American Institute of Mathematical Sciences

Advanced
  • Previous Article
    How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?
  • NACO Home
  • This Issue
  • Next Article
    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 & 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. 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. 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. Google Scholar

[4]

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

[10]

A. Eberhard and C. E. M. Pearce, Class-inclusion properties for convex functions,, in, 39 (2000), 129. Google Scholar

[11]

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

[13]

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

[14]

M. Merkle, Jensen's inequality for multivariate medians,, J. Math. Anal. Appl., 370 (2010), 258. doi: 10.1016/j.jmaa.2010.04.033. Google Scholar

[15]

C. E. M. Pearce, Quasiconvexity, fractional programming and extremal traffic congestion,, in, 74 (2004), 403. 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. 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. 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. doi: 10.1016/j.jmaa.2009.01.059. Google Scholar

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. 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. 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. Google Scholar

[4]

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

[10]

A. Eberhard and C. E. M. Pearce, Class-inclusion properties for convex functions,, in, 39 (2000), 129. Google Scholar

[11]

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

[13]

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

[14]

M. Merkle, Jensen's inequality for multivariate medians,, J. Math. Anal. Appl., 370 (2010), 258. doi: 10.1016/j.jmaa.2010.04.033. Google Scholar

[15]

C. E. M. Pearce, Quasiconvexity, fractional programming and extremal traffic congestion,, in, 74 (2004), 403. 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. 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. 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. doi: 10.1016/j.jmaa.2009.01.059. Google Scholar

[1]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[2]

Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61
[3]

Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353
[4]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119
[5]

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693
[6]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[7]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945
[8]

Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1
[9]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[10]

Ahmet Sahiner, Gulden Kapusuz, Nurullah Yilmaz. A new smoothing approach to exact penalty functions for inequality constrained optimization problems. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 161-173. doi: 10.3934/naco.2016006
[11]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545
[12]

Guodong Ma, Jinbao Jian. A QP-free algorithm of quasi-strongly sub-feasible directions for inequality constrained optimization. Journal of Industrial & Management Optimization, 2015, 11 (1) : 307-328. doi: 10.3934/jimo.2015.11.307
[13]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523
[14]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961
[15]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
[16]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[17]

S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271
[18]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[19]

Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $ S_n$-symmetry. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823
[20]

Daniele Cassani, Bernhard Ruf, Cristina Tarsi. On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 245-250. doi: 10.3934/dcdss.2019017

 Impact Factor: 

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences