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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

S. S. Dragomir 1, and  C. E. M. Pearce 2,

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.
Keywords: Jensen's inequality, quasi-superadditivity., Quasiconvex functions.
Mathematics Subject Classification: 26D07, 26D1.
Citation: S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control and 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.

[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.

[3]

S. S. Dragomir, Two mappings associated with Jensen's inequality, Extracta Math., 8 (1993), 102-105.

[4]

S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Zb. Rad. (Krajujevac), 15 (1994), 65-73.

[5]

S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Math. Balkanica (N.S.), 9 (1995), 3-9.

[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.

[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.

[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.

[9]

S. S. Dragomir, J. E. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.

[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.

[11]

N. Hadjisavvas, Hadamard-type inequalities for quasiconvex functions, J. Inequal. Pure Appl. Math., 4 (2003), 6 pp. (electronic).

[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.

[13]

M. Jovanović, Some inequalities for strong quasiconvex functions, Glas. Mat. Ser. III, 24 (1989), 25-29.

[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.

[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.

[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.

[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.

[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.

show all references

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.

[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.

[3]

S. S. Dragomir, Two mappings associated with Jensen's inequality, Extracta Math., 8 (1993), 102-105.

[4]

S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Zb. Rad. (Krajujevac), 15 (1994), 65-73.

[5]

S. S. Dragomir and D. M. Milošević, Two mappings in connection to Jensen's inequality, Math. Balkanica (N.S.), 9 (1995), 3-9.

[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.

[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.

[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.

[9]

S. S. Dragomir, J. E. Pečarić and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.

[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.

[11]

N. Hadjisavvas, Hadamard-type inequalities for quasiconvex functions, J. Inequal. Pure Appl. Math., 4 (2003), 6 pp. (electronic).

[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.

[13]

M. Jovanović, Some inequalities for strong quasiconvex functions, Glas. Mat. Ser. III, 24 (1989), 25-29.

[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.

[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.

[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.

[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.

[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.

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