Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T01:55:08.403Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Hadamard's inequalities based on general Euler 4-point formulae

Published online by Cambridge University Press:  17 February 2009

M. Klaričić Bakula  and
J. Pečarić
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a general closed 4-point quadrature rule based on Euler-type identities. We use this rule to prove a generalization of Hadamard's inequalities for (2r)-convex functions (r ≥ 1).

Keywords

Euler formulaequadrature rulesHadamard's inequalitiesconvex functions.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 3 , January 2007 , pp. 387 - 404
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical functions with formulae, graphs and mathematical tables, Applied Math. Series 55, 4th printing (National Bureau of Standards, Washington, 1965).Google Scholar
[2]Berezin, I. S. and Zhidkov, N. P., Computing Methods, Vol. 1 (Pergamon Press, Oxford, 1965).Google Scholar
[3]Bullen, P. S., “Error estimates for some elementary quadrature rules”, Univ. Beograd Publ. Elektrotehn. Fak., Ser. Mat. Fiz. 602–633 (1978) 97–103.Google Scholar
[4]Dedić, Lj., Matić, M. and Pečarić, J., “On generalizations of Ostrowski inequality via some Euler-type identities”, Math. Inequal. Appl. 3 (2000) 337–353.Google Scholar
[5]Dedić, Lj., Matić, M. and Pečarić, J., “On Euler trapezoid rule”, Appl. Math. Comput. 123 (2001) 37–62.CrossRefGoogle Scholar
[6]Dedić, Lj., Matić, M., Pečarić, J. and Vukelić, A., “Hadamard-type inequalities via some Euler-type identities—Euler bitrapezoid formulae”, Nonlinear Stud. 8 (2001) 343–372.Google Scholar
[7]Klaričić Bakula, M., Pečarić, J. and Vukelić, A., “Interpolation of periodic functions and applications on some integration formulae of interpolatory type”, Bull. Math. Soc. Sci. Math. Roumanie 48 (2005) 261–275.Google Scholar
[8]Pečarić, J. and Vukelić, A., Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions, Functional Equations, Inequalities and Applications (Kluwer, Dordrecht, 2003).CrossRefGoogle Scholar
[9]Pečarić, J. E., Perić, I. and Vukelić, A., “Sharp integral inequalities based on general Euler two-point formulae”, ANZIAM J. 46 (2005) 555–574.CrossRefGoogle Scholar
[10]Pečarić, J. E., Proschan, F. and Tong, Y. L., Convex functions, partial orderings, and statistical applications (Academic Press, New York, 1992).Google Scholar
[11]Roberts, A. W. and Varberg, D. E., Convex Functions (Academic Press, New York, 1973).Google Scholar