By Peter Bullen
Adding new effects that experience seemed within the final 15 years, Dictionary of Inequalities, moment Edition presents a good way for researchers to find an inequality by means of identify or topic. This variation deals an up to date, alphabetical directory of every inequality with a quick assertion of the outcome, a few reviews, references to comparable inequalities, and resources of data on proofs and different information. The booklet doesn't contain proofs and makes use of easy mathematical terminology up to attainable, permitting readers to entry a consequence or inequality effortlessly.
New to the second one Edition
- More than a hundred new inequalities, together with lately came across ones
- Updated inequalities based on the latest research
- Inclusion of a reputation index
- Updated bibliography that includes URLs for vital references
The publication quite often offers the commonest model of the inequality and later provides extra common effects as extensions or variations. Inequalities that exist at a variety of degrees of generality are awarded within the least difficult shape with the opposite types as extensions or less than a distinct heading. the writer additionally clarifies any non-standard notations and contains cross-references for transliterations.
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Extra info for Dictionary of Inequalities, Second Edition
12], Mitrinovi´c, Peˇcari´c & Volenec [MPV]; Klamkin , . 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).
Iii) The case α = 1/(n + 1), n = 1, 2, . . of (∼B) follows from properties a certain polynomial; see: Polynomial Inequalities Comments (i). ✐ ✐ ✐ ✐ ✐ ✐ “K23178” — 2015/5/15 — ✐ 20 ✐ 2 Backward–Bushell (iv) In general the result follows by a simple application of Taylor’s Theorem. Variants (a) If a, b > 0, a = b, and α > 1 or α < 0, aα bα−1 > αa − (α − 1)b; (1) and (∼1) holds when 0 < α < 1. (b) If a, b and α are as in (a) and α + β = 1 then aα bβ > αa + βb; (2) and if α > 0, β > 0 and α + β = 1 then (∼2) holds.
690, 696]. References Abi-Khuzam’s Inequality If A, B, C are the angles of a triangle then √ 3 2 3 ABC, sin A sin B sin C ≤ 2π with equality if and only if the triangle is equilateral. Reference Klamkin . Absolutely Monotonic Function Inequalities (a) If f, f ∈ C ∞ [a, b] , is absolutely monotonic then f (k) ≥ 0, a < x < b, k ∈ N. (b) If f is absolutely monotonic on ] − ∞, 0 ] then f (k) f (k+2) ≥ f (k+1) , k ∈ N. (c) If f is absolutely monotonic on ] − ∞, 0 [ then det f (i+j−2) 1≤i≤n 1≤j≤n ≥0 and f (i+j−1) det 1≤i≤n 1≤j≤n ≥ 0, n ≥ 1.