Research article

Some Opial type inequalities in (p, q)-calculus

  • Received: 31 March 2020 Accepted: 28 June 2020 Published: 15 July 2020
  • MSC : 26D15, 81P68

  • In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.

    Citation: Chunhong Li, Dandan Yang, Chuanzhi Bai. Some Opial type inequalities in (p, q)-calculus[J]. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377

    Related Papers:

  • In this paper, we establish 5 kinds of integral Opial-type inequalities in (p, q)-calculus by means of H?lder's inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (p, q)-calculus involving one function and its (p, q) derivative. Furthermore, Opial inequalities in (p, q)-calculus involving two functions and two functions with their (p, q) derivatives are given. Our results are (p, q)-generalizations of some known inequalities, such as Opial-type integral inequalities and (p, q)-Wirtinger inequality.


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    [1] Z. Opial, Sur une inegalite, Ann. Pol. Math., 8 (1960), 29-32. doi: 10.4064/ap-8-1-29-32
    [2] W. S. Cheung, Some generalized Opial-type inequalities, J. Math. Anal. Appl., 162 (1991), 317- 321.
    [3] M. Z. Sarikaya, H. Budak, New inequalities of Opial Type for conformable fractional integrals, Turk. J. Math., 41 (2017), 1164-1173. doi: 10.3906/mat-1606-91
    [4] H. L. Keng, On an inequality of Opial, Scientia Sinica, 14 (1965), 789-790.
    [5] B. G. Pachpatte, On Opial-type integral inequalities, J. Math. Anal. Appl., 120 (1986), 547-556. doi: 10.1016/0022-247X(86)90176-9
    [6] B. G. Pachpatte, A note on some new Opial type integral inequalities, Octogan Math. Mag., 7 (1999), 80-84.
    [7] S. H. Saker, M. D. Abdou, I. Kubiaczyk, Opial and Polya type inequalities via convexity, Fasciculi Mathematici, 60 (2018), 145-159. doi: 10.1515/fascmath-2018-0009
    [8] T. Z. Mirković, S. B. Tričković, M. S. Stanković, Opial inequality in q-calculus, J. Inequal. Appl., 2018 (2018), 1-8. doi: 10.1186/s13660-017-1594-6
    [9] N. Alp, C. C. Bilisik, M. Z. Sarikaya, On q-Opial type inequality for quantum integral, Filomat, 33 (2019), 4175-4184. doi: 10.2298/FIL1913175A
    [10] R. Chakrabarti, R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991), 711-718. doi: 10.1088/0305-4470/24/13/002
    [11] I. M. Burban, A. U. Klimyk, (P, Q)-Differentiation, (P, Q) integration, and (P, Q)-hypergeometric functions related to quantum groups, Integr. Transf. Spec. F., 2 (1994), 15-36. doi: 10.1080/10652469408819035
    [12] M. Mursaleen, K. J. Ansari, A. Khan, Erratum to "On (p, q)-analogue of Bernstein Operators" [Appl. Math. Comput. 266 (2015) 874-882], Appl. Math. Comput., 278 (2016), 70-71.
    [13] P. N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, Results Math., 73 (2018), 1-21. doi: 10.1007/s00025-018-0773-1
    [14] M. Nasiruzzaman, A. Mukheimer, M. Mursaleen, Some Opial-type integral inequalities via (p, q)- calculus, J. Inequal. Appl., 2019 (2019), 1-11. doi: 10.1186/s13660-019-1955-4
    [15] M. N. Hounkonnou, J. D. B. Kyemba, R(p, q)-calculus: differentiation and integration, SUT J. Math., 49 (2013), 145-167.
    [16] M. Mursaleen, M. Nasiruzzaman, A. Khan, et al. Some approximation results on Bleimann-ButzerHahn operators defined by (p, q)-integers, Filomat, 30 (2016), 639-648. doi: 10.2298/FIL1603639M
    [17] N. Rao, A. Wafi, Bivariate-Schurer-Stancu operators based on (p, q)-integers, Filomat, 32 (2018), 1251-1258. doi: 10.2298/FIL1804251R
    [18] M. Mursaleen, M. Nasiruzzaman, K. J. Ansari, et al. Generalized (p, q)-Bleimann-Butzer-Hahn operators and some approximation results, J. Inequal. Appl., 2017 (2017), 1-14. doi: 10.1186/s13660-016-1272-0
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