Download e-book for iPad: Dictionary of Inequalities, Second Edition by Peter Bullen

By Peter Bullen

ISBN-10: 148223761X

ISBN-13: 9781482237610

ISBN-10: 1482237628

ISBN-13: 9781482237627

Adding new effects that experience seemed within the final 15 years, Dictionary of Inequalities, moment Edition offers a good way for researchers to find an inequality via identify or topic. This version deals an updated, alphabetical directory of every inequality with a brief assertion of the outcome, a few reviews, references to comparable inequalities, and resources of data on proofs and different info. The ebook doesn't comprise 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 comprises URLs for very important references

The publication in most cases provides the most typical model of the inequality and later supplies extra common effects as extensions or versions. Inequalities that exist at numerous degrees of generality are provided within the least difficult shape with the opposite varieties as extensions or less than a special heading. the writer additionally clarifies any non-standard notations and comprises cross-references for transliterations.

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Extra info for Dictionary of Inequalities, Second Edition

Example text

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 [157]. 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.

15], Hewitt & Stromberg [HS, pp. 36–37]. References then n a21 − n a2i b21 i=2 − n i=2 If a, b are n-tuples with a21 − Acz´ el & Varga’s Inequality 2 n b2i i=2 ≤ a1 b 1 − a2i > 0 ai b i , i=2 with equality if and only if a ∼ b. Comments (i) This result can be considered as an analogue of (C) in a nonEuclidean geometry. As this space is called a Lorentz space this inequality is sometimes called a Lorentz inequality. (ii) The Acz´el inequality was used to prove the Aleksandrov-Fenchel Inequality.

The inequality of Acz´ el & Varga is an analogue of (C). Extension [Kivinukk] the conjugate index If f ∈ Lp ([a, b]), 1 ≤ p ≤ ∞, is convex, and if q is 1 − λ f (a)+λf (b)−f 1 − λ a+λb ≤ 21/p (q+1)−1/q λ(1−λ)(b−a)1+1/q ||f ′′ ||p,[a,b] . (iv) The case p = ∞, λ = 1/2 is (a) above. (v) See also Minc-Sathre Inequality Comment (ii). (vi) There are numerous other inequalities by Alzer. Comments References [GI6, pp. 5–16] [H, pp. 171–173]; Alzer [18],[24], Bullen [79], Kiv- inukk [154]. Analytic Function Inequalities (a) If f is analytic in D and if 0 ≤ r1 ≤ r2 < 1 then, Mp (f ; r1 ) ≤ Mp (f ; r2 ), 0 ≤ p ≤ ∞.

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