Functional Analysis

Download Convex Functions and their Applications: A Contemporary by Constantin Niculescu PDF

By Constantin Niculescu

Thorough advent to an immense region of arithmetic includes contemporary effects contains many workouts

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Prove that a continuous convex function f : [a, b] → R can be extended to a convex function on R if and only if f+ (a) and f− (b) are finite. 6. 10 to prove that the sine function is strictly concave on [0, π]. Infer that x a cot a sin a x ≤ sin x ≤ sin a a a for every a ∈ (0, π/2] and every x ∈ [0, a]. For a = π/2 this yields the classical inequality of Jordan. 7. Let f : [0, 2π] → R be a convex function. Prove that an = 2π 1 π f (t) cos nt dt ≥ 0 for every n ≥ 1. 0 8. (J. L. W. V. Jensen [115]) Prove that a function f : [0, M ] → R is nondecreasing if and only if n n λk f (xk ) ≤ k=1 n λk f k=1 xk k=1 for all finite families λ1 , .

5 The Subdifferential 29 Exercises 1. (Kantorovich’s inequality) Let m, M, a1 , . . , an be positive numbers, with m < M . Prove that the maximum of n f (x1 , . . , xn ) = n ak xk ak /xk k=1 k=1 for x1 , . . ,n} 2 ak − min k∈X ak . k∈ X Remark. The following particular case 1 n n xk k=1 1 n n k=1 1 xk (M + m)2 (1 + (−1)n+1 )(M − m)2 − 4M m 8M mn2 ≤ represents an improvement on Schweitzer’s inequality for odd n. 2. Let ak , bk , ck , mk , Mk , mk , Mk be positive numbers with mk < Mk and mk < Mk for k ∈ {1, .

K=1 In particular, x21 + · · · + x2n x1 + · · · + xn 2 M 2 + ≤ , n n 4 which represents an additive converse to the Cauchy–Buniakovski–Schwarz inequality. 1, the function n n λk xpk − E(x1 , . . , xn ) = k=1 p λk xk , k=1 attains its supremum on [0, M ]n at a point whose coordinates are either 0 or M . Therefore sup E(x1 , . . , xn ) does not exceed M p ·sup {s − sp | s ∈ [0, 1]} = (p − 1) pp/(1−p) M p . 5 The Subdifferential 29 Exercises 1. (Kantorovich’s inequality) Let m, M, a1 , . . , an be positive numbers, with m < M .

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