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 study paintings. The chapters are self-contained and a number of other complicated classes may be taught out of this publication. broad historical past and motivations are given in each one bankruptcy with a finished checklist of references given on the finish.

the subjects coated are wide-ranging and numerous. contemporary advances on Ostrowski style inequalities, Opial variety inequalities, Poincare and Sobolev kind inequalities, and Hardy-Opial kind 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 capability inequalities are studied.

the consequences awarded are normally optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, akin to mathematical research, chance, usual and partial differential equations, numerical research, details idea, etc., are explored intimately, as such this monograph is acceptable for researchers and graduate scholars. it will likely be an invaluable educating fabric at seminars in addition to a useful reference resource in all technological know-how libraries.

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**Example text**

44) are still true. 22. 20 for their cases. 44). 23. 20. We observe for j = 1, . . 50) where (bj − aj )m−1 Γj := m! i=1 × Here we assume (bi − ai ) xj − a j bj − a j Bm j j−1 [ai ,bi ] i=1 ∗ − Bm xj − s j bj − a j ∂mf (s1 , s2 , . . , sj , xj+1 , . . , xn ) ds1 · · · dsj . ∂xm j j ∂mf · · · , xj+1 , . . , xn ∈ L∞ ∂xm j for any (xj+1 , . . 51) j [ai , bi ] i=1 [ai , bi ], any xj ∈ [aj , bj ]. Thus we obtain Γj ≤ (bj − aj )m−1 j−1 Bn j [ai ,bi ] xj − a j bj − a j (bi − ai ) i=1 j m ∂ f × ·, ·, ·, · · · , ·, xj+1 , .

Proof. 8 we have f (x1 , x2 , x3 , x4 ) = m−1 + k=1 − 1 b1 − a 1 b1 f (s1 , x2 , x3 , x4 )ds1 a1 x1 − a 1 (b1 − a1 )k−1 Bk k! b1 − a 1 ∂ k−1 f (b1 , x2 , x3 , x4 ) ∂x1k−1 (b1 − a1 )m−1 ∂ k−1 f (a , x , x , x ) + 1 2 3 4 m! 31) f (s1 , x2 , x3 , x4 )ds1 + T1 (x1 , x2 , x3 , x4 ). 32) a1 f (s1 , x2 , x3 , x4 ) = m−1 Bm b1 = + b1 1 b2 − a 2 b2 f (s1 , s2 , x3 , x4 )ds2 a2 (b2 − a2 )k−1 x2 − a 2 Bk k! 5in Book˙Adv˙Ineq Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities − (b2 − a2 )m−1 ∂ k−1 f (s , a , x3 , x4 ) + k−1 1 2 m!

Here p, q : p1 + q1 = 1. Then f (x) − n−2 · b 1 (g(b) − g(a)) b f (s1 )dg(s1 ) − a b f (k+1) (s1 )dg(s1 ) · a k=0 a p < +∞, 1 (g(b) − g(a)) b ··· P (g(x), g(s1 )) a k · i=1 P (g(si ), g(si+1 )) · ds1 · · · dsk+1 ≤ f (n) b p · a n−2 b ··· |P (g(x), g(s1 ))| · a · P (g(sn−1 ), g(sn )) q,sn i=1 |P (g(si ), g(si+1 ))| · ds1 ds2 · · · dsn−1 . 34) Finally we present L1 Ostrowski type results. 24. 4. Additionally suppose that f (n) Then |θ1,n | ≤ f (n) b 1 · a · P (sn−1 , •) ∞ a < +∞. n−2 b ··· 1 |P (x, s1 )| · i=1 |P (si , si+1 )| · ds1 ds2 · · · dsn−1 .