SOME NEW INTEGRAL INEQUALITIES FOR n- TIMES DIFFERENTIABLE QUASI-CONVEX FUNCTIONS

Year 2017, Volume: 35 Issue: 3, 363 - 368, 01.09.2017

İmdat İşcan Huriye Kadakal Mahir Kadakal

Abstract

In this work, by using an integral identity together with both the Hölder and the Power-Mean integral inequality we establish several new inequalities for n-time differentiable quasi-convex functions. Using this inequalities, we obtain some new inequalities connected with means.

Keywords

Convex function, quasi-convex function, Hölder Integral inequality and Power-Mean Integral inequality

References

• M. Alomari, and M. Darus. “On some inequalities Simpson-type via quasi-convex functions with applications”, RGMIA Res. Rep. Coll 13.1 (2010). M. Alomarı, M. Darus And S. S. Dragomir, “New Inequalities of Hermite-Hadamard Type for Functions whose Second Derivatives Absolute Values are Quasi-Convex”, Tamkang Journal of Mathematics Volume 41, Number 4, 353-359, Winter 2010.
• M. Alomari, S. Hussain, “Two Inequalities of Simpson Type for Quasi-Convex Functions and Applications”, Applied Mathematics E-Notes, 11(2011), 110-117.
• S.-P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable (a,m) -convex functions”, Jour. of Ineq. and Appl., 2012, 2012:267.
• P. Cerone, S.S. Dragomir and J. Roumeliotis, “Some Ostrowski type inequalities for n-time differentiable mappings and applications”, Demonstratio Math., 32 (4) (1999), 697–712.
• P. Cerone, S.S. Dragomir, J. Roumeliotis and J. Sunde, “A new generalization of the trapezoid formula for n-time differentiable mappings and applications”, Demonstratio Math., 33 (4) (2000), 719–736.
• S.S. Dragomir and C.E.M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000, online: http://www.staxo.vu.edu.au/RGMIA/monographs/hermite hadamard.html.
• D.-Y. Hwang, “Some Inequalities for n-time Differentiable Mappings and Applications”, Kyung. Math. Jour., 43 (2003), 335–343.
• S. Hussain, S. Qaisar, “New Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity”, Punjab University, Journal of Mathematics (ISSN 1016-2526), Vol. 45 (2013), pp 33-38.
• D. A. Ion, “Some estimates on the Hermite-Hadamard inequality through quasi-convex functions”, Annals of University of Craiova, Math. Comp. Sci. Ser. Volume 34, 2007, Pages 82–87.
• İ. İşcan and S. Turhan, “Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral”, Moroccan J. Pure and Appl. Anal.(MJPAA), Volume 2(1) (2016), 34-46.
• İ. İşcan, “Hermite-Hadamard type inequalities for harmonically convex functions”, Hacettepe Journal of Mathematics and Statistics, Volume 43 (6) (2014), 935–942.
• W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard inequality to n-time differentiable function which are s -convex in the second sense”, Analysis (Munich), 32 (2012), 209–220.
• S. Maden, H. Kadakal, M. Kadakal and İ. İşcan, “Some new integral inequalities for n-times differentiable convex and concave functions”. Available online at: https://www.researchgate.net/publication/312529563.
• M. A. Latif, S. Hussain and S. S. Dragomir, “Refinements of Hermıte-Hadamard-Type Inequalities for Co-Ordinated Quasi-Convex Functıons”, International Journal Of Mathematical Archive-3(1), 2012, Page 161-171.
• W. Liu, “New Integral Inequalities vıa (α,m)-Convexıty and Quasi-Convexity”, Hacettepe Journal of Mathematics and Statistics Volume 42 (3) (2013), 289-297.
• K. Murota, A. Shioura, “Quasi M-convex and L-convex functions—quasiconvexity in discrete optimization”, Discrete Applied Mathematics 131, (2003), 467-494.
• K. Murato, A. Shioura, “Quasi M-convex and L-convex Functions Quasiconvexity in Discrete Optimization”, RIMS Preprint No. 1306, Kyoto University, December 2000.
• S.H. Wang, B.-Y. Xi and F. Qi, “Some new inequalities of Hermite-Hadamard type for n-time differentiable functions which are m-convex”, Analysis (Munich), 32 (2012), 247–262.
• Ç. Yıldız, “New inequalities of the Hermite-Hadamard type for n-time differentiable functions which are quasiconvex”, Journal of Mathematical Inequalities, 10, 3(2016), 703-711.