Generalized r-Convex Functions and Integral Inequalities

Main Article Content

Muhammad Aslam Noor
Khalida Inayat Noor
Farhat Safdar

Abstract

In this paper, we introduce and investigate a new class of generalized convex functions, known as generalized $r$-convex function. Some new Hermite-Hadamard integral inequalities via generalized $r$-convex functions have been established. Results proved in this paper can be viewed as significant new contributions in this area of research.

Article Details

References

  1. G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl, 335(2007),1294-1308.
  2. M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Simpson's type for s-convex functions with applications, RGMIA Res. Rep. Coll, 12 (4) (2009).
  3. G. Cristescu, L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publishers, Dordrechet, Holland, (2002).
  4. M. R. Delavar and S. S. Dragomir, On η-convexity, Math. Inequal. Appl., 20(1)(2017), 203-216.
  5. S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Victoria University, Australia, (2000).
  6. M. E. Gordji, M. R. Delavar and M. D. Sen, On φ convex functions, J. Math. Inequal, 10(1)(2016), 173-183.
  7. M. E. Gordji, M. R. Delavar and S. S. Dragomir, An inequality related to η-convex functions (II), Int. J. Nonlinear. Anal. Appl, 6(2)(2015), 27-33.
  8. P. M. Gill, C. E. M. Pearce , J. Pecaric, Hadamards inequality for r-convex functions, J. Math. Anal. Appl, 215(1997), 461-470.
  9. J. Hadamard, Etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par Riemann, J. Math. Pure. Appl., 58(1893), 171-215.
  10. C. Hermite, Sur deux limites d'une integrale definie, Mathesis, 3(1883), 82.
  11. D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc, 3(1952), 821-828.
  12. C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications. Springer-Verlag, New York, (2006).
  13. M. A. Noor, General variational inequalities, Appl. Math. Letters,1(1988), 119-121.
  14. M. A. Noor, Some develpments in general variational inequalities, Appl. Math. Comput. 152(2004), 199-277.
  15. M. A. Noor and K. I. Noor, Harmonic variational inequalities, Appl. Math. Inform. Sci. 10(5)(2016), 1811-1814.
  16. M. A. Noor and K. I. Noor, Some implicit methods for solving harmonic variational inequalities , Inter. J. Anal. Appl. 12(1)(2016), 10-14.
  17. M. A. Noor, K. I. Noor and M. U. Awan, Some new estimates of Hermite-Hadamard inequalities via harmonically r-convex functions, Le Mathematiche, LXXI(II)(2016), 117-127.
  18. M. A. Noor, K. I. Noor, M. U. Awan and F. Safdar, On strongly generalized convex functions, Filomat, 31(18)(2017), 5783-5790.
  19. M. A. Noor, K. I. Noor and F. Safdar, Generalized geometrically convex functions and inequalities, J. Inequal. Appl, 2017(2017), Article ID 22.
  20. M. A. Noor, K. I. Noor and F. Safdar, Integral inequaities via generalized convex functions, J. Math. Computer, Sci, 17(4)(2017), 465-476.
  21. M. A. Noor, K. I. Noor, S. Iftikhar, F. Safdar, Integral inequaities for relative harmonic (s,η)-convex functions, Appl. Math. Comp. Sci, 1(1)(2015), 27-34.
  22. M. A. Noor, K. I. Noor and F. Safdar, Integral inequaities via generalized (α,m)-convex functions, J. Nonlinear. Funct. Anal, 2017(2017), Article ID 32.
  23. M. A. Noor, K. I. Noor and F. Safdar, New inequalities for generalized log h-convexd functions, J. Appl. Math. Inform. 36(3-4)(2018), 245-256.
  24. M. A. Noor, K. I. Noor and F. Safdar,Inequalities via generalized beta m-convex functions, J. Math. Anal. 9((2018).
  25. M. A. Noor, K. I. Noor and S. Iftikhar, Inequaities via (p,r)-convex functions, RAD, (2018).
  26. M. A. Noor, K. I. Noor and S. Iftikhar, On harmonic (h,r)-convex functions, Proced. Jangj. Math. Soc. 21(2)(2018), 239-251.
  27. M. A. Noor, K. I. Noor, F. Safdar, M. U. Awan and S. Ullah, Inequaities via generalized log m-convex functions, J. Nonlinear. Sci. Appl, 10(2017), 5789-5802.
  28. M. A. Noor, K. I. Noor, S. Iftikhar and F. Safdar, Generalized (h,r)-harmonic convex functions and inequalities, Int. J. Math. Anal. 16(4)(2018).
  29. N. P. N. Ngoc, N.V. Vinh, P. T. T. Hien, Integral inequalities of Hadamard type for r-Convex functions, Int. Math. Forum, 4(35)(2009), 1723-1728.
  30. J. Pecaric, F. Proschan and Y. T. Tong, Convex Functions, Partial Ordering and Statistical Applications, Academic Pres, New York, (1992).
  31. G. S. Yang, Refinement of Hadamard's inequality for r-convex functions, Indian J. Pure Appl. Math. 32(10)(2001), 1571- 1579.