By Constantin Niculescu

Convex services play an enormous function in just about all branches of arithmetic in addition to different parts of technological know-how and engineering. This booklet is a radical advent to modern convex functionality thought addressed to everybody whose study or educating pursuits intersect with the sector of convexity. It covers a wide number of matters, from the only actual variable case (with all its mathematical gemstones) to a couple of the main complicated subject matters reminiscent of Choquet's idea, the Prékopa-Leindler variety inequalities and their ramifications, in addition to the variational process of partial differential equations and convex programming. Many effects are new and the entire publication displays the authors’ personal event, either in educating and learn. The booklet can function a reference and resource of proposal to researchers in different branches of arithmetic and engineering and it may well even be used for graduate courses.

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**Extra info for Convex Functions and their Applications: A Contemporary Approach (CMS Books in Mathematics)**

**Sample text**

4 above), if the Berwald inequality works for two concave sequences, then it also works for all linear combinations of them, with positive coeﬃcients. Then apply the assertion (ii) of the preceding exercise. 7 Conjugate Convex Functions The aim of this section is to develop a concept of duality between convex functions which makes possible an easy handling of some problems. The basic idea can be traced back to Young’s inequality. If ϕ : [0, ∞) → [0, ∞) is an increasing and continuous function with ϕ(0) = 0 and ϕ(x) → ∞ as x → ∞, then ϕ−1 exists and has the same properties as ϕ.

Proof. Let u, v ∈ I, u = v, and let t ∈ (0, 1). Then (1 − t)u + tv ∈ int I, so that for all λ ∈ ∂f ((1 − t)u + tv) we get f (u) ≥ f ((1 − t)u + tv) + t(u − v) · λ, f (v) ≥ f ((1 − t)u + tv) − (1 − t)(u − v) · λ. By multiplying the ﬁrst inequality by 1 − t, the second one by t and then adding them side by side, we get (1 − t)f (u) + tf (v) ≥ f ((1 − t)u + tv), hence f is a convex function. We shall illustrate the importance of the subdiﬀerential by proving two classical results. 10. 4 (The Hardy–Littlewood–P´ olya inequality) Suppose that f is a convex function on an interval I and consider two families x1 , .

Xn ] = f [x0 , . . , xn−1 ] − f [x1 , . . , xn ] . x0 − xn Thus the 1-convex functions are the nondecreasing functions, while the 2-convex functions are precisely the classical convex functions. 2. As T. Popoviciu noticed in his book [205], if f is n-times diﬀerentiable, with f (n) ≥ 0, then f is n-convex. See [196] and [212] for a more detailed account on the theory of n-convex functions. 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.