By George A. Anastassiou
This monograph offers 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 a number of other complex classes should be taught out of this e-book. vast historical past and motivations are given in each one bankruptcy with a accomplished checklist of references given on the finish. the themes lined are wide-ranging and various. fresh advances on Ostrowski variety inequalities, Opial variety inequalities, Poincare and Sobolev sort inequalities, and Hardy-Opial style inequalities are tested. Works on traditional and distributional Taylor formulae with estimates for his or her remainders and functions in addition to Chebyshev-Gruss, Gruss and comparability of potential inequalities are studied. the implications awarded are usually optimum, that's the inequalities are sharp and attained. functions in lots of components of natural and utilized arithmetic, reminiscent of mathematical research, chance, usual and partial differential equations, numerical research, details thought, etc., are explored intimately, as such this monograph is appropriate 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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Additional info for Advanced inequalities
Xn ) k−1 ∂xj ∂xjk−1 j k−1 ∂ f ≤ ω1 k−1 · · · , xj+1 , . . , xn , bj − aj , all j = 1, . . , n; k = 1, . . , m − 1. 31. 10. 45), j = 1, . . , n. Then for any n (xj , xj+1 , . . 5in Book˙Adv˙Ineq ADVANCED INEQUALITIES 50 we have m−1 |Aj | = |Aj (xj , xj+1 , . . , xn )| ≤ j k−1 · ω1 k=1 xj − a j (bj − aj )k−1 Bk k! bj − a j ∂ f · · · , xj+1 , . . , xn ), bj − aj , for all j = 1, . . , n. 76) k−1 ∂xj Putting together all these above auxilliary results, we derive the following multivariate Ostrowski type inequalities.
11. 10 for m = n = 2, xi ∈ [ai , bi ], i = 1, 2. 9) and x1 − a 1 (f (b1 , x2 ) − f (a1 , x2 )) b1 − a 1 T1 := T1 (x1 , x2 ) := B1 + (b1 − a1 ) 2 b1 x1 − a 1 b1 − a 1 B2 a1 − B2∗ x1 − s 1 b1 − a 1 ∂2f (s1 , x2 )ds1 . 10) Proof. 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 .
M; j = 1, . . , n, be [ai , bi ]; m, n ∈ N, ai , bi ∈ R. 11. 10 for m = n = 2, xi ∈ [ai , bi ], i = 1, 2. 9) and x1 − a 1 (f (b1 , x2 ) − f (a1 , x2 )) b1 − a 1 T1 := T1 (x1 , x2 ) := B1 + (b1 − a1 ) 2 b1 x1 − a 1 b1 − a 1 B2 a1 − B2∗ x1 − s 1 b1 − a 1 ∂2f (s1 , x2 )ds1 . 10) Proof. 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 ).
Advanced inequalities by George A. Anastassiou