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 fruits of the author's final 13 years of analysis paintings. The chapters are self-contained and a number of other complex classes should be taught out of this booklet. vast historical past and motivations are given in each one bankruptcy with a accomplished checklist of references given on the finish. the themes lined are wide-ranging and various. fresh advances on Ostrowski kind inequalities, Opial style inequalities, Poincare and Sobolev sort inequalities, and Hardy-Opial kind 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 potential inequalities are studied. the consequences provided are quite often optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, comparable 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. it is going to be an invaluable instructing fabric at seminars in addition to a useful reference resource in all technological know-how libraries.

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**Extra resources for Advanced Inequalities (Series on Concrete and Applicable Mathematics)**

**Example text**

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 , . . , xn ) ∂xm j − Bm (tj )|pj dtj . 26. 20. 46) and pj , qj > 1 : p1j + q1j = 1; j = 1, . . , n. In particular we suppose that ∂mf (· · · , xj+1 , . . , xn ) ∈ Lqj ∂xm j j [ai , bi ] , i=1 n for any (xj+1 , .

2 Consider p4 (t) := B4 (t) − B4 (λ) = t4 − 2t3 + t2 − λ4 + 2λ3 − λ2 . 10) We need to compute 1 I4 (λ) := 0 |p4 (t)|dt, 0 ≤ λ ≤ 1. 5. We find 14 3 1 16λ5 4 2 5 − 7λ + 3 λ − λ + 30 , I4 (λ) = 5 3 − 16λ + 9λ4 − 26λ + 3λ2 − 1 , 5 3 10 which is continuous in λ ∈ [0, 1]. 12) 1/2 ≤ λ ≤ 1, 1 , 30 7 . 13) Proof. Here all calculations are done by Mathematica 4. The equation p4 (t) = 0 has four real roots, r1 = 1 − λ, r2 = λ, r3 = 1 1− 2 1 + 4λ − 4λ2 and r4 = 1 1+ 2 1 + 4λ − 4λ2 . We find the following orders: i) if 0 ≤ λ ≤ 1/2, then r3 ≤ 0 ≤ r 2 ≤ r 1 ≤ 1 ≤ r 4 , and ii) if 1/2 ≤ λ ≤ 1, then r3 ≤ 0 ≤ r 1 ≤ r 2 ≤ 1 ≤ r 4 .

Sn , sn+1 ) Tj . dsn+1 ds1 . . , dsn + ∂xm n+1 j=1 xn+1 − sn+1 bn+1 − an+1 Hence f (x1 , . . , xn+1 ) = n+1 1 n+1 n+1 i=1 f (s1 , . . , sn+1 ) ds1 . . dsn+1 + [ai ,bi ] (bi − ai ) Tj , j=1 i=1 where Tn+1 (xn+1 ) := Tn+1 := n i=1 n [ai ,bi ] i=1 m−1 1 (bi − ai ) k=1 (bn+1 − an+1 )k−1 xn+1 − an+1 Bk k! bn+1 − an+1 ∂ k−1 f (s1 , . . , sn , bn+1 ) ∂ k−1 f − k−1 s1 , . . , sn , an+1 k−1 ∂xn+1 ∂xn+1 + (bn+1 − an+1 )m−1 n m! i=1 ∗ −Bm Bm n+1 [ai ,bi ] bi − a i i=1 ∂mf s1 , . . , sn , sn+1 ∂xm n+1 xn+1 − sn+1 bn+1 − an+1 ds1 .

### Advanced Inequalities (Series on Concrete and Applicable Mathematics) by George A. Anastassiou

by Ronald

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