By Peter S. Bullen
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Extra resources for Dictionary of Inequalities
47–48]; Borwein & Borwein [BB, pp. 1–5]; Almkvist & Berndt , Borwein & Borwein . References If f (x) = n∈N an xn , 0 ≤ x < 1, An+1 = ak , Φ : [0, ∞[→ [0, ∞[ increasing and convex, and β > −2 then Askey-Karlin Inequalities n k=0 1 0 Φ ◦ |f |(x)(1 − x)β dx 1 0 Comment Reference Φ ◦ |f |(x) dx (1 − x)2 ≤ (β + 1)! ≤ Φ n∈N n∈N Φ(|An+1 |) |An+1 | n+1 n! ; (n + β + 2)! These results have been generalized by Pachpatte. Pachpatte . © 2015 by Taylor & Francis Group, LLC ✐ ✐ ✐ ✐ ✐ ✐ “K23178” — 2015/5/15 — ✐ ✐ 2 Backward–Bushell Backward H¨ older Inequality See: Young’s Convolution Inequality Comments (iii).
References If R = [0, a]×[0, b] and f ∈ C 2 (R), with f (x, 0) = f (0, y) = 0, 0 ≤ x ≤ a, 0 ≤ y ≤ b, then Agarwal’s Inequality ab ′′ |f ||f12 |≤ √ 2 2 R R ′′ 2 |f12 | . The best value of the numerical constant on the right-hand side is √ not known; but it is greater than 3 + 13 24. Comment Reference [GI3, pp. 501–503]. Ahlswede-Daykin Inequality Let fi : X → [0, ∞ [, 1 ≤ i ≤ 4, where X is a distributive lattice. If for all a, b ∈ X f1 (a)f2 (b) ≤ f3 (a ∨ b)f4 (a ∧ b), (1) then for all A, B ⊆ X, f (a) a∈A b∈B f (b) ≤ f (x) x∈A∨B f (y).
Xk )|2 , where λk , k ∈ I, is any set of complex numbers. (iii) If the orthonormal sequence is complete then (1), and (2), becomes an equality known as Parseval’s equality. bj |, (3) with equality if and only if a, bk , 1 ≤ k ≤ m, are linearly dependent. (iv) The case m = 1 of (3) is just (C). (v) For a further extension see: Ostrowski’s Inequalities Hausdorff-Young Inequalities, Paley’s Inequalities. Comments Comments (i), [EM, vol. 1, pp. 373–374; vol. 7, pp. 93–94], [MPF, pp. 391–405]; Courant & Hilbert [CH, pp.
Dictionary of Inequalities by Peter S. Bullen