Some Characterizations of General Preinvex Functions

Main Article Content

Muhammad Uzair Awan
Muhammad Aslam Noor
Vishnu Narayan Mishra
Khalida Inayat Noor


In this paper, we consider a new class of general preinvex functions involving an arbitrary function. We show that the optimality condition for general preinvex functions on general invex set can be characterized by a class of variational-like inequality. We also derive some integral inequalities of Hermite-Hadamard type via general preinvex functions. Some special cases are also discussed. Our results represent a significant refinement of the previously known results. These results may stimulate further research in this area.

Article Details


  1. G. Cristescu and L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publishers, Dordrecht, Holland, (2002).
  2. G. Cristescu, M. A. Noor, M. U. Awan, Bounds of the second degree cumulative frontier gaps of functions with generalized convexity, Carpathian J. Math. 31(2) (2015), 173-180.
  3. Deepmala, A Study on Fixed Point Theorems for Nonlinear Contractions and its Applications, Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India (2014).
  4. S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Victoria University, Australia, 2000.
  5. D. I. Duca and L. Lupsa, Saddle points for vector valued functions: existence, necessary and sufficient theorems, J. Glob. Optim., 53 (2012), 431-440.
  6. C. Fulga and V. Preda, Nonlinear programming with φ-preinvex and local φ-preinvex functions, Eur. J. Oper. Res. 192 (2009), 737-743.
  7. M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80 (1981) 545-550.
  8. A. Ben-Israel and B. Mond, What is invexity? J. Aust. Math. Soc. Ser. B 28 (1986), 1-9.
  9. S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901-908.
  10. L. N. Mishra, H. M. Srivastava, M. Sen, On existence results for some nonlinear functional-integral equations in Banach algebra with applications, Int. J. Anal. Appl., 11 (1) (2016), 1-10.
  11. L. N. Mishra, R. P. Agarwal, On existence theorems for some nonlinear functional-integral equations, Dynamic Systems and Appl., 25 (2016), 303-320.
  12. L. N. Mishra, On existence and behavior of solutions to some nonlinear integral equations with Applications, Ph.D. Thesis, National Institute of Technology, Silchar 788 010, Assam, India (2017).
  13. M. A. Noor, Differentiable nonconvex functions and general variational inequalities, Appl. Math. Comput., 199 (2008), 623-630.
  14. M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets Syst. 64 (1994), 95-104.
  15. M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory 2 (2007), 126-131.
  16. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.
  17. M. A. Noor, On Hermite-Hadamard integral inequalities for involving two log-preinvex functions, J. Inequal. Pure Appl. Math., 3 (2007), 75-81.
  18. M. A. Noor, Variational like inequalities, Optimization 30 (1994), 323-330.
  19. M. A. Noor, M.U. Awan, K. I. Noor: On some inequalities for relative semi-convex functions. J. Inequal. Appl. 2013 (2013), Art. ID 332.
  20. M. A. Noor, K. I. Noor, M. U. Awan, Geometrically relative convex functions, Appl. Mathe. Infor. Sci., 8(2) (2014), 607-616.
  21. M. A. Noor, K. I. Noor, M. U. Awan, Hermite-Hadamard inequalities for relative semi-convex functions and applications, Filomat. 28 (2) (2014), 221-230.
  22. M. A. Noor, K. I. Noor, M. U. Awan, J. Li, On Hermite-Hadamard type inequalities for h-preinvex functions. Filomat. 28 (7) (2014), 1463-1474.
  23. M. A. Noor, K. I. Noor, Integral inequalities for differentiable relative preinvex functions(survey), TWMS J. Pure Appl. Math. 7(1)(2016), 3-19
  24. T. Weir and B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl., 136 (1988), 29-38.
  25. E. A. Youness, E-convex sets, E-convex functions, and E-convex programming, J. Optim. Theory Appl., 102 (1999), 439-450.