By George A. Anastassiou

This monograph provides univariate and multivariate classical analyses of complex 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. wide heritage and motivations are given in every one bankruptcy with a accomplished record of references given on the finish.

the themes lined are wide-ranging and various. contemporary advances on Ostrowski style inequalities, Opial variety inequalities, Poincare and Sobolev style inequalities, and Hardy-Opial variety inequalities are tested. Works on usual and distributional Taylor formulae with estimates for his or her remainders and purposes in addition to Chebyshev-Gruss, Gruss and comparability of ability inequalities are studied.

the consequences awarded are as a rule optimum, that's the inequalities are sharp and attained. purposes in lots of parts of natural and utilized arithmetic, comparable 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 will likely be an invaluable instructing fabric at seminars in addition to a useful reference resource in all technology libraries.

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**Example text**

44) and Aj for j = 1, . . 45), m ∈ N. In particular we suppose that j ∂mf · · · , xj+1 , . . , xn ∈ L∞ ∂xm j j [ai , bi ] , i=1 n for any (xj+1 , . . , xn ) ∈ [ai , bi ], all j = 1, . . , n. Then i=j+1 f |Em (x1 , . . , xn )| = f (x1 , . . , xn ) − n n i=1 ≤ 1 m! (bi − ai ) n j=1 n 1 [ai ,bi ] f (s1 , . . )2 xj − a j 2 |B2m | + Bm (2m)! bj − a j (bj − aj )m j × ∂mf · · · , xj+1 , . . , xn ) ∂xm j . 77) [ai ,bi ] i=1 Proof. 24. 33. 20. Let Em (x1 , . . 44), m ∈ N. Let pj , qj > 1 : pj + qj = 1; j = 1, .

Xn )| ≤ j=1 |Bj |. 47) Also by denoting ∆ := f (x1 , . . , xn ) − 1 n n i=1 (bi − ai ) [ai ,bi ] f (s1 , . . 48) i=1 we get n |∆| ≤ j=1 (|Aj | + |Bj |). 49) Later we will estimate Aj , Bj . 17. Here m ∈ N, j = 1, . . We suppose n 1) f : i=1 2) ∂ f ∂xj [ai , bi ] → R is continuous. are existing real valued functions for all j = 1, . . , n; 3) For each j = 1, . . , n we assume that continuous real valued function. = 1, . . , m − 2. ∂ m−1 f (x1 , . . , xj−1 , ·, xj+1 , . . 5in Book˙Adv˙Ineq Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities 43 m 4) For each j = 1, .

8 we have f (x1 , x2 ) = 1 b1 − a 1 b1 f (s1 , x2 )ds1 + B1 a1 (b1 − a1 ) + 2 b1 B2 a1 b1 x1 − a 1 b1 − a 1 x1 − a 1 (f (b1 , x2 ) − f (a1 , x2 )) b1 − a 1 − B2∗ x1 − s 1 b1 − a 1 ∂2f (s1 , x2 )ds1 ∂x21 1 f (s1 , x2 )ds1 + T1 (x1 , x2 ). 11) b 1 − a 1 a1 And also we obtain b2 1 x2 − a 2 f (s1 , x2 ) = f (s1 , s2 )ds2 + B1 (f (s1 , b2 ) − f (s1 , a2 )) b 2 − a 2 a2 b2 − a 2 = + (b2 − a2 ) 2 b2 B2 a2 x2 − a 2 b2 − a 2 − B2∗ x2 − s 2 b2 − a 2 ∂ 2f (s1 , s2 )ds2 . 13) proving the claim. 12. 10, case n = 3 and m ∈ N, xi ∈ [ai , bi ], i = 1, 2, 3.