By Constantin Niculescu
Convex services play a tremendous function in just about all branches of arithmetic in addition to different components of technological know-how and engineering. This booklet is an intensive creation to modern convex functionality concept addressed to every person whose examine or educating pursuits intersect with the sphere of convexity. It covers a wide number of matters, from the single genuine variable case (with all its mathematical gemstones) to a few of the main complex issues similar to Choquet's conception, the Prékopa-Leindler sort inequalities and their ramifications, in addition to the variational process of partial differential equations and convex programming. Many effects are new and the entire booklet displays the authors’ personal event, either in instructing and examine. The ebook can function a reference and resource of notion to researchers in numerous branches of arithmetic and engineering and it may well even be used for graduate courses.
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Extra resources for Convex Functions and their Applications: A Contemporary Approach (CMS Books in Mathematics)
Sample text
Xn ) ≥ 0 for x1 , . . 9) where f is a continuous and positively homogeneous function of degree 1 (that is, f (λx1 , . . , λxn ) = λf (x1 , . . , xn ) for λ ≥ 0), extend to the context of Banach lattices, via a functional calculus invented by A. J. Yudin and J. L. Krivine. This allows us to replace the real variables of f by positive elements of a Banach lattice. See [147, Vol. 2, pp. 40–43]. Particularly, this is the case of the AM–GM inequality, Rogers–H¨ older’s inequality, and Minkowski’s inequality.
Clearly, H is continuous and H(a) = H(b). 2 is immediate. The same is true when H attains its infimum at an interior point of [a, b]. 2 works for all c in (a, b). 1. Clearly, we may assume that f is also continuous. We shall show (by reductio ad absurdum) that 28 1 Convex Functions on Intervals E(x1 , . . , xk , . . , xn ) ≤ sup{E(x1 , . . , mk , . . , xn ), E(x1 , . . , Mk , . . , xn )} for all (x1 , x2 , . . , xn ) ∈ Ω and all k ∈ {1, . . , n}. In fact, if E(x1 , x2 , . . , xn ) > sup{E(m1 , x2 , .
If a = t0 < t1 < · · · < tn = b is a division of [a, b], then f− (tk−1 ) ≤ f+ (tk−1 ) ≤ f (tk ) − f (tk−1 ) ≤ f− (tk ) ≤ f+ (tk ) tk − tk−1 for all k. Since n f (b) − f (a) = [f (tk ) − f (tk−1 )], k=1 a moment’s reflection shows that b f (b) − f (a) = a b f− (t) dt = f+ (t) dt. 1. 2 There exist convex functions whose first derivative fails to exist on a dense set. For this, let r1 , r2 , r3 , . . be an enumeration of the rational numbers in [0, 1] and put ϕ(t) = {k|rk ≤t} Then 1 . 2k x f (x) = ϕ(t) dt 0 is a continuous convex function whose first derivative does not exist at the points rk .