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 analysis paintings. The chapters are self-contained and a number of other complex classes might be taught out of this publication. wide heritage 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 style inequalities, Opial kind inequalities, Poincare and Sobolev style inequalities, and Hardy-Opial sort inequalities are tested. Works on traditional and distributional Taylor formulae with estimates for his or her remainders and purposes in addition to Chebyshev-Gruss, Gruss and comparability of ability inequalities are studied. the consequences provided are as a rule optimum, that's the inequalities are sharp and attained. purposes in lots of parts of natural and utilized arithmetic, similar to mathematical research, likelihood, usual and partial differential equations, numerical research, info concept, etc., are explored intimately, as such this monograph is appropriate for researchers and graduate scholars. will probably be an invaluable educating fabric at seminars in addition to a useful reference resource in all technological know-how libraries.
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Additional resources for Advanced Inequalities (Series on Concrete and Applicable Mathematics)
Example text
5in Book˙Adv˙Ineq ADVANCED INEQUALITIES 46 n for any (xj+1 , . . , xn ) ∈ |Bj | ≤ [ai , bi ], xj ∈ [aj , bj ]. 51) we get (bj − aj )m−1 m! − × [ai ,bi ] (bi − ai ) i=1 ds1 · · · dsj ∂mf (s1 , . . , sj , xj+1 , . . , xn ) ∂xm j j [ai ,bi ] j−1 j−1 m! i=1 − qj i=1 (bj − aj )m−1 = 1/pj pj xj − s j bj − a j ∗ Bm xj − a j bj − a j Bm j j−1 i=1 = i=j+1 ∗ Bm i=1 (bi − ai ) 1/pj xj − s j bj − a j pj m− q1 j−1 (bj − aj ) m! 59) Bm (λj ) aj ∂mf (· · · , xj+1 , . . 60) j qj , [ai ,bi ] i=1 −1/qj (bi − ai ) i=1 1/pj 1 |Bm (λj ) 0 ∂mf (· · · , xj+1 , .
28. 35. 10, m, n ∈ N, xi ∈ [ai , bi ], i = 1, 2, . . , n. 46), for j = 1, . . , n. Then f (x1 , . . , xn ) − j=1 n (bi − ai ) i=1 n m−1 n ≤ 1 n |Bj | + j=1 xj − a j bj − a j k=1 ω1 [ai ,bi ] f (s1 , s2 , . . , sn )ds1 ds2 · · · dsn i=1 (bj − aj )k−1 Bk k! ∂ k−1 f (. . , xj+1 , . . , xn ), bj − aj ∂xjk−1 .
Xn ) ∂x2r+1 j j 1, [ai ,bi ] i=1 xj − a j 2(2r + 1)! + B2r+1 (2π)2r+1 (1 − 2−2r ) bj − a j . 82) And at last 3) When m = 1, then n |E1f (x1 , . . , xn )| ≤ 1 j−1 j=1 i=1 1 + xj − 2 × Proof. (bi − ai ) ∂f (. . , xj+1 , . . , xn ) ∂xj j 1, [ai ,bi ] i=1 aj + b j 2 . 28. 35. 10, m, n ∈ N, xi ∈ [ai , bi ], i = 1, 2, . . , n. 46), for j = 1, . . , n. Then f (x1 , . . , xn ) − j=1 n (bi − ai ) i=1 n m−1 n ≤ 1 n |Bj | + j=1 xj − a j bj − a j k=1 ω1 [ai ,bi ] f (s1 , s2 , . . , sn )ds1 ds2 · · · dsn i=1 (bj − aj )k−1 Bk k!
Advanced Inequalities (Series on Concrete and Applicable Mathematics) by George A. Anastassiou
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