By George A. Anastassiou
This monograph provides univariate and multivariate classical analyses of complicated inequalities. This treatise is a end result of the author's final 13 years of analysis paintings. The chapters are self-contained and several other complicated classes may be taught out of this e-book. vast heritage and motivations are given in each one bankruptcy with a finished checklist of references given on the finish. the themes lined are wide-ranging and various. contemporary advances on Ostrowski style inequalities, Opial style inequalities, Poincare and Sobolev variety inequalities, and Hardy-Opial sort inequalities are tested. Works on traditional and distributional Taylor formulae with estimates for his or her remainders and functions in addition to Chebyshev-Gruss, Gruss and comparability of potential inequalities are studied. the consequences awarded are in most cases optimum, that's the inequalities are sharp and attained. purposes in lots of parts of natural and utilized arithmetic, corresponding to mathematical research, likelihood, traditional and partial differential equations, numerical research, details concept, etc., are explored intimately, as such this monograph is appropriate for researchers and graduate scholars. will probably be an invaluable instructing fabric at seminars in addition to a useful reference resource in all technological know-how libraries.
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Additional info for Advanced Inequalities (Series on Concrete and Applicable Mathematics)
6, mostly related to multivariate trapezoid and midpoint rules. 59. The surprising fact there is, that only a very small number of sets of boundary conditions is needed comparely to the higher order of differentiation of the involved functions. 69. 8 ([171, p. 17]). Let f ∈ C n ([a, b]), n ∈ N, x ∈ [a, b]. 1) (b − a)k−1 x−a [f (k−1) (b) − f (k−1) (a)] Bk k! b−a with the convention φ0 (x) = 0, and (b − a)n−1 b ∗ x − t x−a Bn − Bn f (n) (t)dt. 2) n! 3) Bk∗ (x + 1) = Bk∗ (x), x ∈ R. 1]). 5) , k ≥ 0.
20. 4. Additionally assume that f (n) p > 1. Here p, q : p1 + 1q = 1. Then |θ1,n | ≤ f (n) b p · a a |P (x, s1 )| · i=1 |P (si , si+1 )| · P (sn−1 , •) q ds1 ds2 · · · dsn−1 . 21. 6. Additionally assume that f p > 1. Here p, q : p1 + q1 = 1. Then b 1 f (t)dg(t) ≤ f (g(b) − g(a)) a means integration with respect to t. f (x) − Here · q,t p · P (g(x), g(t)) 1 (g(b) − g(a)) b a f (t)dg(t) − b · b P (g(x), g(t))dt a p q,t . 22. 8. Additionally assume that f p > 1. Here p, q : 1p + q1 = 1. 32) p < +∞, 1 (g(b) − g(a)) f (t1 )dg(t1 ) a b ≤ f p· a |P (g(x), g(t))| · P (g(t), g(t1 ) q,t1 · dt.
2) is valid again. 2) is a generalized Euler type identity, see also . We set b 1 f (t)dt ∆n (x) := f (x) − b−a a n−1 − k=1 (b − a)k−1 x−a [f (k−1) (b) − f (k−1) (a)], x ∈ [a, b]. 3) Bk k! 2) that (b − a)n−1 x−a x−t ∆n (x) = Bn − Bn∗ f (n) (t)dt. 4) n! b − a b − a [a,b] In this chapter we give sharp, namely attained, upper bounds for |∆4 (x)| and tight upper bounds for |∆n (x)|, n ≥ 5, x ∈ [a, b], with respect to L∞ norm. 1) for higher order derivatives. High computational difficulties in this direction prevent us for shoming sharpness for n ≥ 5 cases.
Advanced Inequalities (Series on Concrete and Applicable Mathematics) by George A. Anastassiou