New Integral Inequalities Via (α,m)-Convexity and Quasi-Convexity
Year 2013,
Volume: 42 Issue: 3, 289 - 297, 01.03.2013
Wenjun Liu
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.
References
- 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.
- 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.
- A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), no. 1, 7–12.
- 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).
- 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).
- 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).
- C. Dinu, Hermite-Hadamard inequality on time scales, J. Inequal. Appl. 2008, Art. ID 287947, 24 pp.
- S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
- 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.
- S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002), no. 1, 55–65.
- 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.
- 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.
- 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.
- D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, An. Univ. Craiova Ser. Mat. Inform. 34 (2007), 83–88.
- U. S. Kirmaci et al., Hadamard-type inequalities for s-convex functions, Appl. Math. Comput. 193 (2007), no. 1, 26–35.
- 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.
- 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)
- 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.
- M. E. ¨ Ozdemir, E. Set and M. Alomari, Integral inequalities via several kinds of convexity, Creat. Math. Inform. 20 (2011), no. 1, 62–73.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.