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Convex Functions

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Functional Equations and Inequalities

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 54))

Abstract

One of the most famous and important functional inequalities is the following one:

$${\text{f}}\left( {\frac{{{\text{x}}\,{\text{ + }}\,{\text{y}}}}{2}} \right) \leqslant \frac{{{\text{f}}\left( {\text{x}} \right)\,+ \,{\text{f}}\left( {\text{y}} \right)}}{2}$$
(1)

This inequality was first considered by Jensen [l5] and he gave to functions fulfilling (1) the name convex. Jensen himself expressed the opinion: “It seems to me that the notion of a convex function is almost as fundamental as those of a positive function or increasing function. If I am not mistaken, this notion should find place in elementary text-books on the theory of real functions.” And he was not mistaken. Nowadays the notion of convex functions belongs to the most important ones in mathematics.

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Authors and Affiliations

  1. University of Katowice, Caroline, USA

    M. Kuczma

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  1. M. Kuczma
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B. Forte (Coordinatore)

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Kuczma, M. (2010). Convex Functions. In: Forte, B. (eds) Functional Equations and Inequalities. C.I.M.E. Summer Schools, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11004-7_8

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