American Institute of Mathematical Sciences

Advanced
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

S. S. Dragomir 1, and  I. Gomm 1,

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.
Keywords: Convex functions, special means., Hermite-Hadamard inequality.
Mathematics Subject Classification: Primary: 26D15; Secondary: 25D1.
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), 7-12. Google Scholar

[2]

S. S. Dragomir, A mapping in connection to Hadamard's inequalities, An. Öster. Akad. Wiss. Math. Natur., (Wien), 128 (1991), 17-20.  Google Scholar

[3]

S. S. Dragomir, Two mappings in connection to Hadamard's inequalities, J. Math. Anal. Appl., 167 (1992), 49-56. 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), 215-222.  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), Art. 35. Available from: http://www.emis.de/journals/JIPAM/article187.html?sid=187 Google Scholar

[6]

S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc., 74 (2006), 471-476. 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), 9 pages.  Google Scholar

[8]

S. S. Dragomir, D. S. Milośević and J. Sándor, On some refinements of Hadamard's inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math., 4 (1993), 21-24. Google Scholar

[9]

S. S. Dragomir and C. E. M. Pearce, "Selected Topics on Hermite-Hadamard Inequalities and Applications," RGMIA Monographs, 2000. Available from: http://rgmia.org/monographs/hermite_hadamard.html Google Scholar

[10]

A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260-288. 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), 1-32.  Google Scholar

[12]

M. Merkle, Remarks on Ostrowski's and Hadamard's inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10 (1999), 113-117.  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), 92-104. 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, Inequalities and Applications, Kluwer Acad. Publ., Dordrecht, (2003), 105-137. Google Scholar

[15]

G. Toader, Superadditivity and Hermite-Hadamard's inequalities, Studia Univ. Babeş-Bolyai Math., 39 (1994), 27-32.  Google Scholar

[16]

G. S. Yang and M. C. Hong, A note on Hadamard's inequality, Tamkang J. Math., 28 (1997), 33-37.  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), 180-187. 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), 7-12. Google Scholar

[2]

S. S. Dragomir, A mapping in connection to Hadamard's inequalities, An. Öster. Akad. Wiss. Math. Natur., (Wien), 128 (1991), 17-20.  Google Scholar

[3]

S. S. Dragomir, Two mappings in connection to Hadamard's inequalities, J. Math. Anal. Appl., 167 (1992), 49-56. 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), 215-222.  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), Art. 35. Available from: http://www.emis.de/journals/JIPAM/article187.html?sid=187 Google Scholar

[6]

S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc., 74 (2006), 471-476. 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), 9 pages.  Google Scholar

[8]

S. S. Dragomir, D. S. Milośević and J. Sándor, On some refinements of Hadamard's inequalities and applications, Univ. Belgrad, Publ. Elek. Fak. Sci. Math., 4 (1993), 21-24. Google Scholar

[9]

S. S. Dragomir and C. E. M. Pearce, "Selected Topics on Hermite-Hadamard Inequalities and Applications," RGMIA Monographs, 2000. Available from: http://rgmia.org/monographs/hermite_hadamard.html Google Scholar

[10]

A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260-288. 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), 1-32.  Google Scholar

[12]

M. Merkle, Remarks on Ostrowski's and Hadamard's inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10 (1999), 113-117.  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), 92-104. 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, Inequalities and Applications, Kluwer Acad. Publ., Dordrecht, (2003), 105-137. Google Scholar

[15]

G. Toader, Superadditivity and Hermite-Hadamard's inequalities, Studia Univ. Babeş-Bolyai Math., 39 (1994), 27-32.  Google Scholar

[16]

G. S. Yang and M. C. Hong, A note on Hadamard's inequality, Tamkang J. Math., 28 (1997), 33-37.  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), 180-187. doi: 10.1006/jmaa.1999.6506.  Google Scholar

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