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Study of Quantum Ostrowski-Type Inequalities for Differentiable Convex Functions

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Ukrainian Mathematical Journal Aims and scope

We prove some new q-Ostrowski-type inequalities for differentiable and bounded functions. Moreover, we present the relationship between the newly established and already known inequalities, which is very interesting for new readers. Some applications to special means of real numbers are given to make the results more valuable.

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References

  1. M. A. Ali, H. Budak, M. Abbas, and Y.-M. Chu, “Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second q𝔶2-derivatives,” Adv. Difference Equat., 2021, 1–12 (2021).

    MathSciNet  MATH  Google Scholar 

  2. M. A. Ali, H. Budak, Z. Zhang, and H. Yildrim, “Some new Simpson’s type inequalities for co-ordinated convex functions in quantum calculus,” Math. Meth. Appl. Sci., 44, 4515–4540 (2021).

    Article  MATH  Google Scholar 

  3. M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza, and Y.-M. Chu, “New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions,” Adv. Difference Equat., 2021, 1–21 (2021).

    MathSciNet  MATH  Google Scholar 

  4. M. A. Ali, Y.-M. Chu, H. Budak, A. Akkurt, and H. Yildrim, “Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables,” Adv. Difference Equat., 2021, 1–26 (2021).

    MathSciNet  MATH  Google Scholar 

  5. M. A. Ali, N. Alp, H. Budak, Y.-M. Chu, and Z. Zhang, “On some new quantum midpoint type inequalities for twice quantum differentiable convex functions,” Open Math., 19, 427–439 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. A. Ali, H. Budak, A. Akkurt, and Y.-M. Chu, “Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus,” Open Math., 19, 440–449 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. A. Ali, H. Budak, K. Nanlaopon, and Z. Abdullah, “Simpson’s and Newton’s inequalities for (α, m)-convex functions via quantum calculus,” Symmetry Quantum Calc. (2021).

  8. N. Alp, M. Z. Sarikaya, M. Kunt, and İ. İşcan, “q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions,” J. King Saud Univ. Sci., 30, 193–203 (2018).

    Article  MATH  Google Scholar 

  9. N. Alp and M. Z. Sarikaya, “Hermite Hadamard’s type inequalities for co-ordinated convex functions on quantum integral,” Appl. Math. E-Notes, 20, 341–356 (2020).

    MathSciNet  MATH  Google Scholar 

  10. S. Bermudo, P. Kórus, and J. N. Valdés, “On q-Hermite–Hadamard inequalities for general convex functions,” Acta Math. Hungar., 162, 364–374 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Brahim, S. Taf, and L. Rihahi, “Some result for Hadamard-type inequalities in quantum calculus,” Matematiche (Catania), 69, 243–258 (2014).

    MathSciNet  MATH  Google Scholar 

  12. H. Budak, “Some trapezoid and midpoint type inequalities for newly defined quantum integrals,” Proyecciones, 40, 199–215 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Budak, M. A. Ali, and M. Tarhanaci, “Some new quantum Hermite–Hadamard-like inequalities for coordinated convex functions,” J. Optim. Theory Appl., 186, 899–910 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Budak, S. Erden, and M. A. Ali, “Simpson and Newton type inequalities for convex functions via newly defined quantum integrals,” Math. Meth. Appl. Sci., 44, 378–390 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Budak, M. A. Ali, N. Alp, and Y.-M. Chu, “Quantum Ostrowski type integral inequalities,” J. Math. Inequal. (2021) (to appear).

  16. P. Cerone and S. S. Dragomir, “Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions,” Demonstr. Math., 37, 299–308 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Kavurmaci, M. Avci, and M. E. O¨ zdemir, “New inequalities of Hermite–Hadamard type for convex functions with applications,” J. Inequal. Appl., 2011, Article 86 (2011).

  18. F. H. Jackson, “On q-definite integrals,” Quart. J. Pure Appl. Math., 41, 193–203 (1910).

    MATH  Google Scholar 

  19. S. Jhanthanam, J. Tariboon, S. K. Ntouyas, and K. Nonlaopon, “On q-Hermite–Hadamard inequalities for differentiable convex functions,” Mathematics, 7, Article 632 (2019).

  20. V. Kac and P. Cheung, Quantum Calculus, Springer-Verlag, New York (2001).

    MATH  Google Scholar 

  21. H. Kalsoom, J.-D. Wu, S. Hussain, and M. A. Latif, “Simpson’s type inequalities for co-ordinated convex functions on quantum calculus,” Symmetry, 11, Article 768 (2019).

  22. Z. Liu, “A note on Ostrowski type inequalities related to some s-convex functions in the second sense,” Bull. Korean Math. Soc., 49, 775–785 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  23. W. Liu and Z. Hefeng, “Some quantum estimates of Hermite–Hadamard inequalities for convex functions,” J. Appl. Anal. Comput., 7, 501–522 (2016).

    MathSciNet  MATH  Google Scholar 

  24. M. A. Noor, K. I. Noor, and M. U. Awan, “Some quantum estimates for Hermite–Hadamard inequalities,” Appl. Math. Comput., 251, 675–679 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  25. M. A. Noor, K. I. Noor, and M. U. Awan, “Some quantum integral inequalities via preinvex functions,” Appl. Math. Comput., 269, 242–251 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Ostrowski, “Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert,” Comment.Math. Helv., 10, 226–227 (1938).

    Article  MATH  Google Scholar 

  27. I. B. Sial, S. Mei, M. A. Ali, and K. Nanlaopon, “On some generalized Simpson’s and Newton’s inequalities for (α, m)-convex functions in q-calculus,” Mathematics, 2021, Article 3266 (2021).

  28. J. Tariboon and S. K. Ntouyas, “Quantum calculus on finite intervals and applications to impulsive difference equations,” Adv. Difference Equat., 2013, 1–19 (2013).

    MathSciNet  MATH  Google Scholar 

  29. M. Vivas-Cortez, M. A. Ali, A. Kashuri, I. B. Sial, and Z. Zhang, “Some new Newton’s type integral inequalities for co-ordinated convex functions in quantum calculus,” Symmetry, 12, Article 1476 (2020).

  30. M. Vivas-Cortez, A. Kashuri, R. Liko, and J. E. Hernádez, “Quantum estimates of Ostrowski inequalities for generalized ϕ-convex functions,” Symmetry, 11, Article 1513 (2019).

  31. P. P. Wang, T. Zhu, and T. S. Du, “Some inequalities using s-preinvexity via quantum calculus,” J. Interdiscipl. Math., 24, 613–636 (2021).

    Article  Google Scholar 

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Authors and Affiliations

  1. Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China

    M. A. Ali

  2. Comenius University in Bratislava and Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia

    M. Fečkan

  3. COMSATS University Islamabad Sahiwal Campus, Sahiwal, Pakistan

    A. Mateen

Authors
  1. M. A. Ali
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  2. M. Fečkan
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  3. A. Mateen
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Corresponding author

Correspondence to M. Fečkan.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 1, pp. 7–27, January, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i1.7313.

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Ali, M.A., Fečkan, M. & Mateen, A. Study of Quantum Ostrowski-Type Inequalities for Differentiable Convex Functions. Ukr Math J 75, 5–28 (2023). https://doi.org/10.1007/s11253-023-02182-x

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  • DOI: https://doi.org/10.1007/s11253-023-02182-x

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