Fractional (p,q)-Calculus on Finite Intervals and Some Integral Inequalities
Abstract
:1. Introduction
2. Preliminaries
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- ;
- (v)
- ;
- (vi)
- , for ;
- (vii)
- .
- (i)
- ;
- (ii)
- ;
- (iii)
- , for .
- (i)
- ;
- (ii)
- ;
- (iii)
- .
- (i)
- ;
- (ii)
- .
3. Main Results
- (i)
- ;
- (ii)
- The fractional -Hölder inequality on :
- The fractional -Hermite–Hadamard integral inequalities on :
- The fractional -Korkine equality on :
- The fractional -Cauchy–Bunyakovsky–Schwarz integral inequality on :
- The fractional -Grüss integral inequality on :
- The fractional -Grüss–Chebyshev integral inequality on :
- The fractional -Polya–Szeqö integral inequality on :
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Jackson, F.H. On a q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
- Jackson, F.H. q-Difference equations. Am. J. Math. 1910, 32, 305–314. [Google Scholar] [CrossRef]
- Fock, V. Zur theorie des wasserstoffatoms. Z. Physik. 1935, 98, 145–154. [Google Scholar] [CrossRef]
- Bangerezaka, G. Variational q-calculus. J. Math. Anal. Appl. 2004, 289, 650–665. [Google Scholar] [CrossRef] [Green Version]
- Asawasamrit, S.; Sudprasert, C.; Ntouyas, S.; Tariboon, J. Some results on quantum Hanh integral inequalities. J. Inequal. Appl. 2019, 2019, 154. [Google Scholar] [CrossRef] [Green Version]
- Bangerezako, G. Variational calculus on q-nonuniform. J. Math. Anal. Appl. 2005, 306, 161–179. [Google Scholar] [CrossRef] [Green Version]
- Exton, H. q-Hypergeometric Functions and Applications; Hastead Press: New York, NY, USA, 1983. [Google Scholar]
- Annyby, H.M.; Mansour, S.K. q-Fractional Calculus and Equations; Springer: Helidelberg, Germany, 2012. [Google Scholar]
- Ernst, T. A Comprehensive Treatment of q-Calculus; Springer: Basel, Switzerland, 2012. [Google Scholar]
- Ernst, T. A History of q-Calculus and a New Method; UUDM Report 2000:16; Department of Mathematics, Uppsala University: Uppsala, Sweden, 2000. [Google Scholar]
- Ferreira, R. Nontrivial solutions for fractional q-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 2010, 1–10. [Google Scholar] [CrossRef]
- Noor, M.A.; Awan, M.U.; Noor, K.I. Quantum Ostrowski inequalities for q-differentiable convex function. J. Math. Inequal. 2016, 10, 1013–1018. [Google Scholar] [CrossRef]
- Aral, A.; Gupta, V.; Agarwal, R.P. Applications of q-Calculus in Operator Theory; Springer Science + Business Media: New York, NY, USA, 2013. [Google Scholar]
- Gauchman, H. Integral inequalities in q-calculus. J. Comput. Appl. Math. 2002, 47, 281–300. [Google Scholar] [CrossRef] [Green Version]
- Kunt, M.; Aljasem, M. Riemann–Liouville fractional quantum Hermite–Hadamard type inequalities for convex functions. Konuralp J. Math. 2020, 8, 122–136. [Google Scholar]
- Dobrogowska, A.; Odzijewicz, A. A second order q-difference equation solvable by factorization method. J. Comput. Appl. Math. 2006, 193, 319–346. [Google Scholar] [CrossRef] [Green Version]
- Gasper, G.; Rahman, M. Some systems of multivariable orthogonal q-Racah polynomials. Ramanujan J. 2007, 13, 389–405. [Google Scholar] [CrossRef] [Green Version]
- Ismail, M.E.H.; Simeonov, P. q-Difference operators for orthogonal polynomials. J. Comput. Appl. Math. 2009, 233, 749–761. [Google Scholar] [CrossRef] [Green Version]
- Bohner, M.; Guseinov, G.S. The h-Laplace and q-Laplace transforms. J. Comput. Appl. Math. 2010, 365, 75–92. [Google Scholar] [CrossRef] [Green Version]
- El-Shahed, M.; Hassan, H.A. Positive solutions of q-difference equation. Proc. Am. Math. Soc. 2010, 138, 1733–1738. [Google Scholar] [CrossRef] [Green Version]
- Ahmad, B. Boundary-value problems for nonlinear third-order q-difference equations. Electron. J. Differ. Equ. 2011, 2011, 94. [Google Scholar]
- Ahmad, B.; Alsaedi, A.; Ntouyas, S.K. A study of second-order q-difference equations with boundary conditions. Adv. Differ. Equ. 2012, 2012, 35. [Google Scholar] [CrossRef] [Green Version]
- Ahmad, B.; Ntouyas, S.K.; Purnaras, I.K. Existence results for nonlinear q-difference equations with nonlocal boundary conditions. Commun. Appl. Nonlinear Anal. 2012, 19, 59–72. [Google Scholar]
- Ahmad, B.; Nieto, J.J. On nonlocal boundary value problems of nonlinear q-difference equation. Adv. Differ. Equ. 2012, 2012, 81. [Google Scholar] [CrossRef] [Green Version]
- Kac, V.; Cheung, P. Quantum Calculus; Springer: New York, NY, USA, 2002. [Google Scholar]
- Tariboon, J.; Ntouyas, S.K. Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 2013, 282. [Google Scholar] [CrossRef] [Green Version]
- Tariboon, J.; Ntouyas, S.K. Quantum integral inequalities on finite intervals. Adv. Differ. Equ. 2014, 2014, 121. [Google Scholar] [CrossRef] [Green Version]
- Necmettin, A.; Mehmet, Z.S.; Mehmet, K.; İmdat, İ. q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex functions and quasi-convex functions. J. King Saud Univ. Sci. 2018, 30, 193–203. [Google Scholar]
- Sudsudat, W.; Ntouyas, S.K.; Tariboon, J. Quantum integral inequalities for convex functions. J. King Saud Univ. Sci. 2015, 9, 781–793. [Google Scholar]
- Muhammad, A.N.; Khalida, I.N.; Muhammad, U.A. Some quantum integral inequalities via preinvex functions. Appl. Math. Comput. 2015, 269, 242–251. [Google Scholar]
- Muhammad, A.K.; Noor, M.; Eze, N.R.; Yu-Ming, C. Quantum Hermite–Hadamard inequality by means of a Green function. Adv. Differ. Equ. 2020, 2020, 99. [Google Scholar]
- Muhammad, A.N.; Khalida, I.N.; Muhammad, U.A. Quantum analogues of Hermite–Hadamard type inequalities for generalized convexity. In Computation, Cryptography, and Network Security; Springer: Cham, Switzerland, 2015; pp. 413–439. [Google Scholar]
- Humaira, K.; Muhammad, I.; Baleanu, D.; Yu-Ming, C. New estimates of-Ostrowski-type inequalities within a class of-polynomial prevexity of functions. J. Funct. Spaces 2020, 2020, 13. [Google Scholar]
- Wenjun, L.; Hefeng, Z. Some quantum estimates of Hermite–Hadamard inequalities for convex functions. Appl. Math. Comput. 2015, 251, 675–679. [Google Scholar]
- Tun, M.; Gov, E.; Balgeçti, S. Simpson type quantum integral inequalities for convex functions. Miskolc Math. Notes 2018, 9, 649–664. [Google Scholar]
- Latif, M.A.; Dragomir, S.S.; Momoniat, E. Some φ-analogues of Hermite–Hadamard inequality for s-convex functions in the second sense and related estimates. Punjab Univ. J. Math. 2016, 48, 147–166. [Google Scholar]
- Hefeng, Z.; Wenjun, L.; Jaekeun, P. Some quantum estimate of Hermite–Hadamard inequalities for quasi-convex functions. Mathematics 2019, 7, 152. [Google Scholar]
- Tariboon, J.; Ntouyas, S.K.; Agarwal, P. New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Adv. Differ. Equ. 2015, 2015, 18. [Google Scholar] [CrossRef] [Green Version]
- Sudsutad, W.; Ntouyas, S.K.; Tariboon, J. Integral inequalities via fractional quantum calculus. J. Inequal. Appl. 2016, 2016, 81. [Google Scholar] [CrossRef] [Green Version]
- Tun, M.; Gov, E. Some integral inequalities via (p,q)-calculus on finite intervals. RGMIA Res. Rep. Coll. 2016, 19, 1–12. [Google Scholar]
- Tun, M.; Gov, E. (p,q)-Integral inequalities. RGMIA Res. Rep. Coll. 2016, 19, 1–13. [Google Scholar]
- Kunt, M.; İşcan, İ.; Alp, N.; Sarakaya, M.Z. (p,q)-Hermite–Hadamard inequalities and (p,q)-estimates for midpoint type inequalities via convex and quasi-convex functions. Rev. R. Acad. Cienc. 2018, 112, 969–992. [Google Scholar]
- Soontharanon, J.; Sitthiwirattham, T. Fractional (p,q)-calculus. Adv. Differ. Equ. 2020, 2020, 35. [Google Scholar] [CrossRef] [Green Version]
- Bukweli-Kyemba, J.D.; Hounkonnou, M.N. Quantum deformed algebra: Coherent states and special functions. arXiv 2013, arXiv:1301.0116v1. [Google Scholar]
- Prabseang, J.; Nonlaopon, K.; Tariboon, J. (p,q)-Hermite–Hadamard inequalities for double integral and (p,q)-differentiable convex functions. Axioms 2019, 8, 68. [Google Scholar] [CrossRef] [Green Version]
- Kalsoom, H.; Amer, M.; Junjua, M.D.; Hassain, S.; Shahxadi, G. (p,q)-estimates of Hermite–Hadamard-type inequalities for coordinated convex and quasi convex function. Mathematics 2019, 7, 683. [Google Scholar] [CrossRef] [Green Version]
- Chakrabarti, R.; Jagannathan, R. A (p,q)-oscillator realization of two-parameter quantum algebras. J. Phys. A Math. Gen. 1991, 24, L711–L718. [Google Scholar] [CrossRef]
- Burban, I. Two-parameter deformation of the oscillator algebra and (p,q)-analog of two-dimensional conformal field theory. J. Nonlinear Math. Phys. 1995, 2, 384–391. [Google Scholar] [CrossRef] [Green Version]
- Burban, I.M.; Klimyk, A.U. (p,q)-differentiation, (p,q)-integration, and (p,q)-hypergeometric functions related to quantum groups. Integral Transform. Spec. Funct. 1994, 2, 15–36. [Google Scholar] [CrossRef]
- Hounkonnou, M.N.; Désiré, J.; Kyemba, B. R (p,q)-calculus: Differentiation and integration. SUT J. Math. 2013, 49, 145–167. [Google Scholar]
- Aral, A.; Gupta, V. Applications of (p,q)-gamma function to Szász durrmeyer operators. Publ. L’Inst. Math. 2017, 102, 211–220. [Google Scholar] [CrossRef]
- Sahai, V.; Yadav, S. Representations of two parameter quantum algebras and (p,q)-special functions. Adv. Differ. Equ. 2007, 335, 268–279. [Google Scholar]
- Sadjang, P.N. On the fundamental theorem of (p,q)-calculus and some (p,q)-taylor formulas. Results Math. 2018, 73, 39. [Google Scholar] [CrossRef]
- Usman, T.; Saif, M.; Choi, J. Certain identities associated with (p,q)-binomial coefficients and (p,q)-Stirling polynomials of the second kind. Symmetry 2020, 12, 1436. [Google Scholar] [CrossRef]
- Sadjang, P.N. On the (p,q)-gamma and the (p,q)-beta functions. arXiv 2015, arXiv:1506.07394. [Google Scholar]
- Sadjang, P.N. On two (p,q)-analogues of the laplace transform. J. Differ. Equ. Appl. 2017, 23, 1562–1583. [Google Scholar]
- Mursaleen, M.; Ansari, K.J.; Khan, A. On (p,q)-analogues of Bernstein operators. Appl. Math. Comput. 2015, 266, 874–882. [Google Scholar] [CrossRef]
- Mursaleen, M.; Ansari, K.J.; Khan, A. Erratum to “On (p,q)-analogues of Bernstein operators”. Appl. Math. Comput. 2016, 278, 70–71. [Google Scholar]
- Kang, S.M.; Rafiq, A.; Acu, A.M.; Ali, F.; Kwun, Y.C. Erratum to “Some approximation properties of (p,q)-Bernstein operators”. J. Inequal. Appl. 2016, 2016, 169. [Google Scholar] [CrossRef] [Green Version]
- Mursaleen, M.; Khan, F.; Khan, A. Approximation by (p,q)-Lorentz polynomials on a compact disk. Complex Anal. Oper. Theory. 2016, 10, 1725–1740. [Google Scholar] [CrossRef] [Green Version]
- Cai, Q.-B.; Zhou, G. On (p,q)-analogues of Kantorovich type Bernstein-Stancu-Schurer operator. Appl. Math. Comput. 2016, 276, 12–20. [Google Scholar] [CrossRef]
- Mursaleen, M.; Ansari, K.J.; Khan, A. Some approximation results of (p,q)-analogues of Bernstein-Stancu operators. Appl. Math. Comput. 2015, 264, 392–402. [Google Scholar] [CrossRef]
- Acar, T.; Aral, A.; Mohiuddine, S.A. On Kantorovich modification of (p,q)-Baskakov operators. J. Inequal. Appl. 2016, 2016, 98. [Google Scholar] [CrossRef] [Green Version]
- Mursaleen, M.; Nasiruzzaman, M.; Khan, F.; Khan, A. (p,q)-analogues of divided difference and Bernstein operators. J. Nonlinear Funct. Anal. 2017, 2017, 25. [Google Scholar]
- Wachs, M.; White, D. (p,q)-Stirling numbers and set partition statistics. J. Combin. Theory Ser. A. 1991, 56, 27–46. [Google Scholar] [CrossRef] [Green Version]
- Wachs, M.L. σ-restricted growth functions and (p,q)-Stirling numbers. J. Combin. Theory Ser. 1994, 68, 470–480. [Google Scholar] [CrossRef] [Green Version]
- Remmel, J.B.; Wachs, M. Rook theory, generalized Stirling numbers and (p,q)-analogues. Electron. J. Combin. 2004, 11, R84. [Google Scholar] [CrossRef] [Green Version]
- Médics, A.D.; Leroux, P. Generalized Stirling numbers, convolution formula and (p,q)-analogues. Can. J. Math. 1995, 11, 474–499. [Google Scholar] [CrossRef] [Green Version]
- Gradimir, V.M.; Vijay, G.N.M. (p,q)-beta functions and applications in approximation. Bol. Soc. Mat. Mex. 2018, 24, 219–237. [Google Scholar]
- Gasper, G.; Rahman, M. Basic Hypergeometric Series; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Anastassiou, G.A. Intelligent Mathematics: Computational Analysis; Springer: New York, NY, USA, 2011. [Google Scholar]
- Cerone, P.; Dragomir, S.S. Mathematical Inequalities; CRC Press: New York, NY, USA, 2010. [Google Scholar]
- Pachpatte, B.G. Analytic Inequalities; Atlantis Press: Paris, France, 2012. [Google Scholar]
- Polya, G.; Szeqö, G. Aufaben und Lehrsatze aus der Analysis, Band 1. Die Grundlehren der Mathematischen Wissenschaften; Springer: Berlin, Germany, 1925. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Neang, P.; Nonlaopon, K.; Tariboon, J.; Ntouyas, S.K. Fractional (p,q)-Calculus on Finite Intervals and Some Integral Inequalities. Symmetry 2021, 13, 504. https://doi.org/10.3390/sym13030504
Neang P, Nonlaopon K, Tariboon J, Ntouyas SK. Fractional (p,q)-Calculus on Finite Intervals and Some Integral Inequalities. Symmetry. 2021; 13(3):504. https://doi.org/10.3390/sym13030504
Chicago/Turabian StyleNeang, Pheak, Kamsing Nonlaopon, Jessada Tariboon, and Sotiris K. Ntouyas. 2021. "Fractional (p,q)-Calculus on Finite Intervals and Some Integral Inequalities" Symmetry 13, no. 3: 504. https://doi.org/10.3390/sym13030504
APA StyleNeang, P., Nonlaopon, K., Tariboon, J., & Ntouyas, S. K. (2021). Fractional (p,q)-Calculus on Finite Intervals and Some Integral Inequalities. Symmetry, 13(3), 504. https://doi.org/10.3390/sym13030504