Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-23T19:07:11.310Z Has data issue: false hasContentIssue false
  • English
  • Français

SHARP INTEGRAL INEQUALITIES BASED ON GENERAL TWO-POINT FORMULAE VIA AN EXTENSION OF MONTGOMERY’S IDENTITY

Part of: Approximations and expansions Inequalities

Published online by Cambridge University Press:  09 March 2010

A. AGLIĆ ALJINOVIĆ ,
J. PEČARIĆ  and
M. RIBIČIĆ PENAVA
A. AGLIĆ ALJINOVIĆ*
Affiliation:
Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10 000 Zagreb, Croatia (email: andrea.aglic@fer.hr)
J. PEČARIĆ
Affiliation:
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10 000 Zagreb, Croatia (email: pecaric@hazu.hr)
M. RIBIČIĆ PENAVA
Affiliation:
Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia (email: mihaela@mathos.hr)
*
✉For correspondence; e-mail: andrea.aglic@fer.hr
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 consider families of general two-point quadrature formulae, using the extension of Montgomery’s identity via Taylor’s formula. The formulae obtained are used to present a number of inequalities for functions whose derivatives are from Lp spaces and Bullen-type inequalities.

Keywords

Taylor’s formulaMontgomery’s identitytwo-point formulaeBullen-type inequalities

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals 41A55: Approximate quadratures
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 1 , July 2009 , pp. 67 - 101
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Aglić Aljinović, A. and Pečarić, J., “On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula”, Tamkang J. Math. 36 (2005) 199–218.CrossRefGoogle Scholar
[2]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
[3]Dedić, Lj., Matić, M. and Pečarić, J., “On Euler midpoint formulae”, ANZIAM J. 46 (2005) 417–438.CrossRefGoogle Scholar
[4]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
[5]Guessab, A. and Schmeisser, G., “Sharp integral inequalities of the Hermite–Hadamard type”, J. Approx. Theory 115 (2002) 260–288.CrossRefGoogle Scholar
[6]Klaričić, M. and Pečarić, J., “Generalized Hadamard’s inequalities based on general 4-point formulae”, accepted for publication in ANZIAM J.Google Scholar
[7]Kovač, S. and Pečarić, J. E., “Generalization of an integral formula of Guessab and Schmeisser”, submitted for publication.Google Scholar
[8]Matić, M., Pearce, C. E. M. and Pečarić, J., “Two-point formulae of Euler type”, ANZIAM J. 44 (2002) 221–245.CrossRefGoogle Scholar
[9]Mitrinović, D. S., Pečarić, J. E. and Fink, A. M., Inequalities for functions and their integrals and derivatives (Kluwer Academic Publishers, Dordrecht, 1994).Google Scholar
[10]Pečarić, J., “On the Čebyšev inequality”, Bul. Inst. Politeh. Timisoara 25 (1980) 10–11.Google Scholar
[11]Pečarić, J., Perić, I. and Vukelić, A., “Sharp integral inequalities based on general Euler two-point formulae”, Math. Inequal. Appl. 4 (2001) 215–221.Google Scholar
[12]Pečarić, J., Perić, I. and Vukelić, A., “Sharp integral inequalities based on general Euler two-point formulae”, ANZIAM J. 46 (2005) 555–574.CrossRefGoogle Scholar
[13]Ralston, A. and Rabinowitz, P., A first course in numerical analysis (Dover Publications, Mineola, New York, 2001).Google Scholar