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Progress in Approximation Theory and Applicable Complex Analysis pp 237–268Cite as

Quantum Integral Inequalities for Generalized Preinvex Functions

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 117))

Abstract

We consider the generalized preinvex functions, which unify the preinvex and φ-convex functions. We give an account of the quantum integral inequalities via the generalized preinvex functions. Results obtained in this chapter represent significant and important refinements of the known results. These inequalities involve Riemann-type quantum integrals. We would like to emphasize that these results reduce to classical results, when q → 1. It is expected that ideas and techniques given here would inspire further research.

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References

  1. Antczak, T.: Mean value in invexity analysis. Nonl. Anal. 60, 1473–1484 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Israel, A., Mond, B.: What is invexity? J. Aust. Math. Soc. 28, 1–9 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cazacu, C.A., Lehto, O.E., Rassias, T.M. (eds.) Analysis and Topology. World Scientific, Singapore, New Jersey, London (1998)

    Google Scholar 

  4. Cristescu, G., Lupsa, L.: Non-connected Convexities and Applications. Kluwer, Dordrecht, Holland (2002)

    Book  MATH  Google Scholar 

  5. Cristescu, G., Noor, M.A., Awan, M.U.: Bounds of the second degree cumulative frontier gaps of functions with generalized convexity. Carpathian J. Math. 31 (2), 173–180 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Daras, N.J., Rassias, M.T. (eds.) Computation, Cryptography, and Network Secuirty. Springer, New York (2015)

    Google Scholar 

  7. Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11 (5), 91–95 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite-Hadamard Inequalities and Applications. Victoria University, Newport, VIC (2000)

    Google Scholar 

  9. Dragomir, S.S., Rassias, T.M.: Ostrowski Type Inequalities and Applications in Numerical Integration. Springer, Netherlands (2002)

    Book  MATH  Google Scholar 

  10. Ernst, T.: A Comprehensive Treatment of q-Calculus. Springer, Basel, Heidelberg, New York, Dordrecht, London (2014)

    MATH  Google Scholar 

  11. Floudas, C.A., Rassias, T.M. (eds.) Optimization in Science and Engineering: In Honor of the 60th Birth of Panos M. Pardalos. Springer, New York (2014)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Gordji, M.E., Delavar, M.R., Sen, M.D.L.: On φ-convex functions. J. Math. Inequal. (2016, accepted)

    Google Scholar 

  14. Hanson, M.A., On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hyers, D.H., Isac, G., Rassias, T.M.: Topics in Nonlinear Analysis and Applications, World Scientific, Singapore, New Jersey, London (1997)

    Book  MATH  Google Scholar 

  16. Ion, D.A.: Some estimates on the Hermite-Hadamard inequality through quasi-convex functions. Annals of University of Craiova. Math. Comp. Sci. Ser. 34, 82–87 (2007)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  18. Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    Book  MATH  Google Scholar 

  19. Mohan, S.R., Neogy, S.K.: On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Niculescu, C.P., Persson, L.-E.: Convex Functions and their Applications. A Contemporary Approach. Springer, New York (2006)

    Book  MATH  Google Scholar 

  21. Noor, M.A.: Variational-like inequalities. Optimization 30, 323–330 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Noor, M.A.: Invex equilibrium problems. J. Math. Anal. Appl. 302, 463–475 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Noor, M.A.: Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2, 126–131 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Noor, M.A.: Advanced Convex Analysis. Lecture Notes. COMSATS Institute of Information Technology, Islamabad (2008–2015)

    Google Scholar 

  25. Noor, M.A., Noor, K.I., Awan, M.U.: Generalized convexity and integral inequalities. Appl. Math. Inf. Sci. 9 (1), 233–243 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Noor, M.A., Noor, K.I., Awan, M.U.: Quantum analogues of Hermite-Hadamard type inequalities for generalized convexity. In: Daras, N., Rassias, M.T. (eds.) Computation, Cryptography and Network Security. Springer, Berlin (2015)

    Google Scholar 

  27. 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 

  28. 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 

  29. Noor, M.A., Noor, K.I., Awan, M.U., Safdar, F.: Quantum integral inequalities via φ-convex functions. Quantum Infor. Rev. 4 (1), 1–7 (2016)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Noor, M.A., Noor, K.I., Awan, M.U.: Some quantum analogues of integral inequalities. Preprint (2016)

    Google Scholar 

  32. Ozdemir, M.E.: On Iyengar-type inequalities via quasi-convexity and quasi-concavity (2012). Arxiv:1209.2574v1 [math.FA]

    Google Scholar 

  33. Pearce, C.E.M., Pecaric, J.E.: Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 13, 51–55 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pecaric, J.E., Prosch, F., Tong, Y.L.: Convex Functions, Partial Orderings, and Statistical Applications. Academic, New York (1992)

    Google Scholar 

  35. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  36. Rassias, T.M. (ed.) Global Analysis on Manifolds. Dedicated to Marston Morse, 1892–1976. Teubner-Texte zur Mathematik, Band 57. Teubner, Leipzig (1983)

    Google Scholar 

  37. Rassias, T.M. (ed.) Nonlinear Analysis. World Scientific, Singapore, New Jersey, London (1987)

    MATH  Google Scholar 

  38. Rassias, T.M. (ed.) Constantin Caratheodory: An International Tribute (with Foreword by Lars V. Ahlfors). World Scientific, Singapore, New Jersey, London (1991)

    Google Scholar 

  39. Rassias, T.M., Pardalos, P.M. (eds.) Mathematics Without Boundaries: Surveys in Pure Mathematics. Springer, New York (2014)

    MATH  Google Scholar 

  40. Rassias, G.M., Rassias, T.M. (eds.) Differential Geometry, Calculus of variations, and Their Applications. Dedicated to Leonhard Euler. Dekker, New York and Basel (1985)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  44. Weir, T., Mond, J.: Preinvex functions in multiobjective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  45. Yang, X.M., Yang, X.Q., Teo, K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory. Appl. 117, 607–625 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their sincere gratitude to Dr. S. M. Junaid Zaidi (H.I., S. I. ), Rector, COMSATS Institute of Information Technology, Pakistan, for providing excellent research and academic environment. The authors would also like to express their sincere gratitude to the referee for his constructive suggestions and interest.

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Authors and Affiliations

  1. COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

    Muhammad Aslam Noor & Khalida Inayat Noor

  2. Department of Mathematics, National Technical University of Athens, Zografou Campus, GR-15780, Athens, Greece

    Themistocles M. Rassias

  3. Government University College, Faisalabad, Pakistan

    Muhammad Uzair Awan

Authors
  1. Muhammad Aslam Noor
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  2. Themistocles M. Rassias
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  3. Khalida Inayat Noor
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  4. Muhammad Uzair Awan
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Corresponding author

Correspondence to Themistocles M. Rassias .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, USA

    Narendra Kumar Govil

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    Ram Mohapatra

  3. Department of Mathematics, Tuskegee University, Tuskegee, Alabama, USA

    Mohammed A. Qazi

  4. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Noor, M.A., Rassias, T.M., Noor, K.I., Awan, M.U. (2017). Quantum Integral Inequalities for Generalized Preinvex Functions. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_12

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