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Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions
2012, 2(2): 279-291. doi: 10.3934/naco.2012.2.279

## Jensen's inequality for quasiconvex functions

 1 Mathematics, School of Engineering & Science, Victoria University, Melbourne, Australia 2 School of Mathematical Sciences, The University of Adelaide, Adelaide, Australia

Received  October 2011 Revised  March 2012 Published  May 2012

Some inequalities of Jensen type and connected results are given for quasiconvex functions on convex sets in real linear spaces.
Citation: S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279
##### References:
 [1] M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex, Tamkang J. Math, 41 (2010), 353-359.  Google Scholar [2] M. Alomari, M. Darus and U. S. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comput. Math. Appl., 59 (2010), 225-232. doi: 10.1016/j.camwa.2009.08.002.  Google Scholar [3] S. S. Dragomir, Two mappings associated with Jensen's inequality, Extracta Math., 8 (1993), 102-105.  Google Scholar [4] S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Zb. Rad. (Krajujevac), 15 (1994), 65-73.  Google Scholar [5] S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Math. Balkanica (N.S.), 9 (1995), 3-9.  Google Scholar [6] S. S. Dragomir and B. Mond, On Hadamard's inequality for a class of functions of Godunova and Levin, Indian J. Math., 39 (1997), 1-9.  Google Scholar [7] S. S. Dragomir and C. E. M. Pearce, On Jensen's inequality for a class of functions of Godunova and Levin, Periodica Math. Hungar., 33 (1996), 93-100. doi: 10.1007/BF02093506.  Google Scholar [8] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard's inequality, Bull. Austral. Math. Soc., 57 (1998), 377-385. doi: 10.1017/S0004972700031786.  Google Scholar [9] S. S. Dragomir, J. E. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.  Google Scholar [10] A. Eberhard and C. E. M. Pearce, Class-inclusion properties for convex functions, in "Progress in Optimization" (Perth 1998), Appl. Optim., Kluwer Acad. Publ., Dordrecht, 39 (2000), 129-133.  Google Scholar [11] N. Hadjisavvas, Hadamard-type inequalities for quasiconvex functions, J. Inequal. Pure Appl. Math., 4 (2003), 6 pp. (electronic).  Google Scholar [12] D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, An. Univ. Craiova Ser. Mat. Inform., 34 (2007), 83-88.  Google Scholar [13] M. Jovanović, Some inequalities for strong quasiconvex functions, Glas. Mat. Ser. III, 24 (1989), 25-29.  Google Scholar [14] M. Merkle, Jensen's inequality for multivariate medians, J. Math. Anal. Appl., 370 (2010), 258-269. doi: 10.1016/j.jmaa.2010.04.033.  Google Scholar [15] C. E. M. Pearce, Quasiconvexity, fractional programming and extremal traffic congestion, in "Frontiers in Global Optimization," Kluwer, Dordrecht, "Nonlinear Optimization and its Applications", 74 (2004), 403-409.  Google Scholar [16] C. E. M. Pearce and A. M. Rubinov, $P$-functions, quasi-convex functions and Hadamard-type inequalities, J. Math. Anal. Applic., 240 (1999), 92-104. doi: 10.1006/jmaa.1999.6593.  Google Scholar [17] A. M. Rubinov and J. Dutta, Hadamard type inequality for quasiconvex functions in higher dimensions, J. Math. Anal. Appl., 270 (2002), 80-91. doi: 10.1016/S0022-247X(02)00050-1.  Google Scholar [18] M. Wagner, Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems, J. Math. Anal. Appl., 355 (2009), 606-619. doi: 10.1016/j.jmaa.2009.01.059.  Google Scholar

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##### References:
 [1] M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex, Tamkang J. Math, 41 (2010), 353-359.  Google Scholar [2] M. Alomari, M. Darus and U. S. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comput. Math. Appl., 59 (2010), 225-232. doi: 10.1016/j.camwa.2009.08.002.  Google Scholar [3] S. S. Dragomir, Two mappings associated with Jensen's inequality, Extracta Math., 8 (1993), 102-105.  Google Scholar [4] S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Zb. Rad. (Krajujevac), 15 (1994), 65-73.  Google Scholar [5] S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Math. Balkanica (N.S.), 9 (1995), 3-9.  Google Scholar [6] S. S. Dragomir and B. Mond, On Hadamard's inequality for a class of functions of Godunova and Levin, Indian J. Math., 39 (1997), 1-9.  Google Scholar [7] S. S. Dragomir and C. E. M. Pearce, On Jensen's inequality for a class of functions of Godunova and Levin, Periodica Math. Hungar., 33 (1996), 93-100. doi: 10.1007/BF02093506.  Google Scholar [8] S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard's inequality, Bull. Austral. Math. Soc., 57 (1998), 377-385. doi: 10.1017/S0004972700031786.  Google Scholar [9] S. S. Dragomir, J. E. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.  Google Scholar [10] A. Eberhard and C. E. M. Pearce, Class-inclusion properties for convex functions, in "Progress in Optimization" (Perth 1998), Appl. Optim., Kluwer Acad. Publ., Dordrecht, 39 (2000), 129-133.  Google Scholar [11] N. Hadjisavvas, Hadamard-type inequalities for quasiconvex functions, J. Inequal. Pure Appl. Math., 4 (2003), 6 pp. (electronic).  Google Scholar [12] D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, An. Univ. Craiova Ser. Mat. Inform., 34 (2007), 83-88.  Google Scholar [13] M. Jovanović, Some inequalities for strong quasiconvex functions, Glas. Mat. Ser. III, 24 (1989), 25-29.  Google Scholar [14] M. Merkle, Jensen's inequality for multivariate medians, J. Math. Anal. Appl., 370 (2010), 258-269. doi: 10.1016/j.jmaa.2010.04.033.  Google Scholar [15] C. E. M. Pearce, Quasiconvexity, fractional programming and extremal traffic congestion, in "Frontiers in Global Optimization," Kluwer, Dordrecht, "Nonlinear Optimization and its Applications", 74 (2004), 403-409.  Google Scholar [16] C. E. M. Pearce and A. M. Rubinov, $P$-functions, quasi-convex functions and Hadamard-type inequalities, J. Math. Anal. Applic., 240 (1999), 92-104. doi: 10.1006/jmaa.1999.6593.  Google Scholar [17] A. M. Rubinov and J. Dutta, Hadamard type inequality for quasiconvex functions in higher dimensions, J. Math. Anal. Appl., 270 (2002), 80-91. doi: 10.1016/S0022-247X(02)00050-1.  Google Scholar [18] M. Wagner, Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems, J. Math. Anal. Appl., 355 (2009), 606-619. doi: 10.1016/j.jmaa.2009.01.059.  Google Scholar
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