Download Advanced Inequalities (Series on Concrete and Applicable by George A. Anastassiou PDF

By George A. Anastassiou

This monograph offers univariate and multivariate classical analyses of complicated inequalities. This treatise is a end result of the author's final 13 years of study paintings. The chapters are self-contained and a number of other complex classes might be taught out of this booklet. wide historical past and motivations are given in every one bankruptcy with a finished record of references given on the finish. the subjects coated are wide-ranging and numerous. fresh advances on Ostrowski style inequalities, Opial variety inequalities, Poincare and Sobolev sort inequalities, and Hardy-Opial kind inequalities are tested. Works on usual and distributional Taylor formulae with estimates for his or her remainders and functions in addition to Chebyshev-Gruss, Gruss and comparability of ability inequalities are studied. the consequences awarded are often optimum, that's the inequalities are sharp and attained. purposes in lots of parts of natural and utilized arithmetic, similar to mathematical research, chance, traditional and partial differential equations, numerical research, info idea, etc., are explored intimately, as such this monograph is appropriate for researchers and graduate scholars. it is going to be an invaluable educating fabric at seminars in addition to a useful reference resource in all technological know-how libraries.

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299 in [158] and Problem 14(c), p. 264 in [224]. And that f (n−1) as implied absolutely continuous it is also of bounded variation. 2) is valid again. 2) is a generalized Euler type identity, see also [171]. 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.

Xj−1 , ·, xj+1 , . . , xn ) j exists and is real valued with the possibility of being infinite only over an at most countable subset of (aj , bj ). 5) Parts #3, #4 are true for all n (x1 , . . , xj−1 , xj+1 , . . , xn ) ∈ [ai , bi ]. i=1 i=j = 1, . . , m − 2,   j−1 ∂ f ·, ·, ·, · · · , ·, xj , xj+1 , . . , xn  qj ·, ·, · · · , · := ∂xj 6) The functions for j = 2, . . , n;   j−1 j−1 are continuous on i=1 [ai , bi ], for each (xj , xj+1 , . . , xn ) ∈ n [ai , bi ]. i=j 7) The functions for each j = 1, .

033333 . 7. 2, case of n = 4. 12) with λ = x−a b−a . Furthermore we have that |∆4 (x)| ≤ (b − a)4 (4) f 720 ∞, ∀x ∈ [a, b]. 8. 2, case of n = 4. 17) is sharp, namely it is attained when x = a, b by the functions (t − a)4 and (t − b)4 . Proof. We have ∆4 (a) = ∆4 (b) = (b − a) 1 f (a) + f (b) − (f (b)−f (a))− 2 12 b−a b f (t)dt. 17) we have |∆4 (a)| = |∆4 (b)| ≤ (b − a)4 (4) f 720 ∞. 19) is attained. 17) sharp. |∆4 (a)| = |∆4 (b)| = The trapezoid and midpoint inequalities follow. 9. 2, case of n = 4.

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