By George A. Anastassiou
This monograph offers univariate and multivariate classical analyses of complex 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 may be taught out of this ebook. large history and motivations are given in each one bankruptcy with a complete checklist of references given on the finish. the subjects lined are wide-ranging and various. contemporary advances on Ostrowski style inequalities, Opial variety inequalities, Poincare and Sobolev sort inequalities, and Hardy-Opial sort inequalities are tested. Works on traditional and distributional Taylor formulae with estimates for his or her remainders and functions in addition to Chebyshev-Gruss, Gruss and comparability of capacity inequalities are studied. the implications offered are ordinarily optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, equivalent to mathematical research, chance, usual 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 educating fabric at seminars in addition to a useful reference resource in all technological know-how libraries.
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Extra info for Advanced Inequalities (Series on Concrete and Applicable Mathematics)
Xn ) = n n i=1 [ai ,bi ] (bi − ai ) f (s1 , s2 , . . 42) where for j = 1, . . , n we have j−1 i=1 × j−1 [ai ,bi ] i=1 − + m−1 1 Tj := Tj (xj , xj+1 , . . , xn ) := (bi − ai ) k=1 xj − a j (bj − aj )k−1 Bk k! bj − a j ∂ k−1 f (s1 , s2 , . . , sj−1 , bj , xj+1 , . . , xn ) ∂xjk−1 ∂ k−1 f (s1 , s2 , . . , sj−1 , aj , xj+1 , . . , xn ) ds1 · · · dsj−1 ∂xjk−1 (bj − aj )m−1 m! i=1 ∗ −Bm [ai ,bi ] (bi − ai ) xj − s j bj − a j Bm j j−1 i=1 xj − a j bj − a j ∂ mf (s1 , s2 , . . , sj , xj+1 , .
33) b3 f (s1 , s2 , s3 , x4 )ds3 a3 (b3 − a3 )k−1 x3 − a 3 Bk k! b3 − a 3 x3 − s 3 b3 − a 3 a2 x2 − a 2 b2 − a 2 ∂ f (s1 , s2 , x3 , x4 )ds2 , ∂xm 2 ∂ k−1 f (s1 , s2 , b3 , x4 ) ∂x3k−1 (b3 − a3 )m−1 ∂ k−1 f (s , s , a , x4 ) + k−1 1 2 3 m! 34) and finally f (s1 , s2 , s3 , x4 ) = m−1 + k=1 − 1 b4 − a 4 b4 f (s1 , s2 , s3 , s4 )ds4 a4 x4 − a 4 (b4 − a4 )k−1 Bk k! b4 − a 4 ∂ k−1 f (s1 , s2 , s3 , b4 ) ∂x4k−1 ∂ k−1 f (b4 − a4 )m−1 (s1 , s2 , s3 , a4 ) + k−1 m! ∂x4 ∗ − Bm x4 − s 4 b4 − a 4 b4 Bm a4 x4 − a 4 b4 − a 4 ∂mf (s1 , s2 , s3 , s4 )ds4 .
N. In particular we assume that ∂mf (. . , xj+1 , . . , xn ) ∈ Lqj ∂xm j j [ai , bi ] , i=1 n for any (xj+1 , . . , xn ) ∈ [ai , bi ], for all j = 1, . . , n. 5in Book˙Adv˙Ineq Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities f |Em (x1 , . . , xn )| 1 ≤ m! n j=1 (bj − aj ) −Bm (tj ) j i=1 1/pj pj dtj − q1 j−1 m− q1 1 j (bi − ai ) xj − a j bj − a j Bm 0 ∂ mf (. . , xj+1 , . . , xn ) ∂xm j . 78) When pj = qj = 2, all j = 1, . . , n, then f |Em (x1 , . .
Advanced Inequalities (Series on Concrete and Applicable Mathematics) by George A. Anastassiou