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 a number of other complex classes could be taught out of this e-book. vast historical past and motivations are given in every one bankruptcy with a accomplished checklist of references given on the finish.
the themes coated are wide-ranging and numerous. contemporary advances on Ostrowski variety inequalities, Opial kind inequalities, Poincare and Sobolev style 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 skill inequalities are studied.
the implications provided are often optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, corresponding to mathematical research, likelihood, traditional and partial differential equations, numerical research, details idea, etc., are explored intimately, as such this monograph is acceptable for researchers and graduate scholars. will probably be an invaluable educating fabric at seminars in addition to a useful reference resource in all technological know-how libraries.
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Extra info for Advanced Inequalities
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 , . . , xn )| ≤ 1 m! )2 2 |B2m | + Bm (2m)! bj − a j × × (bj − aj )m− 2 ∂mf (.
2, case of n = 4. 17) is sharp, namely it is attained when x = a, b by the functions (t − a)4 and (t − b)4 . Proof. 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.
50) and j j−1 m! [ai , bi ] , i=1 (bi − ai ) [ai ,bi ] i=1 ∂mf (s1 , . . , sj , xj+1 , . . , xn ) ds1 · · · dsj ∂xm j · Bm = m! 64) ∞,[aj ,bj ] xj − a j bj − a j ∗ − Bm xj − s j bj − a j ∞,[aj ,bj ] m × ∂ f (· · · , xj+1 , . . 65) j 1, [ai ,bi ] i=1 (by , p. 347) = (bj − aj )m−1 j−1 m! i=1 × (bi − ai ) ∂ mf (· · · , xj+1 , . . , xn ) ∂xm j i) case m = 2r, r ∈ N, then . j [ai ,bi ] 1, i=1 From , pp. 67) ii) case m = 2r + 1, r ∈ N, then Bm (t) − Bm ≤ xj − a j bj − a j = B2r+1 (t) − B2r+1 ∞,[0,1] xj − a j 2(2r + 1)!