How sharp is the Jensen inequality?
- Danilo Costarelli^{1} and
- Renato Spigler^{2}Email author
https://doi.org/10.1186/s13660-015-0591-x
© Costarelli and Spigler; licensee Springer. 2015
Received: 8 October 2014
Accepted: 6 February 2015
Published: 24 February 2015
Abstract
We study how good the Jensen inequality is, that is, the discrepancy between \(\int_{0}^{1} \varphi(f(x)) \,dx\), and \(\varphi ( \int_{0}^{1} f(x) \,dx )\), φ being convex and \(f(x)\) a nonnegative \(L^{1}\) function. Such an estimate can be useful to provide error bounds for certain approximations in \(L^{p}\), or in Orlicz spaces, where convex modular functionals are often involved. Estimates for the case of \(C^{2}\) functions, as well as for merely Lipschitz continuous convex functions φ, are established. Some examples are given to illustrate how sharp our results are, and a comparison is made with some other estimates existing in the literature. Finally, some applications involving the Gamma function are obtained.
Keywords
Jensen inequality convex functions Orlicz spaces convex modular functionals Cauchy-Schwarz inequality Hölder inequalityMSC
26B25 39B721 Introduction
Inequality (1) reduces to an equality whenever either (i) φ is affine, or (ii) \(f(x)\) is a constant. In this paper, we assume that φ is strictly convex on I. Since (1) and (2) reduce to equalities when φ is affine, we expect that the discrepancy between the two sides of such inequalities depends on the departure of φ from the affine behavior. Such a departure should be measured somehow.
In this paper, we derive several estimates for (3), and we will consider the case of convex functions which are in the \(C^{2}\) class, as well as when they are merely Lipschitz continuous. We will also provide a few examples for the purpose of illustration. Moreover, we compared our estimates with the bounds derived from some other results, already known in the literature, in order to test the quality of our estimates. Some applications involving the Gamma function are also made.
2 Some estimates
Remark 2.1
3 Examples
It is easy to provide some simple examples to show how sharp the Jensen inequality can be in practice.
Example 3.1
Remark 3.2
Example 3.3
We stress that in this case, the inequalities showed in (5), (8), and (13) cannot be applied to the functions involved in Example 3.3, since now \(\varphi(x)\) is not a \(C^{2}\), but it is merely Lipschitz continuous.
Example 3.4
The Gamma function, Γ, is known to be strictly convex on the real positive halfline. Keep in mind that it attains its minimum at \(x_{0} \approx 1.4616\), with \(\Gamma(x_{0}) \approx0.8856\); see [19, 20].
In closing, we observe that, in Example 3.4, the estimate in (13) yields the result \(E_{3} \leq0.0824\), which is clearly worse than that obtained from (8). In this case, we are able to improve the estimate given by (13), which was derived from [18].
4 Final remarks and conclusions
The purpose of this paper is to establish estimates concerning the Jensen inequality, which involve convex functions. Such estimates can be useful in many instances, such as, e.g., modular estimates in Orlicz spaces, or \(L^{p}\)-estimates for linear and nonlinear integral operators. For instance, the convex function can be \(\varphi(x) := |x|^{p}\) with \(p \geq1\), namely the convex ‘φ-function’ used of the general theory of Orlicz spaces, used to generate the \(L^{p}\)-spaces; see, e.g., [4, 5]. Thus, the previous estimates, aimed at assessing how sharp the Jensen inequality might be, can be used to obtain estimates for the \(L^{p}\)-norms of certain given functions.
Besides, one can consider the same functions φ with \(p < 0\), for instance, \(\varphi(x) := x^{-1/2}\), or \(\varphi(x) := x^{-1}\), which are smooth and convex on every interval \([a, b]\) with \(0 < a < b \leq+\infty\). Estimates have been derived for both, smooth and Lipschitz continuous convex φ. They depend, respectively, on the uniform norm of the \(\varphi''\) and on the Lipschitz constant of φ, as well as on the \(L^{1}\) and \(L^{2}\) norms of the function f involved in the inequality. From the numerical experiments it is clear that, in general, the estimate in (8) is sharper than the other ones established in this paper. Moreover, the estimate in (8) improves that in (13), established in [18], when the function φ is a \(C^{2}\) convex function. Finally, we stress the usefulness of estimates (12), which allows one to obtain a rough estimate for the error made using the Jensen inequality when φ is merely Lipschitz continuous. Clearly, a major advantage will occur when the \(C^{2}\)-assumption of φ is not satisfied.
Declarations
Acknowledgements
This work was accomplished within the GNFM and GNAMPA research groups of the Italian INdAM.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
- Jensen, JLWV: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 30(1), 175-193 (1906) (French) View ArticleMATHMathSciNetGoogle Scholar
- Kuczma, M: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality. Birkhaüser, Basel (2008) Google Scholar
- Mukhopadhyay, N: On sharp Jensen’s inequality and some unusual applications. Commun. Stat., Theory Methods 40, 1283-1297 (2011) View ArticleMATHGoogle Scholar
- Musielak, J: Orlicz Spaces and Modular Spaces. Lecture Notes in Math., vol. 1034. Springer, Berlin (1983) MATHGoogle Scholar
- Musielak, J, Orlicz, W: On modular spaces. Stud. Math. 28, 49-65 (1959) MathSciNetGoogle Scholar
- Bardaro, C, Musielak, J, Vinti, G: Nonlinear Integral Operators and Applications. de Gruyter Series in Nonlinear Analysis and Applications, vol. 9. de Gruyter, Berlin (2003) View ArticleMATHGoogle Scholar
- Dragomir, SS, Ionescu, NM: Some converse of Jensen’s inequality and applications. Rev. Anal. Numér. Théor. Approx. 23, 71-78 (1994) MATHMathSciNetGoogle Scholar
- Dragomir, SS, Pečarić, J, Persson, LE: Properties of some functionals related to Jensen’s inequality. Acta Math. Hung. 69(4), 129-143 (1995) Google Scholar
- Dragomir, SS: Bounds for the normalized Jensen functional. Bull. Aust. Math. Soc. 74(3), 471-478 (2006) View ArticleMATHGoogle Scholar
- Costarelli, D, Spigler, R: Convergence of a family of neural network operators of the Kantorovich type. J. Approx. Theory 185, 80-90 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Costarelli, D, Vinti, G: Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Boll. Unione Mat. Ital. (9) IV, 445-468 (2011); Special volume dedicated to Prof. Giovanni Prodi MathSciNetGoogle Scholar
- Costarelli, D, Vinti, G: Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numer. Funct. Anal. Optim. 34(8), 819-844 (2013) View ArticleMATHMathSciNetGoogle Scholar
- Costarelli, D, Vinti, G: Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces. Comment. Math. 53(2), 271-292 (2013); Special volume dedicated to Prof. Julian Musielak MATHMathSciNetGoogle Scholar
- Costarelli, D, Vinti, G: Order of approximation for sampling Kantorovich operators. J. Integral Equ. Appl. 26(3), 345-368 (2014) View ArticleMATHMathSciNetGoogle Scholar
- Cluni, F, Costarelli, D, Minotti, AM, Vinti, G: Applications of sampling Kantorovich operators to thermographic images for seismic engineering. J. Comput. Anal. Appl. 19(4), 602-617 (2015) Google Scholar
- van Wijngaarden, A, Scheen, WL: Table of Fresnel integrals. http://www.dwc.knaw.nl/DL/publications/PU00011466.pdf
- Mercer, AMcD: Some new inequalities involving elementary mean values. J. Math. Anal. Appl. 229, 677-681 (1999) View ArticleMATHMathSciNetGoogle Scholar
- Pečarić, JE, Perić, I, Srivastava, HM: A family of the Cauchy type mean-value theorems. J. Math. Anal. Appl. 306, 730-739 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. U.S. Government Printing Office, Washington (1964) MATHGoogle Scholar
- Olver, FWJ, Lozier, DW, Boisvert, RF, Clark, CW (eds.): NIST Digital Library of Mathematical Functions. Cambridge University Press, Cambridge (2010) Google Scholar