By George A. Anastassiou
This monograph provides 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 several other complicated classes should be taught out of this publication. broad historical past and motivations are given in each one bankruptcy with a entire record of references given on the finish.
the subjects lined are wide-ranging and numerous. contemporary advances on Ostrowski style inequalities, Opial kind 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 consequences awarded are ordinarily optimum, that's the inequalities are sharp and attained. functions in lots of parts of natural and utilized arithmetic, similar to mathematical research, chance, traditional and partial differential equations, numerical research, details idea, 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 technology libraries.
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Additional resources for Advanced Inequalities
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 , .
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 , . .
19) is attained. 17) sharp. |∆4 (a)| = |∆4 (b)| = The trapezoid and midpoint inequalities follow. 9. 2, case of n = 4. It holds f (a) + f (b) 2 − (b − a) 1 (f (b) − f (a)) − 12 b−a b f (t)dt a (b − a)4 (4) f ∞, 720 the last inequality is attained by (t − a)4 and (t − b)4 , that is sharp. 5in Book˙Adv˙Ineq ADVANCED INEQUALITIES 26 Furthermore we obtain f a+b 2 + 1 (b − a) (f (b) − f (a)) − 24 b−a ≤ 7 (b − a)4 f (4) 5760 b f (t)dt a ∞. 10. We do obtain the trapezoidal formula (f (a) + f (b)) (b − a) − (f (b) − f (a)) 2 12 b 1 (b − a)3 f (t)dt.