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

Convex capabilities play a massive function in lots of branches of arithmetic, in addition to different parts of technology and engineering. the current textual content is aimed to a radical advent to modern convex functionality concept, which includes a strong and stylish interplay among research and geometry. a wide number of topics are lined, from one genuine variable case (with all its mathematical gemstones) to a few of the main complex subject matters equivalent to the convex calculus, Alexandrov’s Hessian, the variational procedure of partial differential equations, the Prékopa-Leindler variety inequalities and Choquet's thought.

This booklet can be utilized for a one-semester graduate direction on Convex capabilities and functions, and in addition as a important reference and resource of proposal for researchers operating with convexity. the one necessities are a history in complex calculus and linear algebra. every one part ends with workouts, whereas each one bankruptcy ends with reviews overlaying supplementary fabric and ancient info. Many effects are new, and the full e-book 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 software in utilized arithmetic at Craiova. He bought his doctorate from the collage of Bucharest in 1974. He released in Banach house concept, Convexity Inequalities and Dynamical structures, and has acquired numerous prizes either for study and exposition.

Lars Erik Persson is Professor of arithmetic at Luleå college 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 bought his doctorate from Umeå college in 1974. Dr. Persson has released on interpolation of operators, Fourier research, functionality thought, inequalities and homogenization concept. He has got numerous prizes either for learn and teaching.

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Proof. First, we shall prove that f+ (a) ∈ ∂f (a) for each a ∈ int I (and also at the leftmost point of I, provided that f is diﬀerentiable there). 5 The Subdiﬀerential 31 f (x) ≥ f (a) + f+ (a) · (x − a). If x ≤ a, then a similar argument leads us to f (x) ≥ f (a) + f− (a) · (x − a); or f− (a) · (x − a) ≥ f+ (a) · (x − a), because x − a ≤ 0. Analogously, we can argue that f− (a) ∈ ∂f (a) for all a ∈ int I (and also for the rightmost point in I provided that f is diﬀerentiable at that point). 3.

The ﬁrst inequality is that of Jensen. The second can be obtained from f (M1 (g)) ≥ f (g(x)) + (M1 (g) − g(x)) · ϕ(g(x)) for all x ∈ X by integrating both sides over X. 5 Let f be a convex function deﬁned on an open interval I and let ϕ : I → R be a function such that ϕ(x) ∈ ∂f (x) for all x ∈ I. Then n 0≤ n λk f (xk ) − f k=1 n ≤ λk xk k=1 n λk xk ϕ(xk ) − k=1 n λk xk k=1 λk ϕ(xk ) k=1 for all x1 , . . , xn ∈ I and all λ1 , . . , λn ∈ [0, 1], with n k=1 λk = 1. 5 is indicated in Exercise 6. 6 (The complete form of Chebyshev’s inequality) Let (X, Σ, µ) be a ﬁnite measure space, let g : X → R be a µ-integrable function and let ϕ be a nondecreasing function given on an interval that includes the image of g and such that ϕ ◦ g and g · (ϕ ◦ g) are integrable functions.

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 , . . , xn ), E(M1 , x2 , . . , xn )} for some (x1 , x2 , . . , xn ) ∈ Ω, we consider the function h : [m1 , M1 ] → R, h(x) = E(x, x2 , . . , xn ). 2, there exists a ξ ∈ (m1 , x1 ) such that h(x1 ) − h(m1 ) ≤ (x1 − m1 )Dh(ξ). Since h(x1 ) > h(m1 ), it follows that Dh(ξ) > 0, equivalently, Df (ξ) > Df (λ1 ξ + λ2 x2 + · · · + λn xn ). Or, Df = f+ is a nondecreasing function on (a, b), which yields ξ > λ1 ξ + λ2 x2 + · · · + λn xn , and thus ξ > (λ2 x2 + · · · + λn xn )/(λ2 + · · · + λn ).