Rad HAZU, Matematičke znanosti, Vol. 22 (2018), 93-106.

INEQUALITIES VIA (p,r)-CONVEX FUNCTIONS

Muhammad Aslam Noor, Khalida Inayat Noor and Sabah Iftikhar

Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan
e-mail: khalidan@gmail.com

Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan
e-mail: sabah.iftikhar22@gmail.com

Abstract.   The main aim of this paper is to introduce a new class of convex functions, which is called (p,r)-convex functions. We derive some new Hermite-Hadamard type integral inequalities via (p,r)-convex functions. Some special cases are also discussed. These results can be considered as significant improvement of the known results. The technique and ideas of this paper may motivate further research.

2010 Mathematics Subject Classification.   26D15, 26A51.

Key words and phrases.   r-convex functions, p-convex functions, Hermite-Hadamard type inequality.

DOI: https://doi.org/10.21857/m8vqrtz169

References:

1. G. Cristescu and L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, Holland, 2002.
   MathSciNet     CrossRef

2. P. M. Gill, C. E. M. Pearce and J. Pečarić, Hadamard’s inequality for r-convex functions, J. Math. Anal. Appl. 215 (1997), 461–470.
   MathSciNet     CrossRef

3. J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier dune fonction consideree par Riemann, J. Math. Pure Appl. 58 (1893), 171–215.

4. L. V. Hap and N. V. Vinh, On some Hadamard-type inequalities for (h,r)-convex functions, Int. J. Math. Anal. (Ruse) 7(42) (2013), 2067–2075.
   MathSciNet

5. I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe J. Math. Stats. 43 (2014), 935–942.
   MathSciNet

6. N. P. N. Ngoc, N. V. Vinh and P. T. T. Hien, Integral inequalities of Hadamard type for r-convex functions, Int. Math. Forum 4(35) (2009), 1723–1728.
   MathSciNet

7. C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2006.
   MathSciNet     CrossRef

8. M. A. Noor, K. I. Noor and S. Iftikhar, Nonconvex functions and integral inequalities, Punjab Univ. J. Math. (Lahore) 47 (2015), 19–27.
   MathSciNet

9. M. A. Noor, K. I. Noor and U. Awan, Some new estimates of Hermite-Hadamard inequalities via harmonically r-convex functions, Matematiche (Catania) 71 (2016), 117–127.
   MathSciNet

10. M. A. Noor, K. I. Noor and S. Iftikhar, Hermite-Hadamard inequalities for harmonic nonconvex functions, MAGNT Research Report 4 (2016), 24–40.

11. M. A. Noor, K. I. Noor and S. Iftikhar, On harmonic (h,r)-convex functions, Proc. Jangjeon Math. Soc. 21 (2018), 239-251.
    MathSciNet

12. M. A. Noor, K. I. Noor and S. Iftikhar, Integral inequalities for differentiable pharmonic convex functions, Filomat 31 (2017), 6575-6584.
    MathSciNet     CrossRef

13. C. E. M. Pearce, J. Pečarić and V. Šimić, Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl. 220 (1998), 99–109.
    MathSciNet     CrossRef

14. C. E. M. Pearce, J. Pečarić and V. Šimić, On weighted generalized logarithmic means, Houston J. Math. 24 (1998), 459–465.
    MathSciNet

15. J. Park, On the Hermite-Hadamard-like type inequalities for co-ordinated (s,r)-convex mappings, Int. J. Pure Appl. Math. 74(2) (2012), 251-263.

16. J. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.
    MathSciNet

17. M. Z. Sarikaya, H. Yaldiz and H. Bozkurt, On the Hadamard type integral inequalities involving several differentiable (φ-r)-convex functions, 2012, arXiv:1203.2278.

18. K. S. Zhang and J. P. Wan, p-convex functions and their properties, Pure Appl. Math. (Xi'an) 23 (2007), 130–133.
    MathSciNet