AIMS Mathematics, 2020, 5(6): 7122-7144. doi: 10.3934/math.2020456

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Some New $(p_1p_2,q_1q_2)$-Estimates of Ostrowski-type integral inequalities via n-polynomials s-type convexity

1 School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China
2 Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China
3 Department of Mathematics, Faculty of Technical Science, University “Ismail Qemali”, Vlora, Albania
4 Department of mathematics, Government University, Faisalabad, Pakistan
5 Department of Mathematics, Huzhou University, Huzhou 313000, China
6 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

The purpose of this paper is to establish new generalization of Ostrowski type integral inequalities by using $(p,q)$-analogues which are related to the estimates of upper bound for a class of $(p_1p_2,q_1q_2)$-differentiable functions on co-ordinates. We first establish an integral identity for $(p_1p_2,q_1q_2)$-differentiable functions on co-ordinates. The result is then used to derive some estimates of upper bound for the functions whose twice partial $(p_1p_2,q_1q_2)$-differentiable functions are $n$-polynomial $s$-type convex functions on co-ordinates. Some new special cases from the main results are obtained and some known results are recaptured as well. At the end, an application to special means is given as well.
  Article Metrics


1. F. H. Jackson, On a q-definite integrals, Q. J. Pure Appl. Math., 41 (1910), 193-203.

2. M. Adil Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20.    

3. L. Xu, Y. M. Chu, S. Rashid, et al. On new unified bounds for a family of functions with fractional q-calculus theory, J. Funct. Space., 2020 (2020), 1-9.

4. S. Rashid, A. Khalid, G. Rahman, et al. On new modifications governed by quantum Hahn's integral operator pertaining to fractional calculus, J. Funct. Space., 2020 (2020), 1-12.

5. J. M. Shen, S. Rashid, M. A. Noor, et al. Certain novel estimates within fractional calculus theory on time scales, AIMS Math., 5 (2020), 6073-6086.    

6. H. Kalsoom, M. Idrees, D. Baleanu, et al. New estimates of q1q2-Ostrowski-type inequalities within a class of n-polynomial prevexity of function, J. Funct. Space., 2020 (2020), 1-13.

7. M. U. Awan, S. Talib, A. Kashuri, et al. Estimates of quantum bounds pertaining to new q-integral identity with applications, Adv. Differ. Equ., 2020 (2020), 1-15.    

8. J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 1-13.    

9. J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013 (2013), 1-19.    

10. M. A. Noor, M. U. Awan, K. I. Noor, Quantum Ostrowski inequalities for q-differentiable convex functions, J. Math. Inequal., 10 (2016), 1013-1018.

11. H. Kalsoom, J. D. Wu, S. Hussain, et al. Simpson's type inequalities for co-ordinated convex functions on quantum calculus, Symmetry, 11 (2019), 1-16.

12. Y. P. Deng, M. U. Awan, S. H. Wu, et al. Quantum integral inequalities of Simpson-type for Strongly preinvex functions, Mathematics, 7 (2019), 1-14.

13. Y. P. Deng, H. Kalsoom, S. H. Wu, Some new quantum Hermite-Hadamard-type estimates within a class of generalized (s, m)-preinvex functions, Symmetry, 11 (2019), 1-15.

14. M. Tunç, E. Göv, (p, q)-Integral inequalities, RGMIA Res. Rep. Coll., 19 (2016), 1-13.

15. S. Rashid, İ. İşcan, D. Baleanu, et al. Generation of new fractional inequalities via n polynomials s-type convexixity with applications, Adv. Differ. Equ., 2020 (2020), 1-20.    

16. Y. Khurshid, M. A. Khan, Y. M. Chu, Conformable fractional integral inequalities for GG- and GA-convex function, AIMS Math., 5 (2020), 5012-5030.    

17. T. Abdeljawad, S. Rashid, H. Khan, et al. On new fractional integral inequalities for p-convexity within interval-valued functions, Adv. Differ. Equ., 2020 (2020), 1-17.    

18. M. B. Sun, Y. M. Chu, Inequalities for the generalized weighted mean values of g-convex functions with applications, RACSAM, 114 (2020), 1-12.    

19. S. Y. Guo, Y. M. Chu, G. Farid, et al. Fractional Hadamard and Fejér-Hadamard inequaities associated with exponentially (s, m)-convex functions, J. Funct. Space., 2020 (2020), 1-10.

20. M. U. Awan, S. Talib, M. A. Noor, et al. Some trapezium-like inequalities involving functions having strongly n-polynomial preinvexity property of higher order, J. Funct. Space., 2020 (2020), 1-9.

21. T. Abdeljawad, S. Rashid, Z. Hammouch, et al. Some new local fractional inequalities associated with generalized (s, m)-convex functions and applications, Adv. Differ. Equ., 2020 (2020), 1-27.

22. Y. Khurshid, M. Adil Khan, Y. M. Chu, Conformable integral version of Hermite-Hadamard-Fejér inequalities via η-convex functions, AIMS Math., 5 (2020), 5106-5120.    

23. M. U. Awan, N. Akhtar, A. Kashuri, et al. 2D approximately reciprocal ρ-convex functions and associated integral inequalities, AIMS Math., 5 (2020), 4662-4680.    

24. Y. M. Chu, M. U. Awan, M. Z. Javad, et al. Bounds for the remainder in Simpson's inequality via n-polynomial convex functions of higher order using Katugampola fractional integrals, J. Math., 2020 (2020), 1-10.

25. P. Y. Yan, Q. Li, Y. M. Chu, et al. On some fractional integral inequalities for generalized strongly modified h-convex function, AIMS Math., 5 (2020), 6620-6638.    

26. S. S. Zhou, S. Rashid, F. Jarad, et al. New estimates considering the generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 2020 (2020), 1-15.    

27. S. Hussain, J. Khalid, Y. M. Chu, Some generalized fractional integral Simpson's type inequalities with applications, AIMS Math., 5 (2020), 5859-5883.    

28. G. J. Hai, T. H. Zhao, Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function, J. Inequal. Appl., 2020 (2020), 1-17.    

29. J. M. Shen, Z. H. Yang, W. M. Qian, et al. Sharp rational bounds for the gamma function, Math. Inequal. Appl., 23 (2020), 843-853.

30. M. K. Wang, H. H. Chu, Y. M. Li, et al. Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind, Appl. Anal. Discrete Math., 14 (2020), 255- 271.

31. M. K. Wang, Y. M. Chu, Y. M. Li, et al. Asymptotic expansion and bounds for complete elliptic integrals, Math. Inequal. Appl., 23 (2020), 821-841.

32. T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Math., 5 (2020), 4512-4528.    

33. M. Adil Khan, J. Pečarić, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Math., 5 (2020), 4931-4945.    

34. S. Rashid, R. Ashraf, M. A. Noor, et al. New weighted generalizations for differentiable exponentially convex mapping with application, AIMS Math., 5 (2020), 3525-3546.    

35. M. A. Khan, M. Hanif, Z. A. Khan, et al. Association of Jensen's inequality for s-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1-14.    

36. H. Ge-JiLe, S. Rashid, M. A. Noor, et al. Some unified bounds for exponentially tgs-convex functions governed by conformable fractional operators, AIMS Math., 5 (2020), 6108-6123.    

37. I. Abbas Baloch, A. A. Mughal, Y. M. Chu, et al. A variant of Jensen-type inequality and related results for harmonic convex functions, AIMS Math., 5 (2020), 6404-6418.    

38. Z. H. Yang, W. M. Qian, W. Zhang, et al. Notes on the complete elliptic integral of the first kind, Math. Inequal. Appl., 23 (2020), 77-93.

39. A. Iqbal, M. A. Khan, N. Mohammad, et al. Revisiting the Hermite-Hadamard integral inequality via a Green function, AIMS Math., 5 (2020), 6087-6107.    

40. M. U. Awan, N. Akhtar, S. Iftikhar, et al. New Hermite-Hadamard type inequalities for npolynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1-12.    

41. M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 1-33.    

42. H. X. Qi, M. Yussouf, S. Mehmood, et al. Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity, AIMS Math., 5 (2020), 6030-6042.    

43. X. Z. Yang, G. Farid, W. Nazeer, et al. Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex function, AIMS Math., 5 (2020), 6325-6340.    

44. S. Rashid, M. A. Noor, K. I. Noor, et al. Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions, AIMS Math., 5 (2020), 2629- 2645.

45. S. S. Dragomir, J. Pečarić, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.

46. T. Toplu, M. Kadakal, İ. İşcan, On n-polynomial convexity and some related inequalities, AIMS Math., 5 (2020), 1304-1318.    

47. M. A. Latif, S. Hussain, S. S. Dragomir, New Ostrowski type inequalities for co-ordinated convex functions, Transylv. J. Math. Mech., 4 (2012), 125-136.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved