Integral Inequalities via Generalized Geometrically r-Convex Functions

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, called generalized geometrically r-convex functions. Some new Hermite-Hadamard integral inequalities via generalized geometrically r-convex functions have been established. Results proved in this paper can be viewed as new significant 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. C. Baiochi and A. Capelo, Variational and Quas-Variational Inequalities, Wiley, New York, (1984).
  4. J. Crank, Free and Moving Boundary Problems, Clarendon Press, Osford, UK, (1984).
  5. G. Cristescu, L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publishers, Dordrechet, Holland,(2002).
  6. M. R. Delavar and S. S. Dragomir, On η-convexity, Math. Inequal. Appl, 20(1)(2017), 203-216.
  7. S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Victoria University, Australia, (2000).
  8. R. Glowinski, J. L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, (1981).
  9. M. E. Gordji, M. R. Delavar and M. D. Sen, On φ convex functions, J. Math. Inequal, 10(1)(2016), 173-183.
  10. 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.
  11. P. M. Gill, C. E. M. Pearce , J. Pecaric, Hadamards inequality for r-convex functions, J. Math. Anal. Appl, 215(1997), 461470.
  12. 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.
  13. C. Hermite, Sur deux limites d'une integrale definie, Mathesis, 3(1983), 82.
  14. D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc, 3(1952), 821-828.
  15. C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications. Springer Verlag, New York, (2006).
  16. M. A. Noor, On Variational Inequalities, PhD Thesis, Brunel University, London, UK, (1975).
  17. M. A. Noor, General variational inequalities, Appl. Math. Letters, 1(1988), 119-121.
  18. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl, 251(2000), 217-230.
  19. M. A. Noor, Some developments in general variational inequalites, Appl. Math. Comput. 152(2004), 199-277.
  20. M. A. Noor and K. I. Noor, Harmonic variational inequalities, Appl. Math. Inform. Sci. 10(5)(2016), 1811-1814.
  21. M. A. Noor, K. I. Noor and Th. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47(1993), 285-312.
  22. M. A. Noor, K. I. Noor and M. U. Awan, Some new estimates of Hermite-Hadamard inequalities via harmonically convex functions, Le Mathematiche, LXXI(II)(2016), 117-127.
  23. M. A. Noor, K. I. Noor, M. U. Awan and F. Safdar, On strongly generalized convex functions, Filomat, 31(18)(2017), 5783-5790.
  24. M. A. Noor, K. I. Noor and F. Safdar, Generalized geometrically convex functions and inequalities, J. Inequal. Appl, 2017(2017):22.
  25. M. A. Noor, K. I. Noor and F. Safdar, Integral inequaities via generalized convex functions, J. Math. Computer, Sci, 17(4)(2017), 465-476.
  26. 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.
  27. M. A. Noor, K. I. Noor, S. Iftikhar and S. Safdar, Generalized (h, r)-harmonic convex functionsand inequalities, Inter. J. Math. Anal. 16(4)(2018),542-555.
  28. M. A. Noor, K. I. Noor and F. Safdar, Integral inequaities via generalized (α, m)-convex functions, J. Nonlinear. Func. Anal, 2017, (2017), Article ID: 32.
  29. M. A. Noor, K. I. Noor, S. Iftikhar, Inequaities via (p, r)-convex functions, RAD, (2018).
  30. M. A. Noor, K. I. Noor and F. Safdar, New inequalities for generalized log h-convex function, J. Appl. Math. Inform, 36(3-4)(2018), 245-256.
  31. 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.
  32. M. A. Noor, K. I. Noor and F. Safdar, Generalized r-convex functions and integral inequalities. Int. J. Anal. Appl, 16(2018).
  33. C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl, 3(2)(2000), 155-167.
  34. 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.
  35. G. Stampacchia, Formes bilineaires coercivities sur les ensembles convexes, C. R. Acad. Sci. Paris, 258(1964), 4413-4416.