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.
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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,
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:
More generally, if \(\alpha \in \mathbb{R}\), then
or
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
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
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
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
and an equivalent definition of (2.9) is given in [56] as
where
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
and (2.11) can also be written as
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
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
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:
Proof
By simple computation and using Definition 2.3, we have
and
From (3.2) and (3.3), we obtain
Thus the proof is completed. □
Remark 3.1
If \(\alpha =1\), then (3.1) reduces to Lemma 3.2 in [43] as
If \(p=1\), then (3.1) reduces to Lemma 5.2 in [40] as
Moreover, if \(q \to 1\) and \(\alpha =1\), then (3.6) reduces to
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
then the following Riemann–Liouville fractional \((p,q)\)-trapezoid type inequality holds:
where
and
Proof
Using Lemma 3.1 and the convexity of \(\vert {}_{a}D_{p,q}f \vert \), we have
This completes the proof. □
Remark 3.2
If \(p=1\), then (3.8) reduces to
where
and
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:
where \(B_{1}\) and \(B_{2}\) are given in Theorem 3.1and
Proof
Using Lemma 3.1, the convexity of \(\vert {}_{a}D_{p,q}f \vert ^{r}\), and the power mean inequality, we have
Therefore, the proof is completed. □
Remark 3.3
If \(\alpha =1\), then (3.10) reduces to
where
and
which appeared in [43].
Moreover, if \(p=1\), then (3.10) reduces to
where \(\delta _{1}\) and \(\delta _{2}\) are given in Remark 3.2 and
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:
where
Proof
Using Lemma 3.1, the convexity of \(\vert {}_{a}D_{p,q}f \vert ^{r}\), and Hölder’s inequality, we have
This completes the proof. □
Remark 3.4
If \(\alpha =1\), then (3.13) reduces to
where
which appeared in [43].
Moreover, if \(p=1\), then (3.13) reduces to
where
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:
Proof
By direct computation and using Definitions 2.1 and 2.2, we have
On the other hand, in Lemma 3.1, the following integral was given:
Consequently, from (3.17) and (3.18), we have
Therefore, the proof is completed. □
Remark 3.5
If \(\alpha =1\), then (3.16) reduces to
which appeared in [42].
Moreover, if \(p=1\), then (3.16) reduces to
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:
where
Proof
Using Lemma 3.2 and the convexity of \(\vert {}_{a}D_{p,q}f \vert \), we have
This completes the proof. □
Remark 3.6
If \(\alpha =1\), then (3.21) reduces to
where
which appeared in [42].
Moreover, if \(p=1\), then (3.21) reduces to
where
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:
where \(B_{5}\), \(B_{6}\), \(B_{7}\), and \(B_{8}\) are given in Theorem 3.4and
and
Proof
Using Lemma 3.2, the power mean inequality and the convexity of \(\vert {}_{a}D_{p,q}f \vert ^{r}\), we have
This completes the proof. □
Remark 3.7
If \(\alpha =1\), then (3.24) reduces to
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
where \(\delta _{5}\), \(\delta _{6}\), \(\delta _{7}\), and \(\delta _{8} \) are given in Remark (3.6) and
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:
where
and
Proof
Applying Lemma 3.2, Hölder’s inequality, and the convexity of \(\vert {}_{a}D_{p,q}f \vert ^{r}\), we have
This completes the proof. □
Remark 3.8
If \(\alpha =1\), then (3.27) reduces to
which appeared in [42].
Moreover, if \(p=1\), then (3.27) reduces to
where
and
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.
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This work is supported by the Program Management Unit for Human Resources & Institutional Development, Research and Innovation [grant number B05F630104] and Chiang Mai University, Thailand.
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Neang, P., Nonlaopon, K., Tariboon, J. et al. Some trapezoid and midpoint type inequalities via fractional \((p,q)\)-calculus. Adv Differ Equ 2021, 333 (2021). https://doi.org/10.1186/s13662-021-03487-6
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DOI: https://doi.org/10.1186/s13662-021-03487-6