Download PDF by George A. Anastassiou: Advanced inequalities

By George A. Anastassiou

ISBN-10: 9814317624

ISBN-13: 9789814317627

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 several other complex classes will be taught out of this ebook. wide historical past and motivations are given in each one bankruptcy with a finished checklist of references given on the finish. the themes coated are wide-ranging and various. contemporary advances on Ostrowski sort inequalities, Opial style inequalities, Poincare and Sobolev variety inequalities, and Hardy-Opial variety inequalities are tested. Works on usual and distributional Taylor formulae with estimates for his or her remainders and purposes in addition to Chebyshev-Gruss, Gruss and comparability of capability inequalities are studied. the implications awarded are as a rule optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, corresponding to mathematical research, likelihood, traditional and partial differential equations, numerical research, details conception, etc., are explored intimately, as such this monograph is acceptable for researchers and graduate scholars. it will likely be an invaluable educating fabric at seminars in addition to a useful reference resource in all technology libraries.

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46). In particular suppose that   j j ∂mf   · · · , xj+1 , . . , xn ∈ L∞ [ai , bi ] , ∂xm j i=1 n for any (xj+1 , . . , xn ) ∈ [ai , bi ], all j = 1, . . , n. Then for any i=j+1 n (xj , xj+1 , . . , xn ) ∈ [ai , bi ] we have i=j |Bj | = |Bj (xj , xj+1 , . . , xn )| ≤ (bj − aj )m m!   j ∂mf  × · · · , xj+1 , . . )2 xj − a j 2 |B2m | + Bm (2m)! 58) j ∞, [ai ,bi ] i=1 for all j = 1, . . , n. 25. 23. Let pj , qj > 1: j = 1, . . , n, with the assumption that j ∂mf (· · · , xj+1 , . .

Xj+1 , . . , xn ) ∂x2r j × (1 − 2−2r )|B2r | + 2−2r B2r − B2r j 1, [ai ,bi ] i=1 xj − a j bj − a j . 5in Book˙Adv˙Ineq Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities 49 2) When m = 2r + 1, r ∈ N we obtain |Bj | ≤ ∂ 2r+1 f (. . , xj+1 , . . , xn ) ∂x2r+1 j (bj − aj )2r j−1 (2r + 1)! i=1 × (bi − ai ) j 1, [ai ,bi ] i=1 2(2r + 1)! xj − a j + B2r+1 (2π)2r+1 (1 − 2−2r ) bj − a j . 72) 3) When m = 1 we get |Bj | ≤ 1 j−1 i=1 (bi − ai ) ∂f (. . , xj+1 , . .

We use here the sequence {Bk (t), k ≥ 0} of Bernoulli polynomials which is uniquely determined by the following identities: k ≥ 1, Bk (t) = kBk−1 (t), and Bk (t + 1) − Bk (t) = ktk−1 , B0 (t) = 1 k ≥ 0. The values Bk = Bk (0), k ≥ 0 are the known Bernoulli numbers. We need to mention 1 1 B0 (t) = 1, B1 (t) = t − , B2 (t) = t2 − t + , 2 6 3 1 1 B3 (t) = t3 − t2 + t, B4 (t) = t4 − 2t3 + t2 − , 2 2 30 5 1 5 t2 1 5 . B5 (t) = t5 − t4 + t3 − t, and B6 (t) = t6 − 3t5 + t4 − + 2 3 6 2 2 42 ∗ Let {Bk (t), k ≥ 0} be a sequence of periodic functions of period 1, related to Bernoulli polynomials as Bk∗ (t) = Bk (t), Bk∗ (t + 1) = Bk∗ (t), 0 ≤ t < 1, 21 t ∈ R.

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Advanced inequalities by George A. Anastassiou


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