Title: New Inequalities of Hermite-Hadamard Type for n-Times Differentiable s-Convex Functions with Applications
Author(s): Muhammad Amer Latif, Sever S. Dragomir, Ebrahim Momoniat
Pages: 119-131
Cite as:
Muhammad Amer Latif, Sever S. Dragomir, Ebrahim Momoniat, New Inequalities of Hermite-Hadamard Type for n-Times Differentiable s-Convex Functions with Applications, Int. J. Anal. Appl., 13 (2) (2017), 119-131.

Abstract


In this paper, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are s-convex functions. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are s-convex functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal rule and to special means of established results are given.

Full Text: PDF

 

References


  1. M. W. Alomari, M. Darus and U. S. Kirmaci, Some inequalities of Hadamard type inequalities for s-convex , Acts Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 4, 1643-1652.

  2. S. S. Dragomir and S. Fitzpatric, The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32 (1999), no. 4 687-696.

  3. S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998) 91–95.

  4. H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994) 100–111.

  5. J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction consider´ ee par Riemann, J. Math Pures Appl., 58 (1893), 171–215.

  6. Ch. Hermite, Sur deux limites d’une integrale definie, Mathesis 3 (1883), 82.

  7. D. -Y. Hwang, Some inequalities for n-time differentiable mappings and applications, Kyugpook Math. J. 43(2003), 335-343.

  8. I. ˙ I¸ scan, Generalization of different type integral inequalities for s-convex functions via fractional integrals, Appl. Anal. (2014). 93 (9) (2014), 1846-1862.

  9. W.-D. Jiang, D.-W. Niu, Y. Hua, and F. Qi, Generalizations of Hermite-Hadamard inequality to n-time differentiable functions which are s-convex in the second sense, Analysis (Munich) 32 (2012), 1001–1012.

  10. U. S. Kırmacı and M.E. Ozdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153 (2004), 361-368.

  11. U. S. Kırmacı, Improvement and further generalization of inequalities for differentiable mappings and applications, Comp and Math. with Appl., 55 (2008), 485-493.

  12. M. A. Latif and S. S. Dragomir, New inequalities of Hermite-Hadamard type for n-times differentiable convex and concave functions with applications. Filomat 30 (10) (2016), 26092621.

  13. M. Avci, H. Kavurmaci, M. E. Ozdemir, New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Appl. Math. Comp., 217 (2011) 5171–5176.

  14. C. E. M. Pearce and J. Pecaric, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55.

  15. Shan Peng, Wei Wei and JinRong Wang, On the Hermite-Hadamard Inequalities for Convex Functions via Hadamard Fractional Integrals Facta Universitatis (NIS) Ser. Math. Inform. 29 (1) (2014), 55-75.