}d_{p,q}x \\ \geq &\frac{p\overline{F} ( a ) +q\overline{F} ( b ) }{ [ 2 ] _{p,q}}. \end{aligned}$$
(5.8)

The rest of the proof can be done by applying the same lines of the previous theorem and considering inequalities (5.7) and (5.8). Thus, the proof is completed. □

Theorem 17

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) be a differentiable interval-valued convex function on \([ a,b ] \), then the following inequalities hold for the \(I ( p,q ) ^{b}\)-integral:

$$ \max \{ A_{1},A_{2} \} \supseteq \frac{1}{p ( b-a ) }\int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x \supseteq \frac{pF ( a ) +qF ( b ) }{ [ 2 ] _{p,q}} , $$
(5.9)

where

$$ A_{1}=F \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) $$

and

$$ F \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) + \frac{ ( p-q ) ( b-a ) }{ [ 2 ] _{p,q}} F^{\prime } \biggl( \frac{qa+pb}{ [ 2 ] _{p,q}} \biggr) . $$

Proof

From inequalities (5.1) and (5.2), we have the required inequalities (5.9). Thus, the proof is finished. □

Midpoint and trapezoidal type inequalities for \(I ( p,q ) ^{b}\)-integral

In this section, some new inequalities of midpoint and trapezoidal type for interval-valued functions are obtained.

Theorem 18

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) be a \(I ( p,q ) ^{b}\)-differentiable function. If \(\vert {}^{b}D_{p,q}\underline{F} \vert \) and \(\vert {}^{b}D_{p,q}\overline{F} \vert \) are convex functions on \([ a,b ] \), then the following \(I ( p,q ) \) midpoint inequality holds for interval-valued functions:

$$\begin{aligned}& d_{H} \biggl( \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x,F \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr) \\& \quad \leq ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q}F ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}F ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}F ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q}F ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr], \end{aligned}$$
(6.1)

where \(A_{1} ( p,q ) \)\(A_{4} ( p,q ) \) are defined in Theorem 5and \(d_{H}\) is a Pompeiu–Hausdorff distance between the intervals.

Proof

Using the definition of \(d_{H}\) distance between intervals, one can easily obtain that

$$\begin{aligned}& d_{H} \biggl( \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x,F \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr) \\& \quad =d_{H} \biggl( \biggl[ \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\underline{F} ( x ) \,{}^{b}d_{p,q}x, \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x \biggr] , \\& \qquad {} \biggl[ \underline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) , \overline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr] \biggr) \\& \quad =\max \biggl\{ \biggl\vert \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\underline{F} ( x ) \,{}^{b}d_{p,q}x- \underline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert , \\& \qquad {} \biggl\vert \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x- \overline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert \biggr\} . \end{aligned}$$

Now, using the fact that \(\vert {}^{b}D_{p,q}\underline{F} \vert \) is a convex function, and from inequality (4.8), we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\underline{F} ( x ) \,{}^{b}d_{p,q}x-\underline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert \\& \quad \leq ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q} \underline{F} ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}\underline{F} ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}\underline{F} ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q} \underline{F} ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr] . \end{aligned}$$
(6.2)

Similarly, considering that \(\vert {}^{b}D_{p,q}\overline{F} \vert \) is convex on \([ a,b ] \) and using inequality (4.8), we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x-\overline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert \\& \quad \leq ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q} \overline{F} ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}\overline{F} ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}\overline{F} ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q} \overline{F} ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr] . \end{aligned}$$
(6.3)

So, from inequalities (6.2) and (6.3), we have

$$\begin{aligned}& d_{H} \biggl( \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x,F \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr) \\& \quad =\max \biggl\{ \biggl\vert \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\underline{F} ( x ) \,{}^{b}d_{p,q}x- \underline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert , \\& \qquad {} \biggl\vert \frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}\overline{F} ( x ) \,{}^{b}d_{p,q}x- \overline{F} \biggl( \frac{pa+qb}{ [ 2 ] _{p,q}} \biggr) \biggr\vert \biggr\} \\& \quad \leq \max \bigl\{ ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q}\underline{F} ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}\underline{F} ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}\underline{F} ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q} \underline{F} ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr] , \\& \qquad {} ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q} \overline{F} ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}\overline{F} ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}\overline{F} ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q} \overline{F} ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr] \bigr\} \\& \quad = ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{p,q}F ( a ) \bigr\vert A_{1} ( p,q ) + \bigl\vert {}^{b}D_{p,q}F ( b ) \bigr\vert A_{2} ( p,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{p,q}F ( a ) \bigr\vert A_{3} ( p,q ) + \bigl\vert {}^{b}D_{p,q}F ( b ) \bigr\vert A_{4} ( p,q ) \bigr) \bigr] \end{aligned}$$

since

$$\begin{aligned}& \bigl\vert {}^{b}D_{p,q}F ( a ) \bigr\vert =\max \bigl\{ \bigl\vert {}^{b}D_{p,q}\underline{F} ( a ) \bigr\vert , \bigl\vert {}^{b}D_{p,q}\overline{F} ( a ) \bigr\vert \bigr\} , \\& \bigl\vert {}^{b}D_{p,q}F ( b ) \bigr\vert =\max \bigl\{ \bigl\vert {}^{b}D_{p,q}\underline{F} ( b ) \bigr\vert , \bigl\vert {}^{b}D_{p,q}\overline{F} ( b ) \bigr\vert \bigr\} . \end{aligned}$$

Therefore, the proof is completed. □

Corollary 1

If we set \(p=1\) in Theorem 18, then we have the following new q-midpoint inequality for interval-valued functions:

$$\begin{aligned}& d_{H} \biggl( \frac{1}{ ( b-a ) } \int _{a}^{b}F ( x ) \,{}^{b}d_{q}^{I}x,F \biggl( \frac{a+qb}{ [ 2 ] _{q}} \biggr) \biggr) \\& \quad \leq ( b-a ) \bigl[ \bigl( \bigl\vert {}^{b}D_{q}F ( a ) \bigr\vert A_{1} ( 1,q ) + \bigl\vert {}^{b}D_{q}F ( b ) \bigr\vert A_{2} ( 1,q ) \bigr) \\& \qquad {} + \bigl( \bigl\vert {}^{b}D_{q}F ( a ) \bigr\vert A_{3} ( 1,q ) + \bigl\vert {}^{b}D_{q}F ( b ) \bigr\vert A_{4} ( 1,q ) \bigr) \bigr], \end{aligned}$$

where \(\vert {}^{b}D_{q}\underline{F} \vert \) and \(\vert {}^{b}D_{q}\overline{F} \vert \) both are convex functions.

Corollary 2

If we set \(p=1\) and \(q\rightarrow 1^{-}\) in Theorem 18, then we have the following midpoint inequality for interval-valued functions:

$$\begin{aligned}& d_{H} \biggl( \frac{1}{b-a} \int _{a}^{b}F ( x ) \,d^{I}x,F \biggl( \frac{a+b}{2} \biggr) \biggr) \\& \quad \leq ( b-a ) \bigl[ \bigl( \bigl\vert F^{\prime } ( a ) \bigr\vert A_{1} ( 1,1 ) + \bigl\vert F^{ \prime } ( b ) \bigr\vert A_{2} ( 1,1 ) \bigr) \\& \qquad {} + \bigl( \bigl\vert F^{\prime } ( a ) \bigr\vert A_{3} ( 1,1 ) + \bigl\vert F^{\prime } ( b ) \bigr\vert A_{4} ( 1,1 ) \bigr) \bigr] , \end{aligned}$$

where \(\vert \underline{F}^{\prime } ( a ) \vert \) and \(\vert \overline{F}^{\prime } ( a ) \vert \) both are convex functions.

Theorem 19

Let \(F= [ \underline{F},\overline{F} ] : [ a,b ] \rightarrow I_{c}^{+}\) be an \(I ( p,q ) ^{b}\)-differentiable function. If \(\vert {}^{b}D_{p,q}\underline{F} \vert \) and \(\vert {}^{b}D_{p,q}\overline{F} \vert \) are convex functions on \([ a,b ] \), then the following \(I ( p,q ) \) trapezoidal inequality holds for interval-valued functions:

$$\begin{aligned}& d_{H} \biggl( \frac{pF ( a ) +qF ( b ) }{ [ 2 ] _{p,q}},\frac{1}{p ( b-a ) } \int _{pa+ ( 1-p ) b}^{b}F ( x ) \,{}^{b}d_{p,q}^{I}x \biggr) \\& \quad \leq \frac{q ( b-a ) }{ [ 2 ] _{p,q}} \bigl[ \bigl\vert {}^{b}D_{p,q}F ( a ) \bigr\vert A_{5} ( p,q ) + \bigl\vert {}^{b}D_{p,q}F ( b ) \bigr\vert A_{6} ( p,q ) \bigr], \end{aligned}$$
(6.4)

where \(A_{5}\) and \(A_{6}\) are defined in Theorem 6and \(d_{H}\) is the Pompeiu–Hausdorff distance between the intervals.

Proof

From the definition of \(d_{H}\) distance between the intervals and inequality (4.9), and using the strategies followed in the last theorem, one can easily obtain inequality (6.4). □

Corollary 3

If we set \(p=1\) in Theorem 19, then we have the following new q-trapezoidal inequality for interval-valued functions:

$$\begin{aligned}& d_{H} \biggl( \frac{F ( a ) +qF ( b ) }{ [ 2 ] _{q}},\frac{1}{b-a} \int _{a}^{b}F ( x ) \,{}^{b}d_{q}^{I}x \biggr) \\& \quad \leq \frac{q ( b-a ) }{ [ 2 ] _{q}} \bigl[ \bigl\vert {}^{b}D_{q}F ( a ) \bigr\vert A_{5} ( 1,q ) + \bigl\vert {}^{b}D_{q}F ( b ) \bigr\vert A_{6} ( 1,q ) \bigr], \end{aligned}$$

where \(\vert {}^{b}D_{q}\underline{F} \vert \) and \(\vert {}^{b}D_{q}\overline{F} \vert \) both are convex functions.

Corollary 4

If we set \(p=1\) and \(q\rightarrow 1^{-}\) in Theorem 19, then we have the following new trapezoidal inequality for interval-valued functions:

$$\begin{aligned}& d_{H} \biggl( \frac{F ( a ) +F ( b ) }{2}, \frac{1}{b-a}\int _{a}^{b}F ( x ) \,d^{I}x \biggr) \\& \quad \leq \frac{ ( b-a ) }{2} \bigl[ \bigl\vert F^{\prime } ( a ) \bigr\vert A_{5} ( 1,1 ) + \bigl\vert F^{ \prime } ( b ) \bigr\vert A_{6} ( 1,1 ) \bigr], \end{aligned}$$

where \(\vert \underline{F}^{\prime } ( a ) \vert \) and \(\vert \overline{F}^{\prime } ( a ) \vert \) both are convex functions.

Conclusion

In this study, we have introduced the notions of \(( p,q ) \)-derivative and integral for interval-valued functions and discussed their basic properties. We have proved some new Hermite–Hadamard type inequalities for interval-valued convex functions by using newly given concepts of \(( p,q ) \)-derivative and integral. Moreover, we have proved midpoint and trapezoidal estimates for newly established \(( p,q ) \)-Hermite–Hadamard inequalities. It is an interesting and new problem that the upcoming researchers can establish Simpson type inequalities, Newton type inequalities, and Ostrowski type inequalities for interval-valued functions by employing the techniques of this research in their future work.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Funding

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241).

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Ali, M.A., Budak, H., Murtaza, G. et al. Post-quantum Hermite–Hadamard type inequalities for interval-valued convex functions. J Inequal Appl 2021, 84 (2021). https://doi.org/10.1186/s13660-021-02619-6

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Keywords

  • Hermite–Hadamard inequality
  • \((p,q)\)-integral
  • Quantum calculus
  • Interval-valued calculus
  • Interval-valued convex functions