Advanced Inequalities by George A. Anastassiou PDF

By George A. Anastassiou

ISBN-10: 9814317632

ISBN-13: 9789814317634

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 will be taught out of this e-book. huge historical past and motivations are given in every one bankruptcy with a complete record of references given on the finish.

the subjects lined are wide-ranging and various. fresh advances on Ostrowski style inequalities, Opial kind inequalities, Poincare and Sobolev variety 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 capability inequalities are studied.

the consequences awarded are regularly optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, equivalent to mathematical research, likelihood, traditional and partial differential equations, numerical research, info thought, etc., are explored intimately, as such this monograph is acceptable for researchers and graduate scholars. will probably be an invaluable instructing fabric at seminars in addition to a useful reference resource in all technology libraries.

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Dsj−1 xj − a j bj − a j ∂mf s1 , . . , sj , xj+1 , . . , xn , xn+1 ∂xm j ds1 . . dsj . But it holds f (s1 . . , sn , xn+1 ) = 1 (bn+1 − an+1 ) m−1 + k=1 bn+1 f (s1 , . . , sn , sn+1 )dsn+1 an+1 xn+1 − an+1 (bn+1 − an+1 )k−1 Bk k! bn+1 − an+1 ∂ k−1 f (s1 , . . sn , bn+1 ) ∂ k−1 f − k−1 k−1 ∂xn+1 ∂xn+1 + bn+1 − an+1 m! ∗ − Bm m−1 bn+1 Bm an+1 xn+1 − sn+1 bn+1 − an+1 s1 , . . , sn , an+1 xn+1 − an+1 bn+1 − an+1 ∂ m f (s1 , . . , sn , sn+1 ) dsn+1 . ∂xm n+1 Thus we get f (x1 , x2 , . . , xn , xn+1 ) = 1 n i=1 n (bi − ai ) m−1 + k=1 [ai ,bi ] i=1 1 (bn+1 − an+1 ) bn+1 f (s1 , .

This treatement is based on [35]. 1) where f ∈ C ([a, b]), x ∈ [a, b], which is a sharp inequality, see [16]. Other motivations come from [10], [24], [16], [98] and [121]. 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 .

Xn ) ∈ i=j+1 n [ai , bi ]. Then for any (xj , xj+1 , . . , xn ) ∈ [ai , bi ] i=j we have |Bj | = |Bj (xj , xj+1 , . . , xn )| ≤ (bj − aj )m−1 j−1 m! i=1 (bi − ai ) × Bm (t) − Bm ∂mf (. . , xj+1 , . . , xn ) ∂xm j xj − a j bj − a j j 1, [ai ,bi ] i=1 . 70) ∞,[0,1] The special cases follow: 1) When m = 2r, r ∈ N we have |Bj | ≤ (bj − aj )2r−1 j−1 (2r)! i=1 (bi − ai ) ∂ 2r f (. . , 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 (.

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