2012, 2(2): 271-278. doi: 10.3934/naco.2012.2.271

Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions

1. 

Mathematics, School of Engineering & Science, Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia, Australia

Received  October 2011 Revised  March 2012 Published  May 2012

Some new results concerning two mappings associated to the celebrated Hermite-Hadamard integral inequality for convex function with applications for special means are given.
Citation: 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
References:
[1]

A. G. Azpeitia, Convex functions and the Hadamard inequality,, Rev. Colombiana Mat., 28 (1994), 7. Google Scholar

[2]

S. S. Dragomir, A mapping in connection to Hadamard's inequalities,, An. Öster. Akad. Wiss. Math. Natur., 128 (1991), 17. Google Scholar

[3]

S. S. Dragomir, Two mappings in connection to Hadamard's inequalities,, J. Math. Anal. Appl., 167 (1992), 49. doi: 10.1016/0022-247X(92)90233-4. Google Scholar

[4]

S. S. Dragomir, On Hadamard's inequalities for convex functions,, Mat. Balkanica, 6 (1992), 215. Google Scholar

[5]

S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products,, J. Inequal. Pure & Appl. Math., 3 (2002). Google Scholar

[6]

S. S. Dragomir, Bounds for the normalized Jensen functional,, Bull. Austral. Math. Soc., 74 (2006), 471. doi: 10.1017/S000497270004051X. Google Scholar

[7]

S. S. Dragomir and I. Gomm, Bounds for two mappings associated to the Hermite-Hadamard inequality,, Aust. J. Math. Anal. Appl., 8 (2011). Google Scholar

[8]

S. S. Dragomir, D. S. Milośević and J. Sándor, On some refinements of Hadamard's inequalities and applications,, Univ. Belgrad, 4 (1993), 21. Google Scholar

[9]

S. S. Dragomir and C. E. M. Pearce, "Selected Topics on Hermite-Hadamard Inequalities and Applications,", RGMIA Monographs, (2000). Google Scholar

[10]

A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type,, J. Approx. Theory, 115 (2002), 260. doi: 10.1006/jath.2001.3658. Google Scholar

[11]

E. Kikianty and S. S. Dragomir, Hermite-Hadamard's inequality and the p-HH-norm on the Cartesian product of two copies of a normed space,, Math. Inequal. Appl., 13 (2010), 1. Google Scholar

[12]

M. Merkle, Remarks on Ostrowski's and Hadamard's inequality,, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10 (1999), 113. Google Scholar

[13]

C. E. M. Pearce and A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard type inequalities,, J. Math. Anal. Appl., 240 (1999), 92. doi: 10.1006/jmaa.1999.6593. Google Scholar

[14]

J. Pečarić and A. Vukelić, "Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions,", Functional Equations, (2003), 105. Google Scholar

[15]

G. Toader, Superadditivity and Hermite-Hadamard's inequalities,, Studia Univ. Babeş-Bolyai Math., 39 (1994), 27. Google Scholar

[16]

G. S. Yang and M. C. Hong, A note on Hadamard's inequality,, Tamkang J. Math., 28 (1997), 33. Google Scholar

[17]

G. S. Yang and K. L. Tseng, On certain integral inequalities related to Hermite-Hadamard inequalities,, J. Math. Anal. Appl., 239 (1999), 180. doi: 10.1006/jmaa.1999.6506. Google Scholar

show all references

References:
[1]

A. G. Azpeitia, Convex functions and the Hadamard inequality,, Rev. Colombiana Mat., 28 (1994), 7. Google Scholar

[2]

S. S. Dragomir, A mapping in connection to Hadamard's inequalities,, An. Öster. Akad. Wiss. Math. Natur., 128 (1991), 17. Google Scholar

[3]

S. S. Dragomir, Two mappings in connection to Hadamard's inequalities,, J. Math. Anal. Appl., 167 (1992), 49. doi: 10.1016/0022-247X(92)90233-4. Google Scholar

[4]

S. S. Dragomir, On Hadamard's inequalities for convex functions,, Mat. Balkanica, 6 (1992), 215. Google Scholar

[5]

S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products,, J. Inequal. Pure & Appl. Math., 3 (2002). Google Scholar

[6]

S. S. Dragomir, Bounds for the normalized Jensen functional,, Bull. Austral. Math. Soc., 74 (2006), 471. doi: 10.1017/S000497270004051X. Google Scholar

[7]

S. S. Dragomir and I. Gomm, Bounds for two mappings associated to the Hermite-Hadamard inequality,, Aust. J. Math. Anal. Appl., 8 (2011). Google Scholar

[8]

S. S. Dragomir, D. S. Milośević and J. Sándor, On some refinements of Hadamard's inequalities and applications,, Univ. Belgrad, 4 (1993), 21. Google Scholar

[9]

S. S. Dragomir and C. E. M. Pearce, "Selected Topics on Hermite-Hadamard Inequalities and Applications,", RGMIA Monographs, (2000). Google Scholar

[10]

A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type,, J. Approx. Theory, 115 (2002), 260. doi: 10.1006/jath.2001.3658. Google Scholar

[11]

E. Kikianty and S. S. Dragomir, Hermite-Hadamard's inequality and the p-HH-norm on the Cartesian product of two copies of a normed space,, Math. Inequal. Appl., 13 (2010), 1. Google Scholar

[12]

M. Merkle, Remarks on Ostrowski's and Hadamard's inequality,, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10 (1999), 113. Google Scholar

[13]

C. E. M. Pearce and A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard type inequalities,, J. Math. Anal. Appl., 240 (1999), 92. doi: 10.1006/jmaa.1999.6593. Google Scholar

[14]

J. Pečarić and A. Vukelić, "Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions,", Functional Equations, (2003), 105. Google Scholar

[15]

G. Toader, Superadditivity and Hermite-Hadamard's inequalities,, Studia Univ. Babeş-Bolyai Math., 39 (1994), 27. Google Scholar

[16]

G. S. Yang and M. C. Hong, A note on Hadamard's inequality,, Tamkang J. Math., 28 (1997), 33. Google Scholar

[17]

G. S. Yang and K. L. Tseng, On certain integral inequalities related to Hermite-Hadamard inequalities,, J. Math. Anal. Appl., 239 (1999), 180. doi: 10.1006/jmaa.1999.6506. Google Scholar

[1]

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

[2]

Leon Ehrenpreis. Special functions. Inverse Problems & Imaging, 2010, 4 (4) : 639-647. doi: 10.3934/ipi.2010.4.639

[3]

Liran Rotem. Banach limit in convexity and geometric means for convex bodies. Electronic Research Announcements, 2016, 23: 41-51. doi: 10.3934/era.2016.23.005

[4]

Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223

[5]

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

[6]

Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671

[7]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21

[8]

Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004

[9]

Khalida Inayat Noor, Muhammad Aslam Noor. Higher order uniformly close-to-convex functions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1277-1290. doi: 10.3934/dcdss.2015.8.1277

[10]

Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070

[11]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1

[12]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691

[13]

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

[14]

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

[15]

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

[16]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615

[17]

Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9

[18]

Nguyen Thi Bach Kim, Nguyen Canh Nam, Le Quang Thuy. An outcome space algorithm for minimizing the product of two convex functions over a convex set. Journal of Industrial & Management Optimization, 2013, 9 (1) : 243-253. doi: 10.3934/jimo.2013.9.243

[19]

Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2018163

[20]

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

 Impact Factor: 

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]