Abstract
In this paper we prove several fractional quantum integral inequalities for the new q-shifting operator \({_{a}}\Phi_{q}(m) = qm + (1-q)a\) introduced in Tariboon et al. (Adv. Differ. Equ. 2015:18, 2015), such as: the q-Hölder inequality, the q-Hermite-Hadamard inequality, the q-Cauchy-Bunyakovsky-Schawrz integral inequality, the q-Grüss integral inequality, the q-Grüss-Čebyšev integral inequality, and the q-Pólya-Szegö integral inequality.
Similar content being viewed by others
1 Introduction
The quantum calculus is known as the calculus without limits. It substitutes the classical derivative by a difference operator, which allows one to deal with sets of nondifferentiable functions. Quantum difference operators have an interesting role due to their applications in several mathematical areas, such as orthogonal polynomials, basic hyper-geometric functions, combinatorics, the calculus of variations, mechanics, and the theory of relativity. The book by Kac and Cheung [2] covers many of the fundamental aspects of quantum calculus.
In recent years, the topic of q-calculus has attracted the attention of several researchers and a variety of new results can be found in the papers [3–15] and the references cited therein.
In [16] the notions of \(q_{k}\)-derivative and \(q_{k}\)-integral of a continuous function \(f:[t_{k},t_{k+1}]\to{\mathbb {R}}\), have been introduced and their basic properties were proved. As applications existence and uniqueness results for initial value problems of first and second order impulsive \(q_{k}\)-difference equations were investigated. The q-calculus analogs of some classical integral inequalities, such as Hölder, Hermite-Hadamard, Trapezoid, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Grüss and Grüss-Čebyšev were established in [17]. For recent results on quantum inequalities, see [18–20].
In [1] new concepts of fractional quantum calculus were defined, by defining a new q-shifting operator \({_{a}}\Phi_{q}(m) = qm + (1-q)a\). After giving the basic properties the q-derivative and q-integral were defined. New definitions of the Riemann-Liouville fractional q-integral and the q-difference on an interval \([a,b]\) were given and their basic properties were discussed. As applications of the new concepts, one proved existence and uniqueness results for first and second order initial value problems for impulsive fractional q-difference equations.
In this paper we prove several integral inequalities for the new q-shifting operator \({_{a}}\Phi_{q}(m) = qm + (1-q)a\), such as: the q-Hölder inequality, the q-Hermite-Hadamard inequality, the q-Korkine integral equality, the q-Cauchy-Bunyakovsky-Schwarz integral inequality, the q-Grüss integral inequality, the q-Grüss-Čebyšev integral inequality, and the q-Polya-Szegö integral inequality.
2 Preliminaries
To make this paper self-contained, below we recall some well-known facts on fractional q-calculus. The presentation here can be found, for example, in [7, 8].
Let us define a q-shifting operator as
where \(0< q<1\), \(m,a\in\mathbb{R}\). For any positive integer k, we have
The following results can be found in [1].
Property 2.1
For any \(m,n\in\mathbb{R}\) and for all positive integer k, j, the following properties hold:
-
(i)
\({_{a}}\Phi_{q}^{k}(m) = {_{a}}\Phi_{q^{k}}(m)\);
-
(ii)
\({_{a}}\Phi_{q}^{j}({_{a}}\Phi_{q}^{k}(m)) = {_{a}}\Phi _{q}^{k}({_{a}}\Phi_{q}^{j}(m)) = {_{a}}\Phi_{q}^{j+k}(m)\);
-
(iii)
\({_{a}}\Phi_{q}(a) = a\);
-
(iv)
\({_{a}}\Phi_{q}^{k}(m) - a = q^{k}(m-a)\);
-
(v)
\(m - {_{a}}\Phi_{q}^{k}(m) = (1-q^{k})(m-a)\);
-
(vi)
\({_{a}}\Phi_{q}^{k}(m) = m_{\frac{a}{m}}\Phi_{q}^{k}(1)\), for \(m\neq0\);
-
(vii)
\({_{a}}\Phi_{q}(m) - {_{a}}\Phi_{q}^{k}(n) = q(m-{_{a}}\Phi _{q}^{k-1}(n))\).
The q-analog of the Pochhammer symbol is defined by
We also define the power of the q-shifting operator as
More generally, if \(\gamma\in\mathbb{R}\), then
From the above definitions, the following results were proved in [1].
Property 2.2
For any \(\gamma,m,n\in\mathbb{R}\) with \(n\neq a\) and \(k\in\mathbb {N}\cup \{\infty\}\), the following properties hold:
-
(i)
\({_{a}}(n-m)_{q}^{(k)} = (n-a)^{k} (\frac {m-a}{n-a};q )_{k}\);
-
(ii)
\({_{a}}(n-m)_{q}^{(\gamma)} = (n-a)^{\gamma} \prod_{i=0}^{\infty}\frac{1-\frac{m-a}{n-a}q^{i}}{ 1-\frac{m-a}{n-a}q^{\gamma+i}} = (n-a)^{\gamma}\frac{ (\frac {m-a}{n-a};q )_{\infty}}{ (\frac{m-a}{n-a}q^{\gamma};q )_{\infty}}\);
-
(iii)
\({_{a}}(n-{_{a}}\Phi_{q}^{k}(n))_{q}^{(\gamma)} = (n-a)^{\gamma}\frac{(q^{k};q)_{\infty}}{(q^{\gamma+k};q)_{\infty}}\).
The q-number is defined by
If \(a=0\) and \(m=n=1\), then (2.5) is reduced to
The q-gamma function is defined by
Obviously, \(\Gamma_{q}(t+1) = [t]_{q}\Gamma_{q}(t)\). For any \(s,t>0\), the q-beta function is defined by
The q-beta function in terms of the q-gamma function can be written as
Let us give the definitions of Riemann-Liouville fractional q-integral and the q-derivative on the dense interval \([a,b]\).
Definition 2.3
Let \(\alpha\geq0\) and f be a continuous function defined on \([a,b]\). The fractional q-integral of Riemann-Liouville type is given by \(({_{a}}I_{q}^{0}f)(t) = f(t)\) and
Definition 2.4
The fractional q-derivative of Riemann-Liouville type of order \(\alpha\geq0\) of a continuous function f on the interval \([a,b]\) is defined by \(({_{a}}D_{q}^{0}f)(t) = f(t)\) and
where υ is the smallest integer greater than or equal to α.
Lemma 2.5
[1]
Let \(\alpha, \beta\geq0\), and f be a continuous function on \([a,b]\). The Riemann-Liouville fractional q-integral has the following semi-group properties:
Throughout this paper, in some places, the variable s will be shown inside the fractional integral notation as \(({_{a}}I^{\alpha }_{q}f(s) )(t)\), which means
Lemma 2.6
If \(\alpha,\beta\geq0\), then, for \(t \in[a,b]\), the following relation holds:
Proof
From Definition 2.3 and applying Property 2.1(iv), Property 2.2(iii), it follows that
which leads to (2.12) as required. □
Corollary 2.7
Let \(f(t)=t\) and \(g(t)=t^{2}\) for \(t \in[a,b]\), and \(\alpha>0\). Then we have
-
(i)
\(({_{a}}I_{q}^{\alpha}f(s) )(t) =\frac{(t-a)^{\alpha}}{\Gamma_{q}(\alpha+2)} (t+ ([\alpha +1]_{q}-1 )a )\);
-
(ii)
\(({_{a}}I_{q}^{\alpha}g(s) )(t) =\frac{(t-a)^{\alpha}}{\Gamma_{q}(\alpha+3)} ((1+q)(t-a)^{2}+2a(t-a)[\alpha+2]_{q} +a^{2}[\alpha+1]_{q}[\alpha+2]_{q} )\).
3 Main results
Let us start with the fractional q-Hölder inequality on the interval \([a,b]\).
Theorem 3.1
Let \(0< q<1\), \(\alpha>0\), \(p_{1}, p_{2}>1\), such that \(\frac{1}{p_{1}}+\frac {1}{p_{2}}=1\). Then for \(t\in[a,b]\) we have
Proof
From Definition 2.3 and the discrete Hölder inequality, we have
Therefore, inequality (3.1) holds. □
Remark 3.2
If \(\alpha= 1\) and \(a=0\), then (3.1) is reduced to the q-Hölder inequality in [21].
The fractional q-Hermite-Hadamard integral inequality on the interval \([a,b]\) will be proved as follows.
Theorem 3.3
Let \(f:[a,b]\to\mathbb{R}\) be a convex continuous function, \(0< q<1\) and \(\alpha>0\). Then we have
Proof
The convexity of f on \([a,b]\) means that
Multiplying both sides of (3.3) by \({_{0}}(1-{_{0}}\Phi _{q}(s))_{q}^{(\alpha-1)}/\Gamma_{q}(\alpha)\), \(s \in(0,1)\), we get
Taking q-integration of order \(\alpha>0\) for (3.4) with respect to s on \([0,1]\), we have
which means that
From Corollary 2.7(i), we have
Using the definition of fractional q-integration on \([a,b]\), we have
which gives the second part of (3.2) by using (3.6).
To prove the first part of (3.2), we use the convex property of f as follows:
Multiplying both sides of (3.7) by \({_{0}}(1-{_{0}}\Phi _{q}(s))_{q}^{(\alpha-1)}/\Gamma_{q}(\alpha)\), \(s \in(0,1)\), we get
Again on fractional q-integration of order \(\alpha>0\) to the above inequality with respect to t on \([0,1]\) and changing variables, we get
By a direct computation, we have
together with (3.8), we derive the first part of inequality (3.2) as requested. The proof is completed. □
Remark 3.4
If \(\alpha= 1\) and \(q \to1\), then inequality (3.2) is reduced to the classical Hermite-Hadamard integral inequality as
Let us prove the fractional q-Korkine equality on the interval \([a,b]\).
Lemma 3.5
Let \(f, g:[a,b]\to\mathbb{R}\) be continuous functions, \(0< q<1\), and \(\alpha>0\). Then we have
Proof
From Definition 2.3, we have
from which one deduces (3.9). □
Remark 3.6
If \(\alpha= 1\), then Lemma 3.5 is reduced to Lemma 3.1 in [17].
Next, we will prove the fractional q-Cauchy-Bunyakovsky-Schwarz integral inequality on the interval \([a,b]\).
Theorem 3.7
Let \(f, g:[a,b]\to\mathbb{R}\) be continuous functions, \(0< q<1\), and \(\alpha,\beta>0\). Then we have
Proof
From Definition 2.3, we have
Using the classical discrete Cauchy-Schwarz inequality, we have
Therefore, inequality (3.10) holds. □
Remark 3.8
If \(\alpha= 1\), then inequality (3.10) is reduced to the q-Cauchy-Bunyakovsky-Schwarz integral inequality in [17].
Now, we will prove the fractional q-Grüss integral inequality on the interval \([a,b]\).
Theorem 3.9
Let \(f, g:[a,b]\to\mathbb{R}\) be continuous functions satisfying
For \(0< q<1\) and \(\alpha>0\), we have the inequality
Proof
Applying Theorem 3.7, we have
From Lemma 3.5, it follows that
By a simple computation, we have
and an analogous identity for g.
By assumption (3.11) we have \((f(s)-\phi)(\Phi-f(s)) \geq 0\) for all \(s\in[a,b]\), which implies
From (3.15) and using the fact that \((\frac{A+B}{2} )^{2} \geq AB\), \(A, B\in\mathbb{R}\), we have
A similar argument gives
Using inequality (3.13) via (3.14) and the estimations (3.16) and (3.17), we get
Therefore, inequality (3.12) holds, as desired. □
Remark 3.10
If \(\alpha= 1\) and \(q \to1\), then inequality (3.12) is reduced to the classical Grüss integral inequality as
Next, we are going to prove the fractional q-Grüss-Čebyšev integral inequality on the interval \([a,b]\).
Theorem 3.11
Let \(f, g:[a,b]\to\mathbb{R}\) be \(L_{1}\)-, \(L_{2}\)-Lipschitzian continuous functions, so that
for all \(s,r\in[a,b]\), \(0< q<1\), \(L_{1}, L_{2}>0\), and \(\alpha>0\). Then we have the inequality
Proof
Recall the fractional q-Korkine equality as
It follows from (3.18) that
for all \(s,r \in[a,b]\). Taking the double fractional q-integration of order α with respect to \(s, r\in[a,b]\), we get
From Corollary 2.7(ii), with \(t=b\), we have
By direct computation, we have
Thus, from (3.22) and (3.23), we have
By applying (3.24) to (3.20), we get the desired inequality in (3.19). □
Remark 3.12
If \(\alpha= 1\) and \(q \to1\), then inequality (3.19) is reduced to the classical Grüss-Čebyšev integral inequality as
For the final result, we establish the fractional q-Pólya-Szegö integral inequality on the interval \([a,b]\).
Theorem 3.13
Let \(f,g:[a,b]\to\mathbb{R}\) be two positive integrable functions satisfying
Then for \(0< q<1\) and \(\alpha>0\), we have the inequality
Proof
From (3.25), for \(s\in[a,b]\), we have
which yields
and
Multiplying (3.27) and (3.28), we obtain
or
Inequality (3.29) can be written as
Multiplying both sides of (3.30) by \({_{0}}(b-{_{0}}\Phi _{q}(s))_{q}^{(\alpha-1)}/\Gamma_{q}(\alpha)\) and integrating with respect to s from a to b, we get
Applying the AM-GM inequality, \(A+B\geq2\sqrt{AB}\), \(A,B\in\mathbb {R}^{+}\), we have
which leads to
Therefore, inequality (3.26) is proved. □
Remark 3.14
If \(\alpha= 1\) and \(q \to1\), then inequality (3.26) is reduced to the classical Pólya-Szegö integral inequality as
See also [24].
4 Conclusion
In this work, some important integral inequalities involving the new q-shifting operator \({_{a}}\Phi_{q}(m) = qm + (1-q)a\), introduced in [1], are established in the context of fractional quantum calculus. The derived results constitute contributions to the theory of integral inequalities and fractional calculus and can be specialized to yield numerous interesting fractional integral inequalities including some known results. Furthermore, they are expected to lead to some applications in fractional boundary value problems.
References
Tariboon, J, Ntouyas, SK, Agarwal, P: New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Adv. Differ. Equ. 2015, 18 (2015)
Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)
Jackson, FH: q-Difference equations. Am. J. Math. 32, 305-314 (1910)
Al-Salam, WA: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15(2), 135-140 (1966/1967)
Agarwal, RP: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365-370 (1969)
Ernst, T: The history of q-calculus and a new method. UUDM Report 2000:16, Department of Mathematics, Uppsala University (2000)
Ferreira, R: Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 70 (2010)
Annaby, MH, Mansour, ZS: q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012)
Bangerezako, G: Variational q-calculus. J. Math. Anal. Appl. 289, 650-665 (2004)
Ismail, MEH, Simeonov, P: q-Difference operators for orthogonal polynomials. J. Comput. Appl. Math. 233, 749-761 (2009)
Yu, C, Wang, J: Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives. Adv. Differ. Equ. 2013, 124 (2013)
Ahmad, B, Ntouyas, SK: Boundary value problems for q-difference inclusions. Abstr. Appl. Anal. 2011, 292860 (2011)
Ahmad, B: Boundary-value problems for nonlinear third-order q-difference equations. Electron. J. Differ. Equ. 2011, 94 (2011)
Graef, JR, Kong, L: Positive solutions for a class of higher-order boundary value problems with fractional q-derivatives. Appl. Math. Comput. 218, 9682-9689 (2012)
Ahmad, B, Ntouyas, SK, Purnaras, IK: Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations. Adv. Differ. Equ. 2012, 140 (2012)
Tariboon, J, Ntouyas, SK: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 282 (2013)
Tariboon, J, Ntouyas, SK: Quantum integral inequalities on finite intervals. J. Inequal. Appl. 2014, 121 (2014)
Noor, MA, Noor, KI, Awan, MU: Some quantum estimates for Hermite-Hadamard inequalities. Appl. Math. Comput. 251, 675-679 (2015)
Noor, MA, Noor, KI, Awan, MU: Some quantum integral inequalities via preinvex functions. Appl. Math. Comput. 269, 242-251 (2015)
Taf, S, Brahim, K, Riahi, L: Some results for Hadamard-type inequalities in quantum calculus. Matematiche LXIX(2), 243-258 (2014)
Anastassiou, GA: Intelligent Mathematics: Computational Analysis. Springer, New York (2011)
Cerone, P, Dragomir, SS: Mathematical Inequalities. CRC Press, New York (2011)
Pachpatte, BG: Analytic Inequalities. Atlantis Press, Paris (2012)
Pólya, G, Szegö, G: Aufgaben und Lehrsatze aus der Analysis, Band 1. Die Grundlehren der mathematischen Wissenschaften, vol. 19. Springer, Berlin (1925)
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this article. They read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Sudsutad, W., Ntouyas, S.K. & Tariboon, J. Integral inequalities via fractional quantum calculus. J Inequal Appl 2016, 81 (2016). https://doi.org/10.1186/s13660-016-1024-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-016-1024-1