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 study paintings. The chapters are self-contained and a number of other complicated classes may be taught out of this e-book. vast historical past and motivations are given in every one bankruptcy with a entire checklist of references given on the finish. the themes coated are wide-ranging and various. fresh advances on Ostrowski kind inequalities, Opial kind inequalities, Poincare and Sobolev kind inequalities, and Hardy-Opial sort inequalities are tested. Works on traditional and distributional Taylor formulae with estimates for his or her remainders and purposes in addition to Chebyshev-Gruss, Gruss and comparability of skill inequalities are studied. the consequences awarded are in general optimum, that's the inequalities are sharp and attained. purposes in lots of parts of natural and utilized arithmetic, comparable to mathematical research, likelihood, traditional and partial differential equations, numerical research, details idea, etc., are explored intimately, as such this monograph is appropriate 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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**Sample text**

B1 − a 1 k−1 ∂ f ∂ k−1 f (b , x2 , x3 ) − (a1 , x2 , x3 ) k−1 1 ∂x1 ∂x1k−1 + b1 (b1 − a1 )m−1 m! x1 − a 1 b1 − a 1 Bm a1 ∗ − Bm x1 − s 1 b1 − a 1 ∂mf (s1 , x2 , x3 )ds1 . 17) are zero. Proof. 8 we have f (x1 , x2 , x3 ) = b1 1 b1 − a 1 f (s1 , x2 , x3 )ds1 + T1 (x1 , x2 , x3 ). 18) a1 Furthermore we find f (s1 , x2 , x3 ) = m−1 + k=1 + 1 b2 − a 2 b2 f (s1 , s2 , x3 )ds2 a2 x2 − a 2 (b2 − a2 )k−1 Bk k! b2 − a 2 (b2 − a2 )m−1 m! 19) and f (s1 , s2 , x3 ) = m−1 + k=1 + 1 b3 − a 3 b3 f (s1 , s2 , s3 )ds3 a3 (b3 − a3 )k−1 x3 − a 3 Bk k!

Xj+1 , . . , xn ), bj − aj ∂xjk−1 .

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.