By using the Darboux formula obtained as a generalization of the Taylor formula, we deduce some Jensen–Ostrowski-type inequalities. The applications to quadrature rules and f-divergence measures (specifically, for higher-order χ-divergence) are also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. S. Barnett, P. Cerone, S. S. Dragomir, and A. Sofo, “Approximating Csiszár f-divergence by the use of Taylor’s formula with integral remainder,” Math. Inequal. Appl., 5, No. 3, 417–434 (2002).
C. O. Burg, “Derivative-based closed Newton–Cotes numerical quadrature,” Appl. Math. Comput., 218, No. 13, 7052–7065 (2012).
P. Cerone, S. S. Dragomir, and E. Kikianty, “Jensen–Ostrowski type inequalities and applications for f-divergence measures,” Appl. Math. Comput., 266, 304–315 (2015).
P. Cerone, S. S. Dragomir, and E. Kikianty, “On inequalities of Jensen–Ostrowski type,” J. Inequal. Appl., 2015, Article 328 (2015).
P. Cerone, S. S. Dragomir, and E. Kikianty, “Ostrowski and Jensen type inequalities for higher derivatives with applications,” J. Inequal. Spec. Funct., 7, No. 1, 61–77 (2016).
I. I. Csiszár, “On topological properties of f-divergences,” Studia Sci. Math. Hungar., 2, 329–339 (1967).
I. I. Csiszár and J. Körner, Information Theory: Coding Theorem for Discrete Memoryless Systems, Academic Press, New York (1981).
S. S. Dragomir, “Jensen and Ostrowski type inequalities for general Lebesgue integral with applications,” Ann. Univ. Mariae Curie-Skłodowska, Sect. A., 70, No. 2, 29–49 (2016).
S. S. Dragomir, “New Jensen and Ostrowski type inequalities for general Lebesgue integral with applications,” Iran. J. Math. Sci. Inform., 11, No. 2, 1–22 (2016).
S. S. Dragomir, “General Lebesgue integral inequalities of Jensen and Ostrowski type for differentiable functions whose derivatives in absolute value are h-convex and applications,” Ann. Univ. Mariae Curie-Skłodowska, Sect. A., 69, No. 2, 17–45 (2015).
S. S. Dragomir and T. M. Rassias (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Springer, Netherlands (2002).
S. Kullback and R. A. Leibler, “On information and sufficiency,” Ann. Math. Statist., 22, 79–86 (1951).
F. Nielsen and R. Nock, “On the Chi square and higher-order Chi distances for approximating f-divergences,” IEEE Signal Process. Lett., 21, No. 1 (2014).
A. Ostrowski, “Über die absolutabweichung einer differentienbaren funktionen von ihren integralmittelwert,” Comment.Math. Helv., 10, 226–227 (1938).
J. E. Pečarić, F. Proschan, and Y. L. Tong, “Convex functions, partial orderings, and statistical application,” in: Math. Sci. Eng., 187, Academic Press, Boston, MA (1992).
I. Vajda, Theory of Statistical Inference and Information, Dordrecht–Boston, Kluwer AP (1989).
Z. X. Wang and D. R. Guo, Special Functions, World Scientific Publ. Co., Teaneck, NJ (1989).
M. Wiersma, “Quadrature rules with (not too many) derivatives,” Atl. Electron. J. Math., 5, No. 1, 60–67 (2012).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 8, pp. 1123–1140, August, 2017.
Rights and permissions
About this article
Cite this article
Cerone, P., Dragomir, S.S. & Kikianty, E. Jensen–Ostrowski Inequalities and Integration Schemes via the Darboux Expansion. Ukr Math J 69, 1306–1327 (2018). https://doi.org/10.1007/s11253-017-1432-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-017-1432-5