New Integral Inequalities Via (α,m)-Convexity and Quasi-Convexity

Abstract

In this paper, we establish some new integral inequalities involvingBeta function via (α, m)-convexity and quasi-convexity, respectively.Our results in special cases recapture known results.

Keywords

• Hermite’s inequality, Euler Beta function, H¨older’s inequality, (α, m)convexity, quasi-convexity2000 AMS Classification:26D15, 33B15, 26A51, 39B62.

New Integral Inequalities Via (α,m)-Convexity and Quasi-Convexity

Abstract

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Keywords

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References

1. M. Alomari and M. Darus, On the Hadamard’s inequality for log-convex functions on the coordinates, J. Inequal. Appl. 2009, Art. ID 283147 13 pp.
2. M. Alomari, M. Darus and S.S. Dragomir, Inequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are quasi-convex, RGMIA Res. Rep. Coll., 12 (2009), Supp., No. 14.
3. A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), no. 1, 7–12.
4. M. K. Bakula, M. E. ¨ Ozdemir and J. Peˇ cari´ c, Hadamard type inequalities for m-convex and (α, m)-convex functions, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 4, Article 96, 12 pp. (electronic).
5. M. K. Bakula and J. Peˇ cari´ c, Note on some Hadamard-type inequalities, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 3, Article 74, 9 pp. (electronic).
6. M. K. Bakula, J. Peˇ cari´ c and M. Ribiˇ ci´ c, Companion Inequalities to Jensen’s Inequality for m-convex and (α, m)-convex Functions, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 5, Article 194, 15 pp. (electronic).
7. C. Dinu, Hermite-Hadamard inequality on time scales, J. Inequal. Appl. 2008, Art. ID 287947, 24 pp.
8. S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.

9. S. S. Dragomir and 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), no. 5, 91–95.
10. S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002), no. 1, 55–65.
11. S. S. Dragomir and S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32 (1999), no. 4, 687–696.
12. P. M. Gill, C. E. M. Pearce and J. Peˇ cari´ c, Hadamard’s inequality for r-convex functions, J. Math. Anal. Appl. 215 (1997), no. 2, 461–470.
13. V. N. Huy and N. T. Chung, Some generalizations of the Fej´ er and Hermite-Hadamard inequalities in H¨ older spaces, J. Appl. Math. Inform. 29 (2011), no. 3-4, 859–868.
14. D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, An. Univ. Craiova Ser. Mat. Inform. 34 (2007), 83–88.
15. U. S. Kirmaci et al., Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007), no. 1, 26–35.
16. Z. Liu, Generalization and improvement of some Hadamard type inequalities for Lipschitzian mappings, J. Pure Appl. Math. Adv. Appl. 1 (2009), no. 2, 175–181.
17. D. S. Mitrinovi´ c, J. E. Peˇ cari´ c and A. M. Fink, Classical and new inequalities in analysis, Mathematics and its Applications (East European Series), 61, Kluwer Acad. Publ., Dordrecht, 19 V. G. Mihe¸ san, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania) (1993)
18. M. E. ¨ Ozdemir, M. Avcı and E. Set, On some inequalities of Hermite-Hadamard type via m-convexity, Appl. Math. Lett. 23 (2010), no. 9, 1065–1070.
19. M. E. ¨ Ozdemir, E. Set and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inform. 20 (2011), no. 1, 62–73.
20. M. E. ¨ Ozdemir, E. Set and M. Z. Sarıkaya, Some new Hadamard type inequalities for coordinated m-convex and (α, m)-convex functions, Hacet. J. Math. Stat. 40 (2011), no. 2, 219–229.
21. J. E. Peˇ cari´ c, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, 187, Academic Press, Boston, MA, 1992.
22. M. Z. Sarikaya, E. Set and M. E. ¨ Ozdemir, On some new inequalities of Hadamard type involving h-convex functions, Acta Math. Univ. Comenian. (N.S.) 79 (2010), no. 2, 265–272. E. Set, M. E. ¨ Ozdemir and S. S. Dragomir, On the Hermite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl. 2010, Art. ID 148102, 9 pp. E. Set, M. E. ¨ Ozdemir and S. S. Dragomir, On Hadamard-type inequalities involving several kinds of convexity, J. Inequal. Appl. 2010, Art. ID 286845, 12 pp.
23. E. Set, M. Sardari, M. E. ¨ Ozdemir and J. Rooin, On generalizations of the Hadamard inequality for (α, m)-convex functions, RGMIA Res. Rep. Coll., 12 (4) (2009), No. 4.
24. G. Toader, Some generalizations of the convexity, in Proceedings of the colloquium on approximation and optimization (Cluj-Napoca, 1985), 329–338, Univ. Cluj-Napoca, Cluj.
25. K.-L. Tseng, S.-R. Hwang and S. S. Dragomir, New Hermite-Hadamard-type inequalities for convex functions (II), Comput. Math. Appl. 62 (2011), no. 1, 401–418.