By George A. Anastassiou

ISBN-10: 9814317624

ISBN-13: 9789814317627

This monograph provides 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 several other complicated classes could be taught out of this booklet. vast heritage and motivations are given in every one bankruptcy with a complete record of references given on the finish. the themes coated are wide-ranging and various. contemporary advances on Ostrowski style inequalities, Opial kind inequalities, Poincare and Sobolev variety inequalities, and Hardy-Opial sort 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 often optimum, that's the inequalities are sharp and attained. purposes in lots of parts of natural and utilized arithmetic, equivalent to mathematical research, likelihood, usual and partial differential equations, numerical research, details thought, etc., are explored intimately, as such this monograph is acceptable for researchers and graduate scholars. it is going to be an invaluable educating fabric at seminars in addition to a useful reference resource in all technology libraries.

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**Example text**

25. 23. Let pj , qj > 1: j = 1, . . , n, with the assumption that j ∂mf (· · · , xj+1 , . . 5in Book˙Adv˙Ineq ADVANCED INEQUALITIES 46 n for any (xj+1 , . . , xn ) ∈ |Bj | ≤ [ai , bi ], xj ∈ [aj , bj ]. 51) we get (bj − aj )m−1 m! − × [ai ,bi ] (bi − ai ) i=1 ds1 · · · dsj ∂mf (s1 , . . , sj , xj+1 , . . , 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!

3) Bk k! 2) that (b − a)n−1 x−a x−t ∆n (x) = Bn − Bn∗ f (n) (t)dt. 4) n! b − a b − a [a,b] In this chapter we give sharp, namely attained, upper bounds for |∆4 (x)| and tight upper bounds for |∆n (x)|, n ≥ 5, x ∈ [a, b], with respect to L∞ norm. 1) for higher order derivatives. High computational difficulties in this direction prevent us for shoming sharpness for n ≥ 5 cases. 2. Let f : [a, b] → R be such that f (n−1) , n ≥ 1, is a continuous function and f (n) (x) exists and is finite for all but a countable set of x in (a, b) and that f (n) ∈ L∞ ([a, b]).

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, . . , n. 27. 25. Assume for j = 1, . . , n that j ∂mf (· · · , xj+1 , .

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

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