Abstract
We discuss a rather general condition under which the inequality of Jensen works for certain convex combinations of points not all in the domain of convexity of the function under attention. Based on this fact, an extension of the Hardy–Littlewood–Pólya theorem of majorization is proved and a new insight is given into the problem of risk aversion in mathematical finance.
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Bîrsan, M., Neff, P., Lankeit, J.: Sum of squared logarithms—an inequality relating positive definite matrices and their matrix logarithm, J. Inequal. Appl., paper no. 168. (2013)
Borwein J., Girgensohn R.: A class of exponential inequalities. Math. Inequal. Appl. 6(3), 397–411 (2003)
Czinder, P., Páles, Z.: An extension of the Hermite-Hadamard inequality and an application for Gini and Stolarsky means. J. Inequal. Pure Appl. Math. 5(2), Article no. 42. (2004)
Dragomirescu M., Ivan C.: Convexity-like inequalities for averages in a convex set. Aequ. Math. 45, 179–194 (1993)
Florea, A., Niculescu, C.P.: A Hermite-Hadamard inequality for convex-concave symmetric functions, Bull. Math. Soc. Sci. Math. Roumanie 50(98), no. 2, pp. 149–156 (2007)
Florea A., Păltănea E.: On a class of punctual convex functions. Math. Inequal. Appl. 17(1), 389–399 (2014)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Mathematical Library, 2nd Edition, 1952, Reprinted (1988)
Lieb E.H., Pedersen G.K.: Convex multivariable trace functions. Rev. Math. Phys. 14, 631–648 (2002)
Malamud, S.M.: Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc. 357, pp. 4043–4064 (2005) Posted on arXiv:math/0304158 at July 6, (2003)
Marshall A.W., Olkin I., Arnold B.: Inequalities: theory of majorization and its application, 2nd Edition. Springer, New York (2011)
Mihai, M., Niculescu, C.P.: A simple proof of the Jensen type inequality of Fink and Jodeit. Mediterr. J. Math. doi:10.1007/s00009-014-0480-4
Niculescu, C.P., Persson, L.-E.: Convex Functions and their Applications. A Contemporary Approach, CMS Books in Mathematics vol. 23. Springer, New York (2006)
Niculescu C.P., Rovenţa I.: An approach of majorization in spaces with a curved geometry. J. Math. Anal. Appl. 411(1), 119–128 (2014)
Niculescu C.P., Spiridon C.: New Jensen-type inequalities. J. Math. Anal. Appl. 401(1), 343–348 (2013)
Niculescu, C.P., Stephan, H.: A generalization of Lagrange’s algebraic identity and connections with Jensen’s inequality. Weierstraß—Institut für Angewandte Analysis und Stochastik, Preprint no. 1756/2012.
Niculescu, C.P., Stephan, H.: Lagrange’s Barycentric Identity From An Analytic Viewpoint, Bull. Math. Soc. Sci. Math. Roumanie 56(104), no. 4, pp. 487–496 (2013)
Pearce C.E.M., Pečarić J.: On convexity-like inequalities. Revue Roumaine Math. Pure Appl. Math. 42(1-2), 133–138 (1997)
Pečarić J., Proschan F., Tong Y.L.: Convex functions, partial orderings, and statistical applications. Academic Press, New York (1992)
Simon B.: Convexity. An analytic viewpoint. Cambridge University Press, Cambridge (2011)
Steele, J.M.: The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. Cambridge University Press, Cambridge (2004)
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Niculescu, C.P., Rovenţa, I. Relative convexity and its applications. Aequat. Math. 89, 1389–1400 (2015). https://doi.org/10.1007/s00010-014-0319-x
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DOI: https://doi.org/10.1007/s00010-014-0319-x