By Constantin P. Niculescu, Lars-Erik Persson (auth.)

Convex capabilities play a massive position in lots of branches of arithmetic, in addition to different parts of technological know-how and engineering. the current textual content is aimed to a radical advent to modern convex functionality concept, which includes a robust and stylish interplay among research and geometry. a wide number of topics are coated, from one genuine variable case (with all its mathematical gemstones) to a couple of the main complicated issues similar to the convex calculus, Alexandrov’s Hessian, the variational technique of partial differential equations, the Prékopa-Leindler kind inequalities and Choquet's idea.

This e-book can be utilized for a one-semester graduate path on Convex capabilities and purposes, and in addition as a important reference and resource of concept for researchers operating with convexity. the single must haves are a history in complex calculus and linear algebra. every one part ends with routines, whereas each one bankruptcy ends with reviews overlaying supplementary fabric and old details. Many effects are new, and the total booklet displays the authors’ personal event, either in educating and research.

About the authors:

Constantin P. Niculescu is a Professor within the division of arithmetic on the college of Craiova, Romania. Dr. Niculescu directs the Centre for Nonlinear research and Its functions and likewise the graduate application in utilized arithmetic at Craiova. He obtained his doctorate from the college of Bucharest in 1974. He released in Banach area idea, Convexity Inequalities and Dynamical structures, and has acquired a number of prizes either for examine and exposition.

Lars Erik Persson is Professor of arithmetic at Luleå collage of know-how and Uppsala collage, Sweden. he's the director of middle of utilized arithmetic at Luleå, a member of the Swedish nationwide Committee of arithmetic on the Royal Academy of Sciences, and served as President of the Swedish Mathematical Society (1996-1998). He acquired his doctorate from Umeå college in 1974. Dr. Persson has released on interpolation of operators, Fourier research, functionality idea, inequalities and homogenization thought. He has bought numerous prizes either for examine and teaching.

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Exercises 1. (An application of the second derivative test of convexity) (i) Prove that the functions log((eax −1)/(ex −1)) and log(sinh ax/ sinh x) are convex on R if a ≥ 1. √ √ (ii) Prove that the function b log cos(x/ b) − a log cos(x/ a) is convex on (0, π/2) if b ≥ a ≥ 1. 2. Suppose that 0 < a < b < c (or 0 < b < c < a, or 0 < c < b < a). 2 to infer the following inequalities: 26 1 Convex Functions on Intervals abα + bcα + caα > acα + baα + cbα for α ≥ 1; ab bc ca > ac cb ba ; a(c − b) b(a − c) c(b − a) (iii) + + > 0.

1. 13). 13) we get x b (y − a) dµ(y) + a (b − y) dµ(y) = f (a) + f (b) − 2f x a+b , 2 which yields 0≤ 1 b−a (x − a)(b − x) dµ(x) ≤ f (a) + f (b) − 2f I a+b . 15) In the general case we apply the above reasoning to the restriction of f to the interval [a + ε, b − ε] and then pass to the limit as ε → 0. The uniqueness of µ is a consequence of the fact that f = µ in the sense of distribution theory. 6 Integral Representation of Convex Functions 39 for all ϕ ∈ Cc2 (I), which yields f (x)ϕ (x) dx = I G(x, y)ϕ (x) dx dµ(y) = I×I ϕ(y) dµ(y), I due to the symmetry of G and the Fubini–Tonelli theorem.

Z−a A symmetric argument will then yield the existence of f+ (a) and the availability of the relation f− (a) ≤ f+ (a). 1 yields f (u) − f (x) f (v) − f (x) f (v) − f (y) ≤ ≤ , u−x v−x v−y so letting u → x+ and v → y−, we obtain that f+ (x) ≤ f− (y). 22 1 Convex Functions on Intervals Because f admits ﬁnite lateral derivatives at each interior point, it will be continuous at each interior point. 3, every continuous convex function f (deﬁned on a nondegenerate compact interval [a, b]) admits derivatives f+ (a) and f− (b) at the endpoints, but they can be inﬁnite, −∞ ≤ f+ (a) < ∞ and − ∞ < f− (b) ≤ ∞.