Some trapezoid and midpoint type inequalities via fractional $(p,q)$-calculus

Abstract

Fractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractional q-calculus has been investigated and applied in a variety of research subjects including the fractional q-trapezoid and q-midpoint type inequalities. Fractional $(p,q)$-calculus on finite intervals, particularly the fractional $(p,q)$-integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional $(p,q)$-integral on finite intervals. Then, the obtained results are used to derive some fractional $(p,q)$-trapezoid and $(p,q)$-midpoint type inequalities.

1 Introduction

The ordinary calculus of Newton and Leibniz is well known to be investigated extensively and intensively to produce a large number of related formulas and properties as well as applications in a variety of fields ranging from natural sciences to social sciences. In the early eighteenth century, the well-known mathematician Leonhard Euler (1707–1783) established quantum calculus or q-calculus, which is the study of calculus without limits, in the way of Newton’s work for infinite series. Later, F. H. Jackson initiated a study of q-calculus in a symmetrical manner in 1910 and introduced q-derivative and q-integral in [1], see [2] for more details.

Many physical and mathematical problems have led to the necessity of studying q-calculus; for instance, Fock [3] studied the symmetry of hydrogen atoms using the q-difference equation. In addition, in modern mathematical analysis, q-calculus has lots of applications such as combinatorics, orthogonal polynomials, basic hypergeometric functions, number theory, quantum theory, mechanics, and theory of relativity, see also [4–24] and the references cited therein. The book by Kac and Cheung [25] covers the basic theoretical concepts of q-calculus.

As one of the major driving forces behind the modern approach of real analysis, inequalities have played a vital role in almost all branches of mathematics along with other fields of science. In 2015, Noor et al. [26] established q-analogue of classical integral identity to obtain q-trapezoid type inequalities for convex functions. Moreover, in 2016, Necmettin, Mehmet, and İmdat [27] proved the correctness of left part of q-Hermite–Hadamard and gave some q-midpoint type integral inequalities through q-differentiable convex function and q-differentiable quasi-convex functions. With these results, many researchers have extended some important topics of q-calculus together with applications in many fields, such as q-integral inequalities, see [28–37] for more details.

Since the exploration has been continued to generalize the existing results through creative thoughts and novel techniques of fractional calculus, in 2015, Tariboon, Ntouyas, and Agarwal [38] proposed a new q-shifting operator ${}_{a}\Phi _{q}{(m)}= qm+(1-q)a$ for studying new concepts of fractional q-calculus. In 2016, Sudsutad, Ntouyas, and Tariboon [39] studied some fractional q-integral inequalities. In 2020, Kunt and Aljasem [40] proved Riemann–Liouville fractional q-trapezoid and q-midpoint type inequalities for convex functions. Furthermore, in 2021, Neang et al. [41] introduced fractional $(p,q)$-calculus on finite intervals and proved some well-known integral inequalities.

In 2018, as one of the most attractive areas, Kunt et al. [42] proved $(p,q)$-Hermite–Hadamard inequalities and gave some $(p,q)$-midpoint type integral inequalities via $(p,q)$-differentiable convex and $(p,q)$-differentiable quasi-convex functions. In 2019, Latif et al. [43] proved some $(p,q)$-trapezoid integral inequalities for convex and quasi-convex functions. Based on these results, many authors have generalized and developed $(p,q)$-calculus, which is used efficiently in many fields, and some results on the study of $(p,q)$-calculus can be found in [44–71].

Motivated by some of the above studies and applications, in this paper, we study two identities for continuous functions in the form of fractional $(p,q)$-integral on finite intervals. Then, the obtained results are used to derive some fractional $(p,q)$-trapezoid and $(p,q)$-midpoint type inequalities.

2 Preliminaries

In this section, we recall some well-known facts on fractional $(p,q)$-calculus, which can be found in [10, 11, 38, 53, 55]. Throughout this paper, let $[a,b] \subset \mathbb{R}$ be an interval with $a < b$, and $0< q< p\leq 1$ be constants,

$$\begin{aligned} &[k]_{p,q} = \textstyle\begin{cases} \frac{p^{k}-q^{k}}{p-q}, & k \in \mathbb{N}, \end{cases}\displaystyle \\ &[k]_{p,q}! = \textstyle\begin{cases} [k]_{p,q}[k-1]_{p,q} \cdots [1]_{p,q} = \prod_{i=1}^{k} \frac{p^{i}-q^{i}}{p-q}, & k \in \mathbb{N}, \\ 1,& k=0. \end{cases}\displaystyle \end{aligned}$$

(2.1)

Property 2.1

([38])

Let ${}_{a}\Phi _{q}{(m)}= qm+(1-q)a$. For any $m, n \in \mathbb{R}$ and for all positive integers j, k, we have

• (i) ${{}_{a}\Phi ^{k}_{q}(m)} = {{}_{a}\Phi _{q^{k}}(m)}$;

• (ii) ${{}_{a}\Phi ^{j}_{q}({{}_{a}\Phi ^{k}_{q}(m)})} = {{}_{a}\Phi ^{k}_{q}({{}_{a} \Phi ^{j}_{q}(m)})} = {{}_{a}\Phi ^{j+k}_{q}(m)}$;

• (iii) ${{}_{a}\Phi _{q}(a)}= a$;

• (iv) ${{}_{a}\Phi ^{k}_{q}(m)}-a = q^{k}(m-a)$;

• (v) $m-{{}_{a}\Phi ^{k}_{q}(m)} = (1-q^{k})(m-a)$;

• (vi) ${{}_{a}\Phi ^{k}_{q}(m)}= m{\,{}_{a/m}\Phi ^{k}_{q}(1)}$ for $m \neq 0$;

• (vii) ${{}_{a}\Phi _{q}(m)}- {{}_{a}\Phi ^{k}_{q}(n)} = q (m- {{}_{a} \Phi ^{k-1}_{q}(n)} )$.

Property 2.2

([38])

For any $\gamma , n, m \in \mathbb{R}$ with $n \neq a$ and $k \in \mathbb{N} \cup \{0\}$, we have

• (i) $(n-m)^{(k)}_{a}= (n-a)^{k}{ (\frac{m-a}{n-a};q )}_{k}$;

• (ii) ${(n-m)^{(\gamma )}}_{a}={(n-a)^{\gamma }} \prod_{i=0}^{ \infty }{ \frac{1-{\frac{m-a}{n-a}{q^{i}}}}{1-{\frac{m-a}{n-a}}{q^{\gamma +i}}}}={(n-a)^{ \gamma }} \frac{(\frac{m-a}{n-a};q)_{\infty }}{(\frac{m-a}{n-a}q^{\gamma };q)_{\infty }}$;

• (iii) $(n-{{}_{a}\Phi ^{k}_{q}(n)} )^{\gamma }_{a} = (n-a)^{\gamma }{\frac{(q^{k};q)_{\infty }}{(q^{\gamma +k};q)_{\infty }}}$.

For $m,n \in \mathbb{R}$, the $(p,q)$-analogue of the power function ${}_{a}{(m-n)^{k}_{p,q}}$ with $k \in \mathbb{N}\cup \{0\}$ is defined follows:

$$\begin{aligned}& {}_{a}{(m-n)^{(0)}_{p,q}}: =1,\qquad {}_{a}{(m-n)^{(k)}_{p,q}}:= \prod_{i=0}^{k-1}{ \bigl({}_{a} \Phi ^{i}_{p}{(m)}-{}_{a} \Phi ^{i}_{q}{(n)} \bigr)}, \end{aligned}$$

(2.2)

$$\begin{aligned}& {}_{a}{(m-n)^{(k)}_{p,q}} = {(m-a)^{k}}\prod_{i=0}^{k-1}p^{i} \biggl(1- \biggl(\frac{n-a}{m-a} \biggr){ \biggl(\frac{q}{p} \biggr)^{i}} \biggr). \end{aligned}$$

(2.3)

More generally, if $\alpha \in \mathbb{R}$, then

$$ {}_{a}{(m-n)^{(\alpha )}_{p,q}}= {(m-a)^{\alpha }} \prod_{i=0}^{ \infty } \frac{p^{i}}{p^{\alpha +i}} \frac{1- (\frac{n-a}{m-a} ) (\frac{q}{p} )^{i}}{1- (\frac{n-a}{m-a} ) (\frac{q}{p} )^{\alpha +i}}, $$

(2.4)

or

$$ {}_{a}{(m-n)^{(\alpha )}_{p,q}}= {(m-a)^{\alpha }p^{\binom{\alpha }{2}}} \prod_{i=0}^{\infty } \frac{1- (\frac{n-a}{m-a} ) (\frac{q}{p} )^{i}}{1- (\frac{n-a}{m-a} ) (\frac{q}{p} )^{\alpha +i}}. $$

(2.5)

Property 2.3

([41])

For $\alpha > 0$, the following formulas hold:

• (i) ${}_{a}\Phi ^{k}_{q/p}{(m)} - a = (\frac{q}{p} )^{k}{(m-a)}$;

• (ii) ${}_{a} (m-{}_{a}\Phi ^{k}_{q/p}{(m)} )^{(\alpha )}_{p,q} = {(m-a)^{\alpha }} \prod_{i=0}^{\infty } \frac{p^{i}}{p^{\alpha +i}} \frac{1- (\frac{q}{p} )^{k} (\frac{q}{p} )^{i}}{1- (\frac{q}{p} )^{k} (\frac{q}{p} )^{(\alpha +i)}}= (m-a)^{\alpha } (1- (\frac{q}{p} )^{k} )^{( \alpha )}_{p,q}$.

Definition 2.1

([72])

If $f : [a,b] \rightarrow \mathbb{R}$ is a continuous function, then the $(p,q)$-derivative of f on $[a,\frac{1}{p}(b-a)+a ]$ at x is defined by

$$\begin{aligned}& _{a}D_{p,q} f (x) = \frac{f(px+(1-p)a)-f(qx+(1-q)a)}{(p-q)(x-a)}, \quad x \neq a, \\& {}_{a}D_{p,q} f (a) = \lim_{x \to a}{{}_{a}D_{p,q}f(x)}. \end{aligned}$$

(2.6)

Obviously, a function f is $(p,q)$-differentiable on $[a,\frac{1}{p}(b-a)+a ]$ if $_{a}D_{p,q}f(x)$ exists for all $x \in [a,\frac{1}{p}(b-a)+a ]$. In Definition 2.1, if $a=0$, then $_{0}D_{p,q}f = D_{p,q}f$, where $D_{p,q}f$ is defined by

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

(2.7)

Furthermore, if $p=1$ in (2.7), then it reduces to $D_{q}f$, which is q-derivative of the function f, see [25, 73] for more details.

Definition 2.2

([72])

If $f : [a,b] \to \mathbb{R}$ is a continuous function, then the $(p,q)$-integral is defined by

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

(2.8)

for $x \in [a,\frac{1}{p}(b-a)+a ]$. If $a=0$ and $p=1$ in (2.8), then we have the classical q-integral, see [25] for more details.

Theorem 2.1

([72])

The following formulas hold for $t \in [a,b]$:

• (i) ${{}_{a}D_{p,q}\int _{a}^{t}{f(s)} \,{}_{a} d_{p,q}s} = {f(t)}$;

• (ii) $\int _{a}^{b}\,{}_{a}D_{p,q}{f(s)}\,{}_{a}d_{p,q}s= f(t)-f(a)$;

• (iii) $\int _{c}^{t}\,{}_{a}D_{p,q}{f(s)}\,{}_{a}d_{p,q}s= f(t)-f(c)$ for $c \in (a,t)$.

Theorem 2.2

([72])

If $f, g: [a,b]\to \mathbb{R}$ are continuous functions and $\lambda \in \mathbb{R}$, then the following formulas hold:

• (i) $\int _{a}^{t} [f(s)+g(s) ]\,{}_{a}d_{p,q}s= \int _{a}^{t}{f(s)}\,{}_{a}d_{p,q}s+ \int _{a}^{t}{g(s)}\,{}_{a}d_{p,q}s$;

• (ii) $\int _{a}^{t}{\lambda f(s)}\,{}_{a}d_{p,q}s= \lambda \int _{a}^{t}{f(s)}\,{}_{a}d_{p,q}s$;

• (iii) $\int _{a}^{t}{f(ps+(1-p)a)}\,{}_{a}D_{p,q}{g(s)}\,{}_{a}d_{p,q}s= (fg ){(s)}|^{t}_{a}- \int _{a}^{t}{g(qs+(1-q)a)}\,{}_{a}D_{p,q}{(f(s))}\,{}_{a}d_{p,q}s$.

For $t \in \mathbb{R}\setminus \{0,-1,-2,\dots \}$, the $(p,q)$-gamma function is defined by

$$ \Gamma _{p,q}(t)= \frac{ (p-q )_{p,q}^{(t-1)}}{(p-q)^{t-1}}, $$

(2.9)

and an equivalent definition of (2.9) is given in [56] as

$$ \Gamma _{p,q}{(t)} = p^{\frac{t(t-1)}{2}} \int _{0}^{\infty }x^{t-1}E^{-qx}_{p,q}\, d_{p,q}x, $$

(2.10)

where

$$ E^{-qx}_{p,q} = \sum_{n=0}^{\infty } \frac{q^{\binom{n}{2}}}{[n]_{p,q}} (-qx )^{n}. $$

Obviously, $\Gamma _{p,q}{(t+1)}= [t]_{p,q}\Gamma _{p,q}{(t)}$. For $s, t > 0$, the definition of the $(p,q)$-beta function is defined by

$$ B_{p,q}{(s,t)}= \int _{0}^{1}{u^{s-1}}_{0} \bigl(1-{}_{0} \Phi _{q}{(u)} \bigr)^{(t-1)}_{p,q}\,{}_{0}d_{p,q}u, $$

(2.11)

and (2.11) can also be written as

$$ B_{p,q}{(s,t)} =p^{ (t-1 ) (2s+t-2 )/2} \frac{\Gamma _{p,q}{(s)}\Gamma _{p,q}{(t)}}{\Gamma _{p,q}{(s+t)}}, $$

(2.12)

see [74, 75] for more details.

Definition 2.3

([41])

Let f be a function defined on $[a,b]$, and let $\alpha > 0$. The Riemann–Liouville fractional $(p,q)$-integral is defined by

$$\begin{aligned} & \bigl({}_{a}I^{\alpha }_{p,q}f \bigr) (t) \\ &\quad = \frac{1}{p^{\binom{\alpha }{2}}\Gamma _{p,q}{(\alpha )}} \int _{a}^{t}{ \,{}_{a} \bigl(t-{}_{a}\Phi _{q}(s) \bigr)^{(\alpha -1)}_{p,q}} {f \biggl( \frac{s}{p^{\alpha -1}}+ \biggl(1-\frac{1}{p^{\alpha -1}} \biggr)a \biggr)}\,{}_{a}d_{p,q}s \\ &\quad =\frac{(p-q)(t-a)}{p^{\binom{\alpha }{2}}{}\Gamma _{p,q}{(\alpha )}} \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}{\,{}_{a} \bigl( t-{}_{a}\Phi ^{n+1}_{q/p}{(t)} \bigr)^{( \alpha -1)}_{p,q}}f \biggl(\frac{q^{n}}{p^{\alpha +n}}t+ \biggl(1- \frac{q^{n}}{p^{\alpha +n}} \biggr)a \biggr) \end{aligned}$$

(2.13)

for $t\in [a,p^{\alpha }(b-a)+a ]$.

Theorem 2.3

([41])

If $f:[a,b] \to \mathbb{R}$ is a convex differentiable function and $\alpha > 0$, then we have

$$\begin{aligned} f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr) &\leq \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl({}_{a}I^{\alpha }_{p,q}f(s) \bigr) \bigl(p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \\ &\leq \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}}. \end{aligned}$$

(2.14)

3 Main results

In this section, we give two identities for continuous functions in the form of fractional Riemann–Liouville $(p,q)$-integral type which will be used to prove the fractional Riemann–Liouville $(p,q)$-trapezoid and $(p,q)$-midpoint type inequalities.

Lemma 3.1

Let $f: [a,b] \to \mathbb{R}$ be a continuous function and $\alpha > 0$. If ${}_{a}D_{p,q}f$ is $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$, then the following equality holds:

$$\begin{aligned} &\frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{ \alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \\ &\quad = \frac{(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \bigl( [\alpha +1]_{p,q} \bigl( 1- \Phi _{q}(t) \bigr)^{\alpha }_{p,q}- p^{\alpha } \bigr) \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t. \end{aligned}$$

(3.1)

Proof

By simple computation and using Definition 2.3, we have

$$\begin{aligned} A_{1} &= \frac{b-a}{p^{\binom{\alpha }{2}}} \int _{0}^{1} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{p,q}^{(\alpha )} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &= \frac{b-a}{p^{\binom{\alpha }{2}}} \int _{0}^{1} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{p,q}^{(\alpha )} \frac{f ( (1-pt)a +ptb ) -f ( (1-qt)a +qtb )}{(p-q)(b-a)t} \,{}_{0}d_{p,q}t \\ &= \frac{1}{p^{\binom{\alpha }{2}}(p-q)} \int _{0}^{1} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{p,q}^{(\alpha )} \frac{f ( (1-pt)a +ptb )}{t} \,{}_{0}d_{p,q}t \\ &\quad {} - \frac{1}{p^{\binom{\alpha }{2}}(p-q)} \int _{0}^{1} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{p,q}^{(\alpha )} \frac{f ( (1-qt)a +qtb )}{t} \,{}_{0}d_{p,q}t \\ &=\frac{1}{p^{\binom{\alpha }{2}}} \sum_{n= 0}^{\infty } \frac{q^{n}}{p^{n+1}} \bigl(1- {}_{0}\Phi ^{n+1}_{q/p}(1) \bigr)_{p,q}^{( \alpha )} \frac{f ( (1-{}_{0}\Phi ^{n}_{q/p}(1) )a +{}_{0}\Phi ^{n}_{q/p}(1) b )}{\frac{q^{n}}{p^{n+1}}} \\ & \quad {} -\frac{1}{p^{\binom{\alpha }{2}}}\sum_{n= 0}^{\infty } \frac{q^{n}}{p^{n+1}} \bigl(1- {}_{0}\Phi ^{n+1}_{q/p}(1) \bigr)_{p,q}^{( \alpha )} \frac{f ( (1-{}_{0}\Phi ^{n+1}_{q/p}(1) )a +{}_{0}\Phi ^{n+1}_{q/p}(1) b )}{\frac{q^{n}}{p^{n+1}}} \\ &= \sum_{n=0}^{\infty } \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n+1}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &\quad {} - \sum_{n=0}^{\infty } \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n+1}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n+1} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n+1}b \biggr)} \\ &= \sum_{n=0}^{\infty } \biggl( 1- \biggl( \frac{q}{p} \biggr)^{\alpha +n} \biggr) \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &\quad {} - \sum_{n=0}^{\infty } \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n+1} \biggr) \frac{ ( (\frac{q}{p} )^{n+2}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n+1}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n+1} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n+1}b \biggr)} \\ &=\sum_{n=0}^{\infty } \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &\quad {} -\sum_{n=0}^{\infty } \frac{ ( (\frac{q}{p} )^{n+2}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n+1}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n+1} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n+1}b \biggr)} \\ &\quad {} - \Biggl[\sum_{n=0}^{\infty } \biggl(\frac{q}{p} \biggr)^{ \alpha +n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &\quad {} - \sum_{n=0}^{\infty } \biggl( \frac{q}{p} \biggr)^{n+1} \frac{ ( (\frac{q}{p} )^{n+2}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n+1}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n+1} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n+1}b \biggr)} \Biggr] \\ &= \frac{ ( (\frac{q}{p} )^{1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha }; \frac{q}{p} )_{\infty }}f{(b)} -f(a) - \Biggl[\sum_{n=0}^{\infty } \biggl(\frac{q}{p} \biggr)^{\alpha +n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &\quad {} - \sum_{n=1}^{\infty } \biggl( \frac{q}{p} \biggr)^{n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \Biggr] \\ &= \frac{ ( (\frac{q}{p} )^{1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha }; \frac{q}{p} )_{\infty }}f{(b)} -f(a) - \Biggl[\sum_{n=0}^{\infty } \biggl(\frac{q}{p} \biggr)^{\alpha +n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ & \quad {} - \sum_{n=0}^{\infty } \biggl( \frac{q}{p} \biggr)^{n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n } \biggr)a + \biggl( \frac{q}{p} \biggr)^{n }b \biggr)} + \frac{ ( (\frac{q}{p} )^{1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha }; \frac{q}{p} )_{\infty }}f{(b)} \Biggr] \\ &= -f(a) + \biggl(1- \biggl( \frac{q}{p} \biggr)^{\alpha } \biggr)\sum_{n=0}^{ \infty } \biggl( \frac{q}{p} \biggr)^{n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &= -f(a) + \frac{[\alpha ]_{p,q}(p-q)}{p^{\alpha }}\sum_{n=0}^{ \infty } \biggl(\frac{q}{p} \biggr)^{n} \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{\alpha +n}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &= -f(a) + \frac{[\alpha ]_{p,q}\Gamma _{p,q}(\alpha )}{p^{\alpha ^{2}}(b-a)^{\alpha }} \frac{(p-q)p^{\alpha }(b-a)}{p^{\binom{\alpha }{2}}\Gamma _{p,q}(\alpha )} \sum _{n=0}^{\infty }\frac{q^{n}}{p^{n+1}} p^{\alpha (\alpha -1)}(b-a)^{ \alpha -1}p^{\binom{\alpha -1}{2}} \\ &\quad {} \times \frac{ ( (\frac{q}{p} )^{n+1}; \frac{q}{p} )_{\infty }}{ ( (\frac{q}{p} )^{(\alpha -1)+(n+1)}; \frac{q}{p} )_{\infty }}f{ \biggl( \biggl( 1- \biggl( \frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ &= -f(a) + \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \biggl[ \frac{1}{p^{\binom{\alpha }{2}}\Gamma _{p,q}(\alpha )} \int _{a}^{{}_{a} \Phi _{p^{\alpha }}(b)} \,{}_{a} \bigl({}_{a}\Phi _{p^{\alpha }}(b)-{}_{a} \Phi _{q}(t) \bigr)_{p,q}^{(\alpha -1)} \\ & \quad {} \times f \biggl( \frac{t}{p^{\alpha -1} }+ \biggl(1- \frac{1}{p^{\alpha -1}} \biggr)a \biggr)\,{}_{a}d_{p,q}t \biggr] \\ &= -f(a) + \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}} (b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{ \alpha } \bigr)a \bigr), \end{aligned}$$

(3.2)

and

$$\begin{aligned} A_{2}&= \frac{p^{\alpha }(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &=\frac{p^{\alpha }(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \frac{f ( (1-pt)a +ptb ) -f ( (1-qt)a +qtb )}{(p-q)(b-a)t} \,{}_{0}d_{p,q}t \\ &= \biggl[\frac{p^{\alpha }}{(p-q)[\alpha +1]_{p,q}} \int _{0}^{1} \frac{f ( (1-pt)a +ptb )}{t} \,{}_{0}d_{p,q}t \\ & \quad {} -\frac{p^{\alpha }}{(p-q)[\alpha +1]_{p,q}} \int _{0}^{1} \frac{f ( (1-qt)a +qtb )}{t} \,{}_{0}d_{p,q}t \biggr] \\ &= \frac{p^{\alpha }}{[\alpha +1]_{p,q}} \Biggl[\sum_{n=0}^{\infty } f{ \biggl( \biggl( 1- \biggl(\frac{q}{p} \biggr)^{n} \biggr)a + \biggl( \frac{q}{p} \biggr)^{n}b \biggr)} \\ & \quad {} - \sum_{n=0}^{\infty }f{ \biggl( \biggl( 1- \biggl( \frac{q}{p} \biggr)^{n+1 } \biggr)a + \biggl(\frac{q}{p} \biggr)^{n+1}b \biggr)} \Biggr] \\ &=\frac{p^{\alpha }f(b)-p^{\alpha }f(a) }{[\alpha +1]_{p,q}}. \end{aligned}$$

(3.3)

From (3.2) and (3.3), we obtain

$$\begin{aligned} &\frac{(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \biggl( \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}} } \bigl( 1-{}_{0} \Phi _{q}(s) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr) \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &\quad = A_{1}-A_{2} \\ &\quad =\frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{ \alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}}. \end{aligned}$$

(3.4)

Thus the proof is completed. □

Remark 3.1

If $\alpha =1$, then (3.1) reduces to Lemma 3.2 in [43] as

$$\begin{aligned} &\frac{1}{p(b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x- \frac{pf(a)+qf(a)}{p+q} \\ &\quad = \frac{q(b-a)}{p+q} \int _{0}^{1} \bigl( 1- (p+q)t \bigr) \,{}_{a}D_{p,q}f \bigl( tb + (1-t)a \bigr) \,{}_{a}d_{p,q}t. \end{aligned}$$

(3.5)

If $p=1$, then (3.1) reduces to Lemma 5.2 in [40] as

$$\begin{aligned} &\frac{\Gamma _{q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b)- \frac{ ( [\alpha +1]_{ q} -1 )f(a) + f(b)}{[\alpha +1]_{ q}} \\ &\quad = \frac{(b-a)}{[\alpha +1]_{ q}} \int _{0}^{1} \bigl( [\alpha +1]_{ q} \bigl( 1- \Phi _{q}(t) \bigr)^{(\alpha )}_{ q}- 1 \bigr) \,{}_{a}D_{ q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{ q}t. \end{aligned}$$

(3.6)

Moreover, if $q \to 1$ and $\alpha =1$, then (3.6) reduces to

$$ \frac{f(a) +f(b)}{2} - \frac{1}{b-a} \int _{a}^{b} f(x)\,dx = \frac{b-a}{2} \int _{0}^{1} (1-2t)f' \bigl(ta +(1-t)b \bigr) \,dt, $$

(3.7)

which can be found in [76].

Theorem 3.1

Let $f: [a,b] \to \mathbb{R}$ be a continuous function, $\alpha > 0$, and ${}_{a}D_{p,q}f$ be $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$. If $\vert {}_{a}D_{p,q}f \vert$ is convex on

$$ \biggl(a,\frac{1}{p}(b-a)+a \biggr), $$

then the following Riemann–Liouville fractional $(p,q)$-trapezoid type inequality holds:

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \bigl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert B_{1} + \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert B_{2} \bigr), \end{aligned}$$

(3.8)

where

$$ B_{1}= \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert (1-t ) \,{}_{0}d_{p,q}t $$

and

$$ B_{2}= \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert t \,{}_{0}d_{p,q}t. $$

Proof

Using Lemma 3.1 and the convexity of $\vert {}_{a}D_{p,q}f \vert$, we have

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2} }(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{ \alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \\ &\qquad {} \times \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert (1-t) + \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert t \bigr] \,{}_{0}d_{p,q}t \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert (1-t) \biggr] \,{}_{0}d_{p,q}t \\ & \qquad {} +\frac{(b-a)}{[\alpha +1]_{p,q}} \biggl[ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert t \biggr] \,{}_{0}d_{p,q}t. \end{aligned}$$

This completes the proof. □

Remark 3.2

If $p=1$, then (3.8) reduces to

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{ q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b)- \frac{ ( [\alpha +1]_{ q} - 1 )f(a) + f(b)}{[\alpha +1]_{ q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{ q}} \bigl( \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert \delta _{1} + \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert \delta _{2} \bigr), \end{aligned}$$

(3.9)

where

$$ \delta _{1}= \int _{0}^{1} \bigl\vert [\alpha +1]_{ q} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)_{ q}^{(\alpha )} - 1 \bigr\vert (1-t ) \,{}_{0}d_{ q}t $$

and

$$ \delta _{2}= \int _{0}^{1} \bigl\vert [\alpha +1]_{ q} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)_{ q}^{(\alpha )}- 1 \bigr\vert t \,{}_{0}d_{ q}t, $$

which appeared in [40].

Theorem 3.2

Let $f: [a,b] \to \mathbb{R}$ be a continuous function, $\alpha > 0$, and ${}_{a}D_{p,q}f$ be $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$. If $\vert {}_{a}D_{p,q}f \vert ^{r}$ is convex on $(a,\frac{1}{p}(b-a)+a )$ for $r\geq 0$, then the following Riemann–Liouville fractional $(p,q)$-trapezoid type inequality holds:

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}}B^{1-1/r}_{3} \bigl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r}B_{1} + \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r}B_{2} \bigr)^{1/r}, \end{aligned}$$

(3.10)

where $B_{1}$ and $B_{2}$ are given in Theorem 3.1and

$$ B_{3}= \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \,{}_{0}d_{p,q}t. $$

Proof

Using Lemma 3.1, the convexity of $\vert {}_{a}D_{p,q}f \vert ^{r}$, and the power mean inequality, we have

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {} \times \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert ^{r} \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {} \times \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0}\Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r}(1-t) \\ &\qquad {}+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} t \bigr] \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ & \qquad {} \times \biggl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert (1-t) \,{}_{0}d_{p,q}t \\ &\qquad {} + \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert (1-t) \,{}_{0}d_{p,q}t \biggr]^{1/r}. \end{aligned}$$

Therefore, the proof is completed. □

Remark 3.3

If $\alpha =1$, then (3.10) reduces to

$$\begin{aligned} & \biggl\vert \frac{1}{p(b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x- \frac{pf(a)+qf(a)}{p+q} \biggr\vert \\ &\quad = \frac{q(b-a)}{p+q} \biggl[ \frac{2(p+q-1)}{(p+q)^{2}} \biggr]^{1-1/r} \bigl[\lambda _{1}(p,q) \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} + \lambda _{2}(p,q) \bigl\vert \,{}_{a}D_{p,q}f(a) \bigr\vert ^{r} \bigr]^{1/r}, \end{aligned}$$

(3.11)

where

$$ \lambda _{1}(p,q)= \frac{q [(p^{3}-2+2p)+ (2p^{2}+2)q + pq^{2} pq^{2} ] + 2p^{2}-2p}{(p+q)^{3}(p^{2}+pq+q^{2})} $$

and

$$\begin{aligned} \lambda _{2}(p,q)&= \frac{1}{(p+q)^{3}(p^{2}+pq+q^{2})} \bigl\{ q \bigl[ \bigl(5p^{3}-4p^{2}-2p+2 \bigr) + \bigl(6p^{2}-4p-2 \bigr)q \\ &\quad {} + (5p-2)q^{2}+2q^{3} \bigr]+ \bigl(2p^{4}-2p^{3}-2p^{3}-2p^{2}+2p \bigr) \bigr\} , \end{aligned}$$

which appeared in [43].

Moreover, if $p=1$, then (3.10) reduces to

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{ q}(\alpha +1)}{(b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b)- \frac{ ( [\alpha +1]_{ q} - 1 )f(a) + f(b)}{[\alpha +1]_{ q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{ q}}M^{1-1/r}_{3} \bigl( \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert ^{r}M_{1} + \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert ^{r}M_{2} \bigr)^{1/r}, \end{aligned}$$

(3.12)

where $\delta _{1}$ and $\delta _{2}$ are given in Remark 3.2 and

$$ \delta _{3}= \int _{0}^{1} \bigl\vert [\alpha +1]_{ q} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)_{ q}^{(\alpha )} - 1 \bigr\vert \,{}_{0}d_{ q}t, $$

which appeared in [40].

Theorem 3.3

Let $f: [a,b] \to \mathbb{R}$ be a continuous function, $\alpha > 0$ and ${}_{a}D_{p,q}f$ be $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$. If $\vert {}_{a}D_{p,q}f \vert ^{r}$ is convex on $[a,\frac{1}{p}(b-a)+a ]$ for $r > 1$ and $1/r +1/p = 1$, then the following Riemann–Liouville fractional $(p,q)$-trapezoid type inequality holds:

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}}B^{1/s}_{4} \biggl( \frac{(p+q-1) \vert {}_{a}D_{p,q}f(a) \vert ^{r}+ \vert {}_{a}D_{p,q}f(b) \vert ^{r}}{p+q} \biggr)^{1/r}, \end{aligned}$$

(3.13)

where

$$ B_{4}= \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0}\Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert ^{s} \,{}_{0}d_{p,q}t. $$

Proof

Using Lemma 3.1, the convexity of $\vert {}_{a}D_{p,q}f \vert ^{r}$, and Hölder’s inequality, we have

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr)- \frac{ ( [\alpha +1]_{p,q} - p^{\alpha } )f(a) +p^{\alpha }f(b)}{[\alpha +1]_{p,q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert ^{s} \,{}_{0}d_{p,q}t \biggr)^{1/s} \\ &\qquad {} \times \biggl( \int _{0}^{1} \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert ^{r} \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert ^{s} \,{}_{0}d_{p,q}t \biggr)^{1/s} \\ &\qquad {} \times \biggl( \int _{0}^{1} \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r}(1-t) + \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} t \bigr] \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{p,q}} \biggl( \int _{0}^{1} \biggl\vert \frac{[\alpha +1]_{p,q}}{p^{\binom{\alpha }{2}}} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)^{(\alpha )}_{p,q}- p^{\alpha } \biggr\vert ^{s} \,{}_{0}d_{p,q}t \biggr)^{1/s} \\ &\qquad {} \times \biggl( \frac{(p+q-1) \vert {}_{a}D_{p,q}f(a) \vert ^{r}+ \vert {}_{a}D_{p,q}f(b) \vert ^{r} }{p+q} \biggr)^{1/r}. \end{aligned}$$

This completes the proof. □

Remark 3.4

If $\alpha =1$, then (3.13) reduces to

$$\begin{aligned} & \biggl\vert \frac{1}{p(b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x- \frac{pf(a)+qf(a)}{p+q} \biggr\vert \\ &\quad =\frac{q(b-a)}{p+q} [ \lambda _{3} ]^{1/s} \biggl( \frac{ \vert {}_{a}D_{p,q}f(b) \vert ^{r}+ (p+q-1) \vert {}_{a}D_{p,q}f(a) \vert ^{r}}{p+q} \biggr)^{1/r}, \end{aligned}$$

(3.14)

where

$$ \lambda _{3} = \int _{0}^{1} \bigl\vert 1-(p+q)t \bigr\vert ^{s} \,{}_{0}d_{p,q}t, $$

which appeared in [43].

Moreover, if $p=1$, then (3.13) reduces to

$$\begin{aligned} & \biggl\vert \frac{\Gamma _{q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b)- \frac{ ( [\alpha +1]_{ q} - )f(a) + f(b)}{[\alpha +1]_{ q}} \biggr\vert \\ &\quad \leq \frac{(b-a)}{[\alpha +1]_{ q}}\delta ^{1/s}_{4} \biggl( \frac{q \vert {}_{a}D_{ q}f(a) \vert ^{r}+ \vert {}_{a}D_{ q}f(b) \vert ^{r}}{1+q} \biggr)^{1/r}, \end{aligned}$$

where

$$ \delta _{4}= \int _{0}^{1} \bigl\vert [\alpha +1]_{ q} \bigl( 1- {}_{0} \Phi _{q}(t) \bigr)_{ q}^{(\alpha )} - 1 \bigr\vert ^{s} \,{}_{0}d_{q}t, $$

(3.15)

which appeared in [40].

Now we will prove the following lemma to obtain the Riemann–Liouville fractional $(p,q)$-midpoint type inequalities.

Lemma 3.2

Let $f: [a,b] \to \mathbb{R}$ be a continuous function and $\alpha > 0$. If ${}_{a}D_{p,q}f$ is $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$, then the following equality holds:

$$\begin{aligned} &f \biggl( \frac{ ([\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \\ &\quad = (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl(1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr) \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \biggr]. \end{aligned}$$

(3.16)

Proof

By direct computation and using Definitions 2.1 and 2.2, we have

$$\begin{aligned} A_{3}& = \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ & = \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \frac{f ( (1-pt)a +ptb ) -f ( (1-qt)a +qtb )}{(p-q)(b-a)t} \,{}_{0}d_{p,q}t \\ &= \frac{1}{(p-q)(b-a)} \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \frac{f ( (1-pt)a +ptb )}{t} \,{}_{0}d_{p,q}t \\ &\quad {} - \frac{1}{(p-q)(b-a)} \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \frac{f ( (1-qt)a +qtb )}{t} \,{}_{0}d_{p,q}t \\ & = \frac{p^{\alpha }}{(b-a)[\alpha +1]_{p,q}}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}} \frac{f ( (1-\frac{q^{n}p^{\alpha }}{p^{n}[\alpha +1]_{p,q}} )a +\frac{q^{n}p^{\alpha }}{p^{n}[\alpha +1]_{p,q}}b )}{\frac{q^{n}p^{\alpha }}{p^{n+1}[\alpha +1]_{p,q}}} \\ & \quad {} -\frac{p^{\alpha }}{(b-a)[\alpha +1]_{p,q}}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}} \frac{f ( (1-\frac{q^{n+1}p^{\alpha }}{p^{n+1}[\alpha +1]_{p,q}} )a +\frac{q^{n+1}p^{\alpha }}{p^{n+1}[\alpha +1]_{p,q}}b )}{\frac{q^{n}p^{\alpha }}{p^{n+1}[\alpha +1]_{p,q}}} \\ &= \frac{1}{(b-a)} \Biggl[ \sum_{n=0}^{\infty } f \biggl( \biggl(1- \frac{q^{n}p^{\alpha }}{p^{n}[\alpha +1]_{p,q}} \biggr)a + \frac{q^{n}p^{\alpha }}{p^{n}[\alpha +1]_{p,q}}b \biggr) \\ &\quad {} -\sum_{n=0}^{\infty } f \biggl( \biggl(1- \frac{q^{n+1}p^{\alpha }}{p^{n+1}[\alpha +1]_{p,q}} \biggr)a + \frac{q^{n+1}p^{\alpha }}{p^{n+1}[\alpha +1]_{p,q}}b \biggr) \Biggr] \\ &= \frac{1}{(b-a)} \biggl[ f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)-f(a) \biggr]. \end{aligned}$$

(3.17)

On the other hand, in Lemma 3.1, the following integral was given:

$$\begin{aligned} A_{1}& = \frac{b-a}{p^{\binom{\alpha }{2}}} \int _{0}^{1} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{p,q}^{(\alpha )} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ & =-f(a) + \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr). \end{aligned}$$

(3.18)

Consequently, from (3.17) and (3.18), we have

$$\begin{aligned} &A_{3} +A_{1} \\ &\quad = (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl(1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr) \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \biggr] \\ &\quad = {(b-a)} \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &\qquad {} - \int _{0}^{1} \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \biggr] \\ &\quad = f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( \,{}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr). \end{aligned}$$

Therefore, the proof is completed. □

Remark 3.5

If $\alpha =1$, then (3.16) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ qa+pb}{p+q} \biggr)- \frac{ 1}{p (b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x \biggr\vert \\ &\quad = q(b-a) \biggl[ \int _{0}^{\frac{p }{ p+q}}t \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p }{p+q}}^{1} \biggl( t-\frac{1}{q} \biggr) \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{p,q}t \biggr], \end{aligned}$$

(3.19)

which appeared in [42].

Moreover, if $p=1$, then (3.16) reduces to

$$\begin{aligned} &f \biggl( \frac{ ( [\alpha +1]_{ q}- 1 )a+ b}{[\alpha +1]_{ q}} \biggr)- \frac{\Gamma _{ q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( \,{}_{a}I^{ \alpha }_{ q}f \bigr) (b) \\ &\quad = (b-a) \biggl[ \int _{0}^{\frac{ 1}{[\alpha +1]_{ q}}} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \,{}_{a}D_{ q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{ q}t \\ &\qquad {}- \int _{\frac{ 1}{[\alpha +1]_{ q}}}^{1} \bigl(1- {}_{0} \Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \,{}_{a}D_{ q}f \bigl((1-t)a +tb \bigr) \,{}_{0}d_{ q}t, \biggr] \end{aligned}$$

(3.20)

which appeared in [40].

Theorem 3.4

Let $f: [a,b] \to \mathbb{R}$ be a continuous function, $\alpha > 0$, and ${}_{a}D_{p,q}f$ be $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$. If $\vert {}_{a}D_{p,q}f \vert$ is convex on $(a,\frac{1}{p}(b-a)+a )$, then the following Riemann–Liouville fractional $(p,q)$-midpoint type inequality holds:

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \biggr\vert \\ & \quad \leq (b-a) \bigl[ B_{5} \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert + B_{6} \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert + B_{7} \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert + B_{8} \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert \bigr], \end{aligned}$$

(3.21)

where

$$\begin{aligned}& B_{5} = \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert (1-t) \,{}_{0}d_{p,q}t \biggr], \\& B_{6} = \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert t \,{}_{0}d_{p,q}t \biggr], \\& B_{7} = \biggl[ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert (1-t) \,{}_{0}d_{p,q}t \biggr], \\& B_{8} = \biggl[ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert t \,{}_{0}d_{p,q}t \biggr]. \end{aligned}$$

Proof

Using Lemma 3.2 and the convexity of $\vert {}_{a}D_{p,q}f \vert$, we have

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \biggr\vert \\ &\quad \leq (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \biggr] \\ &\quad \leq (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert (1-t)+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert t \bigr] \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert (1-t)+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert t \bigr] \,{}_{0}d_{p,q}t \biggr] \\ &\quad \leq (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert (1-t) \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert t \,{}_{0}d_{p,q}t \biggr] \\ &\qquad {} + (b-a) \biggl[ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert (1-t) \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert t \,{}_{0}d_{p,q}t \biggr]. \end{aligned}$$

This completes the proof. □

Remark 3.6

If $\alpha =1$, then (3.21) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ qa+pb}{p+q} \biggr)- \frac{ 1}{p (b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x \biggr\vert \\ &\quad \leq q(b-a) \bigl[ \lambda _{4}(p,q) \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert + \lambda _{5}(p,q) \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert \\ &\qquad {}+\lambda _{6}(p,q) \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert + \lambda _{7}(p,q) \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert \bigr], \end{aligned}$$

(3.22)

where

$$\begin{aligned}& \lambda _{4}(p,q) = \frac{ p^{3}}{(p+q)^{3}(p^{2}+pq+q^{2})}, \qquad \lambda _{5}(p,q) = \frac{p^{2}(p^{2}+pq+q^{2})-p^{3}}{(p+q)^{3}(p^{2}+pq+q^{2})}, \\& \lambda _{6}(p,q) = \frac{2p^{3}}{(p+q)^{3}(p^{2}+pq+q^{2})}, \qquad \lambda _{7}(p,q) = \frac{ p^{4}+p^{3}q+p^{2}q^{2}-2p^{3}}{(p+q)^{3}(p^{2}+pq+q^{2})}, \end{aligned}$$

which appeared in [42].

Moreover, if $p=1$, then (3.21) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{q}- 1 )a+ b}{[\alpha +1]_{ q}} \biggr)- \frac{\Gamma _{ q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b) \biggr\vert \\ &\quad \leq (b-a) \bigl[ \delta _{5} \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert + \delta _{6} \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert +\delta _{7} \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert + \delta _{8} \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert \bigr], \end{aligned}$$

(3.23)

where

$$\begin{aligned}& \delta _{5} = \biggl[ \int _{0}^{\frac{ 1}{[\alpha +1]_{q}}} \bigl\vert 1- \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert (1-t) \,{}_{0}d_{ q}t \biggr], \\& \delta _{6} = \biggl[ \int _{0}^{\frac{1}{[\alpha +1]_{ q}}} \bigl\vert 1- \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{q}^{(\alpha )} \bigr\vert t \,{}_{0}d_{ q}t \biggr], \\& \delta _{7} = \biggl[ \int _{\frac{1}{[\alpha +1]_{q}}}^{1} \bigl\vert - \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert (1-t) \,{}_{0}d_{ q}t \biggr], \\& \delta _{8} = \biggl[ \int _{\frac{1}{[\alpha +1]_{ q}}}^{1} \bigl\vert - \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert t \,{}_{0}d_{ q}t \biggr], \end{aligned}$$

which appeared in [40].

Theorem 3.5

Let $f: [a,b] \to \mathbb{R}$ be a continuous function, $\alpha > 0$ and ${}_{a}D_{p,q}f$ be $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$. If $\vert {}_{a}D_{p,q}f \vert ^{r}$ is convex on $(a,\frac{1}{p}(b-a)+a )$ for $r\geq 0$, then the following Riemann–Liouville fractional $(p,q)$-midpoint type inequality holds:

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \biggr\vert \\ &\quad \leq (b-a) \bigl[ B^{1-1/r}_{9} \bigl(B_{5} \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} + B_{6} \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \bigr)^{1/r} \\ &\qquad {}+B^{1-1/r}_{10} \bigl(B_{7} \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} + B_{8} \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \bigr)^{1/r} \bigr], \end{aligned}$$

(3.24)

where $B_{5}$, $B_{6}$, $B_{7}$, and $B_{8}$ are given in Theorem 3.4and

$$ B_{9} = \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t $$

and

$$ B_{10} = \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t. $$

Proof

Using Lemma 3.2, the power mean inequality and the convexity of $\vert {}_{a}D_{p,q}f \vert ^{r}$, we have

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \biggr\vert \\ &\quad \leq (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \biggr] \\ &\quad \leq (b-a) \biggl[ \biggl( \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {}\times \biggl( \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert \,{}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert ^{r} \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\qquad {}+ \biggl( \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {}\times \biggl( \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert ^{r} \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr] \\ &\quad \leq (b-a) \biggl[ \biggl( \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {}\times \biggl( \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r}(1-t)+ \bigl\vert \,{}_{a}D_{p,q}f(b) \bigr\vert ^{r}t \bigr] \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\qquad {}+ \biggl( \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {}\times \biggl( \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl[ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r}(1-t)+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r}t \bigr] \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr] \\ &\quad \leq (b-a) \biggl[ \biggl( \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {}\times \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert (1-t) \,{}_{0}d_{p,q}t \\ &\qquad {}+\bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert t \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr] \\ &\qquad {} + (b-a) \biggl[ \biggl( \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \,{}_{0}d_{p,q}t \biggr)^{1-1/r} \\ &\qquad {}\times \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert (1-t) \,{}_{0}d_{p,q}t \\ &\qquad {}+\bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert t \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr]. \end{aligned}$$

This completes the proof. □

Remark 3.7

If $\alpha =1$, then (3.24) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ qa+pb}{p+q} \biggr)- \frac{ 1}{p (b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x \biggr\vert \\ &\quad \leq q(b-a) \biggl( \frac{p^{2}}{(p+q)^{3}} \biggr)^{1-1/r} \bigl[ \bigl(\lambda _{4}(p,q) \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} + \lambda _{5}(p,q) \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert {r} \bigr)^{1/r} \\ &\qquad {}+ \bigl(\lambda _{6}(p,q) \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} + \lambda _{7}(p,q) \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \bigr)^{1/r} \bigr], \end{aligned}$$

(3.25)

where $\lambda _{4}(p,q)$, $\lambda _{5}(p,q)$, $\lambda _{6}(p,q)$, and $\lambda _{7}(p,q)$ are given in Remark (3.6), which appeared in [42].

Moreover, if $p=1$, then (3.24) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{ q}-1 )a+ b}{[\alpha +1]_{ q}} \biggr)- \frac{\Gamma _{ q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b) \biggr\vert \\ &\quad \leq (b-a) \bigl[ \delta ^{1-1/r}_{9} \bigl( \delta _{5} \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert ^{r} + \delta _{6} \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert ^{r} \bigr)^{1/r} \\ &\qquad {}+\delta ^{1-1/r}_{10} \bigl(\delta _{7} \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert ^{r} + \delta _{8} \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert ^{r} \bigr)^{1/r} \bigr], \end{aligned}$$

(3.26)

where $\delta _{5}$, $\delta _{6}$, $\delta _{7}$, and $\delta _{8}$ are given in Remark (3.6) and

$$\begin{aligned}& \delta _{9} = \int _{0}^{\frac{1}{[\alpha +1]_{ q}}} \bigl\vert 1- \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert \,{}_{0}d_{q}t, \\& \delta _{10} = \int _{\frac{1}{[\alpha +1]_{ q}}}^{1} \bigl\vert - \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert \,{}_{0}d_{q}t, \end{aligned}$$

which appeared in [40].

Theorem 3.6

Let $f: [a,b] \to \mathbb{R}$ be a continuous function, $\alpha > 0$, and ${}_{a}D_{p,q}f$ be $(p,q)$-integrable on $(a,\frac{1}{p}(b-a)+a )$. If $\vert {}_{a}D_{p,q}f \vert ^{r}$ is convex on $[a,\frac{1}{p}(b-a)+a ]$ for $r > 1$ and $1/r +1/s = 1$, then the following Riemann–Liouville fractional $(p,q)$-midpoint type inequality holds:

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \biggr\vert \\ &\quad \leq (b-a) \biggl[ (B_{11} )^{1/s} \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \biggl( \frac{p^{\alpha }(p+q)[\alpha +1]_{p,q}-p^{\alpha }}{(p+q) ([\alpha +1]_{p,q} )^{2}} \biggr) \\ &\qquad {}+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \biggl( \frac{p^{\alpha }}{(p+q) ([\alpha +1]_{p,q} )^{2}} \biggr) \biggr)^{1/r} \biggr] \\ &\qquad {} + (b-a) \biggl[ (B_{12} )^{1/s} \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \biggl( \frac{p+q-1}{p+q}- \frac{p^{\alpha }(p+q)[\alpha +1]_{p,q}-p^{2\alpha }}{(p+q) ( [\alpha +1]_{p,q} )^{2}} \biggr) \\ &\qquad {}+\bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \biggl(\frac{1}{p+q}- \frac{p^{2\alpha }}{(p+q) ( [\alpha +1]_{p,q} )^{2}} \biggr) \biggr)^{1/r} \biggr], \end{aligned}$$

(3.27)

where

$$ B_{11} = \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{s} \,{}_{0}d_{p,q}t $$

and

$$ B_{12} = \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{s} \,{}_{0}d_{p,q}t. $$

Proof

Applying Lemma 3.2, Hölder’s inequality, and the convexity of $\vert {}_{a}D_{p,q}f \vert ^{r}$, we have

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{p,q}-p^{\alpha } )a+p^{\alpha }b}{[\alpha +1]_{p,q}} \biggr)- \frac{\Gamma _{p,q}(\alpha +1)}{p^{\alpha ^{2}}(b-a)^{\alpha }} \bigl( {}_{a}I^{\alpha }_{p,q}f \bigr) \bigl( p^{\alpha }b+ \bigl(1-p^{\alpha } \bigr)a \bigr) \biggr\vert \\ &\quad \leq (b-a) \biggl[ \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \\ &\qquad {}+ \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert \,{}_{0}d_{p,q}t \biggr] \\ &\quad \leq (b-a) \biggl[ \biggl( \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{p} \,{}_{0}d_{p,q}t \biggr)^{1/p} \\ &\qquad {}\times \biggl( \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert ^{r} \,{}_{0}d_{p,q}t \biggr)^{1/r} \\ &\qquad {}+ \biggl( \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{p} \,{}_{0}d_{p,q}t \biggr)^{1/p} \\ &\qquad {}\times \biggl( \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \bigl\vert {}_{a}D_{p,q}f \bigl((1-t)a +tb \bigr) \bigr\vert ^{r} \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr] \\ & \quad \leq (b-a) \biggl[ \biggl( \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{p} \,{}_{0}d_{p,q}t \biggr)^{1/p} \\ &\qquad {}\times \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}} (1-t) \,{}_{0}d_{p,q}t + \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \int _{0}^{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}t \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr] \\ &\qquad {} + (b-a) \biggl[ \biggl( \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{p} \,{}_{0}d_{p,q}t \biggr)^{1/p} \\ &\qquad {}\times \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} (1-t) \,{}_{0}d_{p,q}t \\ &\qquad {}+\bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \int _{ \frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} t \,{}_{0}d_{p,q}t \biggr)^{1/r} \biggr] \\ &\quad \leq (b-a) \biggl[ \biggl( \int _{0}^{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}} \biggl\vert 1- \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{p} \,{}_{0}d_{p,q}t \biggr)^{1/p} \\ &\qquad {}\times \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \biggl( \frac{p^{\alpha }(p+q)[\alpha +1]_{p,q}-p^{\alpha }}{(p+q) ([\alpha +1]_{p,q} )^{2}} \biggr) \\ &\qquad {}+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \biggl( \frac{p^{\alpha }}{(p+q) ([\alpha +1]_{p,q} )^{2}} \biggr) \biggr)^{1/r} \biggr] \\ & \qquad {} + (b-a) \biggl[ \biggl( \int _{\frac{p^{\alpha }}{[\alpha +1]_{p,q}}}^{1} \biggl\vert - \frac{ (1- {}_{0}\Phi _{q}(t) )_{p,q}^{(\alpha )}}{p^{\binom{\alpha }{2}}} \biggr\vert ^{p} \,{}_{0}d_{p,q}t \biggr)^{1/p} \\ &\qquad {}\times \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \biggl( \frac{p+q-1}{p+q}- \frac{p^{\alpha }(p+q)[\alpha +1]_{p,q}-p^{2\alpha }}{(p+q) ( [\alpha +1]_{p,q} )^{2}} \biggr) \\ &\qquad {}+\bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \biggl( \frac{1}{p+q}- \frac{p^{2\alpha }}{(p+q) ( [\alpha +1]_{p,q} )^{2}} \biggr) \biggr)^{1/r} \biggr]. \end{aligned}$$

This completes the proof. □

Remark 3.8

If $\alpha =1$, then (3.27) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ qa+pb}{p+q} \biggr)- \frac{ 1}{p (b-a)} \int _{a}^{pb+(1-p)a} f(x) \,{}_{a}d_{p,q}x \biggr\vert \\ &\quad \leq q(b-a) \biggl[ \biggl( \biggl(\frac{p}{p+q} \biggr)^{s+1} \biggl( \frac{p-q}{p^{s+1}-q^{s+1}} \biggr) \biggr)^{1/s} \biggl( \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \biggl( \frac{p^{3}+2p^{2}q+pq^{2}-p^{2}}{(p+q)^{3}} \biggr) \\ &\qquad {}+ \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \biggl( \frac{ p^{2}}{(p+q)^{3}} \biggr) \biggr)^{1/r} \\ &\qquad {}+ \biggl( \int _{\frac{p}{p+q}}^{1} \biggl( \frac{1}{q}-t \biggr)^{s} \,{}_{0}d_{p,q}t \biggr)^{1/s} \biggl( \bigl\vert {}_{a}D_{p,q}f(b) \bigr\vert ^{r} \biggl( \frac{2pq+q^{2}}{(p+q)^{3}} \biggr) \\ &\qquad {}+ \bigl\vert {}_{a}D_{p,q}f(a) \bigr\vert ^{r} \biggl( \frac{ p^{2}q+2pq^{2}-2pq-q^{2}+q^{3}}{(p+q)^{3}} \biggr) \biggr)^{1/r} \biggr], \end{aligned}$$

which appeared in [42].

Moreover, if $p=1$, then (3.27) reduces to

$$\begin{aligned} & \biggl\vert f \biggl( \frac{ ( [\alpha +1]_{ q}-1 )a+ b}{[\alpha +1]_{ q}} \biggr)- \frac{\Gamma _{ q}(\alpha +1)}{ (b-a)^{\alpha }} \bigl( {}_{a}I^{ \alpha }_{ q}f \bigr) (b) \biggr\vert \\ &\quad \leq (b-a) \biggl[ (\delta _{11} )^{1/s} \biggl( \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \biggl( \frac{ (1+q)[\alpha +1]_{ q}-1}{(1+q) ([\alpha +1]_{ q} )^{2}} \biggr) \\ &\qquad {}+ \bigl\vert {}_{a}D_{ q}f(b) \bigr\vert ^{r} \biggl( \frac{1}{(1+q) ([\alpha +1]_{ q} )^{2}} \biggr) \biggr)^{1/r} \biggr] \\ &\qquad {} + (b-a) \biggl[ (\delta _{12} )^{1/s} \biggl( \bigl\vert {}_{a}D_{ q}f(a) \bigr\vert ^{r} \biggl(\frac{q}{1+q}- \frac{(1+q)[\alpha +1]_{ q}-1}{(1+q) ( [\alpha +1]_{ q} )^{2}} \biggr) \\ &\qquad {}+\bigl\vert {}_{a}D_{ q}f(b) \bigr\vert ^{r} \biggl(\frac{1}{1+q}- \frac{1}{(1+q) ( [\alpha +1]_{ q} )^{2}} \biggr) \biggr)^{1/r} \biggr], \end{aligned}$$

where

$$ \delta _{11} = \int _{0}^{\frac{1}{[\alpha +1]_{ q}}} \bigl\vert 1- \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert ^{s} \,{}_{0}d_{ q}t $$

and

$$ \delta _{12} = \int _{\frac{1}{[\alpha +1]_{ q}}}^{1} \bigl\vert - \bigl(1- {}_{0}\Phi _{q}(t) \bigr)_{ q}^{(\alpha )} \bigr\vert ^{s} \,{}_{0}d_{ q}t, $$

which appeared in [40].

4 Conclusions

In this work, we studied two identities for continuous functions in the form of fractional Riemann–Liouville $(p,q)$-integral. Based on these two identities, some fractional Riemann–Liouville $(p,q)$-trapezoid and $(p,q)$-midpoint type inequalities are given. From this idea, as well as the techniques of this paper, we hope that it will inspire interested readers working in this field.