Quantum Trapezium-Type Inequalities Using Generalized ϕ-Convex Functions
Abstract
:1. Introduction
2. Some Quantum Trapezium-Type Inequalities
3. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Ernst, T. The History of Q-Calculus and a New Method; Department of Mathematics, Uppsala University: Stockholm, Sweden, 2000. [Google Scholar]
- Jackson, F.H. On a q-definite integrals. Q. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
- Ernst, T. The different tongues of q-calculus. Proc. Eston. Acad. Sci. 2008, 57, 81–99. [Google Scholar] [CrossRef]
- Ernst, T. A Comprehensive Treatment of q-Calculus; Springer: Basel, Switzerland, 2012. [Google Scholar]
- Gauchman, H. Integral Inequalities in q-Calculus. Comp. Math. Appl. 2004, 47, 281–300. [Google Scholar] [CrossRef] [Green Version]
- Kac, V.; Cheung, P. Quantum Calculus; Universitext; Springer: New York, NY, USA, 2002. [Google Scholar]
- Ismail, M.E.H. Classical and Quantum Orthogonal Polynomials in One Variable; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Ismail, M.E.H.; Mansour, Z.C.I. q-analogues of Freud weights and nonlinear difference equations. Adv. Appl. Math. 2010, 45, 518–547. [Google Scholar] [CrossRef] [Green Version]
- Brahim, K.; Taf, S.; Nefzi, B. New Integral Inequalities in Quantum Calculus. Int. J. Anal. Appl. 2015, 7, 50–58. [Google Scholar]
- Mirković, T.Z.; Tričković, S.B.; Stanković, M.S. Opial inequality in q-calculus. J. Ineq. Appl. 2018, 2018, 1–8. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Niculescu, C.P. An invitation to convex function theory. In Order Structures in Functional Analysis; Editura Academiei Romane: Bucharest, Romania, 2001. [Google Scholar]
- Bennett, C.; Sharpley, R. Interpolation of Operators; Academic Press: Boston, MA, USA, 1998. [Google Scholar]
- Nguyen, N.A.; Gulan, M.; Olaru, S.; Averbe, P.R. Convex Lifting: Theory and Control Applications. IEEE Trans. Autom. Control 2010, 63, 1–16. [Google Scholar] [CrossRef]
- Mititelu, Ş.; Trenţă, S. Efficiency conditions in vector control problems governed by multiple integrals. J. Appl. Math. Comp. 2018, 57, 647–665. [Google Scholar] [CrossRef]
- Trenţă, S. Multiobjective Fractional Variational Problem on Higher-Order Jet Bundles. Comm. Math. Stat. 2016, 4, 323–340. [Google Scholar] [CrossRef]
- Trenţă, S. On a New Class of Vector Variational Control Problems. Num. Funct. Anal. Optim. 2018, 39, 1594–1603. [Google Scholar] [CrossRef]
- Trenţă, S. KT-geodesic pseudoinvex control problems governed by multiple integrals. J. Nonlinear Convex Anal. 2019, 20, 73–84. [Google Scholar]
- Jensen, J.L.W.V. Om konvexe Funktioner og Uligheder mellen Middelvaerdier. Nyt. Tidsskr. Math. 1905, 16, 49–68. [Google Scholar]
- Jensen, J.L.W.V. Sur les fonctions convexes et les inegalités entre les valeurs moyennes. Acta Math. 1906, 30, 175–193. [Google Scholar] [CrossRef]
- Tomar, M.; Agarwal, P.; Jleli, M.; Samet, B. Certain Ostrowski type inequalities for generalized s-convex functions. J. Nonlinear Sci. Appl. 2017, 10, 5947–5957. [Google Scholar] [CrossRef] [Green Version]
- Alomari, M.; Darus, M.; Dragomir, S.S.; Cerone, P. Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Appl. Math. Lett. 2010, 23, 1071–1076. [Google Scholar] [CrossRef]
- Akdemir, A.O.; Özdemir, M.E. Some Hadamard-type inequalities for coordinated P-convex functions and Godunova-Levin functions. AIP Conf. Proc. 2010, 1309, 7–15. [Google Scholar]
- Liu, W.; Wen, W.; Park, J. Hermite–Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals. J. Nonlinear Sci. Appl. 2016, 9, 766–777. [Google Scholar] [CrossRef]
- Bracamonte, M.; Giménez, J.; Vivas-Cortez, M.J. Hermite–Hadamard-Fejér Type Inequalities for Strongly (s,m)-Convex Functions with Modulus c, in Second Sense. Appl. Math. Inf. Sci. 2016, 10, 2045–2053. [Google Scholar] [CrossRef]
- Hernández Hernández, J.E. On Some New Integral Inequalities Related with the Hermite–Hadamard Inequality via h-Convex Functions. MAYFEB J. Math. 2017, 4, 1–12. [Google Scholar]
- Hernández Hernández, J.E. On log-(m,h1,h2)-convex functions and related integral inequalities. Int. J. Open Prob. Compt. Math. 2019, 12, 43–59. [Google Scholar]
- Vivas-Cortez, M.J.; García, C.; Hernández Hernández, J.E. Ostrowski-Type Inequalities for Functions Whose Derivative Modulus is Relatively (m,h1,h2)-Convex. Appl. Math. Inf. Sci. 2019, 13, 369–378. [Google Scholar] [CrossRef]
- Vivas-Cortez, M.J.; García, C.; Hernández Hernández, J.E. Ostrowski Type Inequalities for Functions Whose Derivative Modulus is Relatively Convex. Appl. Math. Inf. Sci. 2019, 13, 121–127. [Google Scholar] [CrossRef]
- Vivas-Cortez, M.J.; Hernández Hernández, J.E. Ostrowski and Jensen type inequalities via (s,m)-convex functions in the second sense. Boletin de la Sociedad Matemática Mexicana 2019, in press. [Google Scholar] [CrossRef]
- Vivas-Cortez, M.J.; Medina Viloria, J. Hermite–Hadamard Type Inequalities for Harmonically Convex Functions on n-Coordinates. Appl. Math. Inf. Sci. Lett. 2018, 6, 1–6. [Google Scholar]
- Vivas-Cortez, M.J.; García, C. Ostrowski Type Inequalities for Functions Whose Derivatives are (m,h1,h2)-Convex. Appl. Math. Inf. Sci. 2017, 11, 79–86. [Google Scholar] [CrossRef]
- Delavar, M.R.; De La Sen, M. Some generalizations of Hermite–Hadamard type inequalities. SpringerPlus 2016, 5, 1–9. [Google Scholar]
- Hernández Hernández, J.E. Some fractional integral inequalities of Hermite Hadamard and Minkowski type. Selecciones Matemáticas (Universidad de Trujillo) 2019, 6, 41–48. [Google Scholar] [CrossRef] [Green Version]
- Kashuri, A.; Liko, R. Some new Hermite–Hadamard type inequalities and their applications. Stud. Sci. Math. Hung. 2019, 56, 103–142. [Google Scholar] [CrossRef]
- Noor, M.A. Some new classes of non-convex functions. Nonlinear Funct. Anal. Appl. 2006, 11, 165–171. [Google Scholar]
- Set, E.; Noor, M.A.; Awan, M.U.; Gözpinar, A. Generalized Hermite–Hadamard type inequalities involving fractional integral operators. J. Inequal. Appl. 2017, 169, 1–10. [Google Scholar] [CrossRef] [Green Version]
- Vivas-Cortez, M.J.; Hernández Hernández, J.E. Hermite–Hadamard Inequalities type for Raina’s fractional integral operator using η-convex functions. Revista de Matemática Teoría y Aplicaciones 2019, 26, 1–19. [Google Scholar]
- Vivas-Cortez, M.J.; Liko, R.; Kashuri, A.; Hernández Hernández, J.E. New Quantum Estimates of Trapezium-Type Inequalities for Generalized ϕ-Convex Functions. Mathematics 2019, 7, 1047. [Google Scholar] [CrossRef] [Green Version]
- Cortez, M.V. Relative strongly h-convex functions and integral inequalities. Appl. Math. Inf. Sci. Lett. 2016, 4, 39–45. [Google Scholar] [CrossRef]
- Raina, R.K. On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Math. J. 2005, 21, 191–203. [Google Scholar]
- Alp, N.; Sarikaya, M.Z.; Kunt, M.; Iscan, I. q-Hermite–Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud Univ. Sci. 2018, 30, 193–203. [Google Scholar] [CrossRef] [Green Version]
- Liu, W.J.; Zhuang, H.F. Some quantum estimates of Hermite–Hadamard inequalities for convex functions. J. Appl. Anal. Comput. 2017, 7, 501–522. [Google Scholar]
- Noor, M.A.; Noor, K.I.; Awan, M.U. Some quantum estimates for Hermite–Hadamard inequalities. Appl. Math. Comput. 2015, 251, 675–679. [Google Scholar] [CrossRef]
- Noor, M.A.; Noor, K.I.; Awan, M.U. Quantum analogues of Hermite–Hadamard type inequalities for generalized convexity. In Computation, Cryptography and Network Security; Daras, N., Rassias, M.T., Eds.; Springer: Cham, Switzerland, 2015; pp. 413–439. [Google Scholar]
- Sudsutad, W.; Ntouyas, S.K.; Tariboon, J. Quantum integral inequalities for convex functions. J. Math. Inequal. 2015, 9, 781–793. [Google Scholar] [CrossRef] [Green Version]
- Tariboon, J.; Ntouyas, S.K. Quantum integral inequalities on finite intervals. J. Inequal. Appl. 2014, 121, 1–13. [Google Scholar] [CrossRef] [Green Version]
- Zhuang, H.; Liu, W.; Park, J. Some quantum estimates of Hermite–Hadamard inequalities for quasi-convex functions. Mathematics 2019, 7, 152. [Google Scholar] [CrossRef] [Green Version]
© 2020 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
Vivas-Cortez, M.J.; Kashuri, A.; Liko, R.; Hernández, J.E. Quantum Trapezium-Type Inequalities Using Generalized ϕ-Convex Functions. Axioms 2020, 9, 12. https://doi.org/10.3390/axioms9010012
Vivas-Cortez MJ, Kashuri A, Liko R, Hernández JE. Quantum Trapezium-Type Inequalities Using Generalized ϕ-Convex Functions. Axioms. 2020; 9(1):12. https://doi.org/10.3390/axioms9010012
Chicago/Turabian StyleVivas-Cortez, Miguel J., Artion Kashuri, Rozana Liko, and Jorge E. Hernández. 2020. "Quantum Trapezium-Type Inequalities Using Generalized ϕ-Convex Functions" Axioms 9, no. 1: 12. https://doi.org/10.3390/axioms9010012