Harmonic Beta-Preinvex Functions and Inequalities

Article Sidebar

PDF
Cite as:
Muhammad Aslam Noor, Khalida Inayat Noor, Sabah Iftikhar, Harmonic Beta-Preinvex Functions and Inequalities, Int. J. Anal. Appl., 13 (2) (2017), 144-160.

Main Article Content

Muhammad Aslam Noor, Khalida Inayat Noor, Sabah Iftikhar

Abstract

In this paper, we introduce and study a new class of harmonic convex functions which is called harmonic beta-preinvex functions. We establish some estimates, involving the Euler Beta function and the Hypergeometric function of the integral $\int_a^{a+\eta(b,a)}(x-a)^p(a+\eta(b,a)-x)^qf(x)\mathrm{d}x$ for the class of functions whose certain powers of the absolute value are generalized harmonic preinvex function. Some special cases are also discussed. Results obtained in this paper can be viewed as significant contribution in this fascinating and dynamic field.

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. A. Ben-Isreal and B. Mond. What is invexity? J. Austral. Math. Soc. Ser. B, 28(1)(1986), 1-9.
  3. G. Cristescu and L. Lupsa, Non-connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, Holland, (2002).
  4. G. Cristescu, Improved integral inequalities for product of convex functions, J. Inequal. Pure Appl. Math., 6(2)(2005) Art. ID 35.
  5. 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.
  6. M. A. Hanson. On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80(1981), 545-550.
  7. C. Hermite, Sur deux limites d'une integrale definie. Mathesis, 3(1883), 82.
  8. I, Iscan, Hermite-Hadamard type inequalities for harmonically convex functions. Hacett, J. Math. Stats., 43(6)(2014), 935-942.
  9. I, Iscan, M. Aydin and S. Dikmenoglu, New integral inequalities via harmonically convex functions, Math. Stat., 3(5)(2015), 134-140.
  10. M. Liu, New integral inequalities involving Beta function via P-convexity, Miskol. Math. Notes, 15(2)(2014), 585-591.
  11. S. R. Mohan and S. K. Neogy, On ivex sets and preinvex functions, J. Math. Anal. Appl. 189(1995), 901-908.
  12. C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, (2006).
  13. M. A. Noor, Variational-like inequalities, Optimization, 30(1994), 323-330.
  14. M. A. Noor and K. I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316(2006), 697-706.
  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, Hadamard integral inequalities for product of two preinvex functions, Nonl. Anal. Forum, 14(2009), 167-173.
  17. On Hadamard integral inequalities involving two log-preinvex functions, J. Inequal. Pure Appl. Math. 8(1)(2007), 1-14.
  18. M. A. Noor, K. I. Noor and M. U. Awan, Fractional Hermite-Hadamrd inequalities for for two kinds of s-preinvex functions, Nonlinear Sci. Lett. A. 8(1)(2017), 11-24.
  19. M. A. Noor, K. I. Noor and M. U. Awan, Some quantum integral inequalities via preinvex functions, Appl. Math. Comput. 269(2015), 242-251.
  20. M. A. Noor, K. I. Noor and S. Khan, Some inequalities for geometrically preinvex functions, Proc. Jungjeon Math. Soc. 19(1)(2016), 67-81.
  21. M. A. Noor, S. Khan and K. I. Noor, Integral inequalities for geometrically log-preinvex functions, Appl. Math. Inform. Sci. Lett. 4(3)(2016), 103-110.
  22. M. A. Noor, K. I. Noor, M. U. Awan and S. Costache, Some integral inequalities for harmonically h-convex functions, U.P.B. Sci. Bull. Serai A, 77(1)(2015), 5-16.
  23. M. A. Noor, K. I. Noor and S. Iftikhar, Hermite-Hadamard inequalities for harmonic preinvex functions, SAUSSUREA, 6(2)(2016), 34-53.
  24. M. A. Noor, K. I. Noor and S. Iftikhar, Integral inequalities for differentiable relative harmonic preinvex functions, TWMS J. Pure Appl. Math. 7(1) (2016), 3-19.
  25. M. A. Noor, K. I. Noor and S. Iftikhar, Hermite-Hadamard inequalities for harmonic nonconvex functions, MAGNT Res. Report. 4(1)(2016), 24-40.
  26. M. A. Noor, K. I. Noor and S. Iftikhar, Some Newtons's type inequalities for harmonic convex functions. J. Adv. Math. Stud., 9(1)(2016), 7-16.
  27. M. A. Noor, K. I. Noor, S. Iftikhar and K. Al-Bany, Inequalities for MT-harmonic convex functions, J. Adv. Math. Stud., 9(2)(2016), 194-207.
  28. M. A. Noor, K. I. Noor and S. Iftikhar, Integral inequalities of Hermite-Hadamard type for harmonic (h,s)-convex functions, Int. J. Anal. Appl., 11(1)(2016), 61-69.
  29. M. A. Noor, K. I. Noor and S. Iftikhar, Some integral inequalities for beta-preinvex functions, Inter. J. Anal. Appl. 13(1)(2017), 41-53.
  30. M. A. Noor, K. I. Noor and S. Iftikhat, Harmonic beta-convex functions and integral inequalities, Preint(2017).
  31. M. A. Noor, K. I. Noor, S. Iftikhar and F. Safdar, Integral inequalities for relative harmonic (s,η)-convex functions, Appl. Math. Comput. Sci. 1(1)(2016), 27-34.
  32. B. G. Pachpatte, On some inequalities for convex functions. RGMIA Res. Rep. Coll., 6(E)(2003), 1-8.
  33. M. E. Ozdemir, E. Set, and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inform., 20(1)(2011), 6273.
  34. J. Pecaric, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Acdemic Press, New york, (1992).
  35. D. D. Stancu, G. Coman, and P. Blaga, Analiza numericˇ a si teoria aproxima arii. ClujNapoca: Presa Universitara Clujeana, II(2002).
  36. M. Tunc, U. Sanal and E. Gov, Some Hermite-Hadamard inequalities for beta-convex and its fractional applications, New Trends Math. Sci., 3(4)(2015), 18-33.