By George A. Anastassiou

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 several other complex classes might be taught out of this booklet. vast heritage and motivations are given in each one bankruptcy with a entire checklist of references given on the finish. the themes lined are wide-ranging and various. fresh advances on Ostrowski sort inequalities, Opial kind inequalities, Poincare and Sobolev sort inequalities, and Hardy-Opial variety 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 implications awarded are often optimum, that's the inequalities are sharp and attained. purposes in lots of components of natural and utilized arithmetic, resembling mathematical research, chance, usual and partial differential equations, numerical research, info thought, 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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44), m ∈ N. In particular we assume for j = 1, . . , n that ∂mf (. . , xj+1 , . . , xn ) ∈ L1 ∂xm j for any (xj+1 , . . , xn ) ∈ f |Em (x1 , . . , xn )| ≤ n i=j+1 [ai , bi ]. 1 m! n j [ai , bi ] , i=1 Then (bj − aj )m−1 j−1 j=1 i=1 ∂mf (. . , xj+1 , . . , xn ) ∂xm j (bi − ai ) j 1, [ai ,bi ] i=1 Bm (t) − Bm xj − a j bj − a j . 5in Book˙Adv˙Ineq ADVANCED INEQUALITIES 52 f |E2r (x1 , . . , xn )| ≤ 1 (2r)! n (bj − aj )2r−1 ∂ 2r f (. . , xj+1 , . . , xn ) ∂x2r j j−1 j=1 i=1 (bi − ai ) j 1, [ai ,bi ] i=1 xj − a j bj − a j × (1 − 2−2r )|B2r | + 2−2r B2r − B2r .

We have ∆4 (a) = ∆4 (b) = (b − a) 1 f (a) + f (b) − (f (b)−f (a))− 2 12 b−a b f (t)dt. 17) we have |∆4 (a)| = |∆4 (b)| ≤ (b − a)4 (4) f 720 ∞. 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 ∞.

Xn ) ∈ [ai , bi ] we have i=j |Bj | = |Bj (xj , xj+1 , . . , xn )| (bj − aj ) ≤ m! m− q1 j − Bm (tj ) (bi − ai ) i=1 1/pj pj dtj −1/qj j−1 1 Bm 0 ∂mf (· · · , xj+1 , . . , xn ) ∂xm j xj − a j bj − a j . 5in Book˙Adv˙Ineq Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities 47 When pj = qj = 2, then we obtain 1 (bj − aj )m− 2 |Bj | ≤ m! )2 xj − a j 2 |B2m | + Bm (2m)! bj − a j (bi − ai ) m × ∂ f (· · · , xj+1 , . . , xn ) ∂xm j . 63) are true for all j = 1, .