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\(\left( p,q\right) \)-Hermite–Hadamard inequalities and \(\left( p,q\right) \)-estimates for midpoint type inequalities via convex and quasi-convex functions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we prove the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, some new \(\left( p,q\right) \)-Hermite–Hadamard inequalities, and generalized \(\left( p,q\right) \)-Hermite–Hadamard inequality. By using the left hand part of the correct \(\left( p,q\right) \)-Hermite–Hadamard inequality, we have a new equality. Finally using the new equality, we give some \(\left( p,q\right) \)-midpoint type integral inequalities through \(\left( p,q\right) \)-differentiable convex and \(\left( p,q\right) \)-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works.

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References

  1. Alomari, M., Darus, M., Dragomir, S.S.: Inequalities of Hermite–Hadamard’s type for functions whose derivatives absolute values are quasi-convex. RGMIA Res. Rep. Coll. 12(Supplement), 1–11

  2. Alp, N., Sarıkaya, M.Z., Kunt, M., İşcan, İ.: q-Hermite–Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud Univ. Sci. doi:10.1016/j.jksus.2016.09.007

    Article  Google Scholar 

  3. Ernst, T.: A Comprehensive Treatment of \(q\)-Calculus. Springer, Basel (2012)

  4. Gauchman, H.: Integral inequalities in \(q\)-calculus. Comput. Math. Appl. 47, 281–300 (2004)

    Article  MathSciNet  Google Scholar 

  5. Jagannathan, R., Srinivasa, R.K.: Tow-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series (2006). arXiv:math/0602613v

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

    MATH  Google Scholar 

  7. Kırmacı, U.S.: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147, 137–146 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Kac, V., Cheung, P.: Quantum calculus. Springer, Berlin, New York (2001)

    MATH  Google Scholar 

  9. Latif, M.A., Kunt, M., Dragomir, S.S., İşcan, İ.: Some (p,q) estimates for Hermite-Hadamard inequalities via convex and quasi-convex functions. doi:10.13140/RG.2.1.1280.2806

  10. Liu, W., Zhuang, H.: Some quantum estimates of Hermite–Hadamard inequalities for convex functions. J. Appl. Anal. Comput. 7(2), 501–522 (2017)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  13. Sadjang, P.N.: On the fundamental theorem of \(\left( p,q\right) \)-calculus and some \(\left( p,q\right) \)-Taylor formulas (2013). arXiv:1309.3934v1

  14. Sudsutad, W., Ntouyas, S.K., Tariboon, J.: Quantum integral inequalities for convex functions. J. Math. Inequal. 9(3), 781–793 (2015)

    Article  MathSciNet  Google Scholar 

  15. Tunç, M., Göv, E.: \(\left( p,q\right) \)-Integral inequalities. RGMIA Res. Rep. Coll. 19, 1–13, Art. 97 (2016)

  16. Tunç, M., Göv, E.: Some integral inequalities via \(\left( p,q\right) \)-calculus on finite intervals. RGMIA Res. Rep. Coll. 19, 1–12, Art. 95 (2016)

  17. Tunç, M., Göv, E.: \(\left( p,q\right) \)-integral inequalities for convex functions. RGMIA Res. Rep. Coll. 19, 1–12, Art. 98 (2016)

  18. Tariboon, J., Ntouyas, S.K.: Quantum integral inequalities on finite intervals. J. Inequal. Appl. 121, 1–13 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Tariboon, J., Ntouyas, S.K.: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 282, 1–19 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Zhuang, H., Liu, W., Park, J.: Some quantum estimates of Hermite–Hadmard inequalities for quasi-convex functions. Miskolc Math. Notes 17(2)

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey

    Mehmet Kunt

  2. Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200, Giresun, Turkey

    İmdat İşcan

  3. Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

    Necmettin Alp & Mehmet Zeki Sarıkaya

Authors
  1. Mehmet Kunt
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  2. İmdat İşcan
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  3. Necmettin Alp
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  4. Mehmet Zeki Sarıkaya
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Correspondence to Mehmet Kunt.

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The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Kunt, M., İşcan, İ., Alp, N. et al. \(\left( p,q\right) \)-Hermite–Hadamard inequalities and \(\left( p,q\right) \)-estimates for midpoint type inequalities via convex and quasi-convex functions. RACSAM 112, 969–992 (2018). https://doi.org/10.1007/s13398-017-0402-y

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  • DOI: https://doi.org/10.1007/s13398-017-0402-y

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