A Treatise On The Differential Calculus with numerous by Isaac Todhunter

By Isaac Todhunter

This Elibron Classics ebook is a facsimile reprint of a 1864 variation through Macmillan and Co., Cambridge and London.

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1 Henee, (5) holds if we have ( " 1: Xf-l ;= )II(P-I) (" + 1: yf-l )1/(1)-1) (7) i = I I ~ [ "JIM-I) ,1: (x i + Yi)P-l ,= I . ° This, however, is MINKOWSKI'S inequality, valid for 1 < The inequality sign in (1) is reversed for ~ p ~ 1. P~ 2. § 24. An Inequality of Dresher An extension of BECKENBACH'S inequality was obtained by [1] by means of moment-space teehniques: Theorem 10. p . DRESHER ( 1) This result ean be derived through quasi linearization, as in § 23. DANSKIN [2], who employed a eombination of the Hölder and Minkowski inequalities.

CHASSAN [6J. § 45. Refinements of the Cauchy-Buniakowsky-Schwarz Inequalities Having established the nonnegativity of the functional I (u, v) = (f u dt) (f 2 V2 dt) - (f uv dtr (1) we naturally are interested in obtaining a more precise lower bound than zero. We can do this whenever the functions or functionals under consideration are quadratic in the following fashion. Reverting to inner products, consider the function J (u, v) = (u, u) (v, v) - (u, V)2, (2) assumed nonnegative for all u and v.

Similarly, for a strictly f(u) = min [f(v) + (u-v)f'(v)J. v (1) (2) The general resuIt is Theorem 11. Let f(x) = f (Xl' x 2 , ••• , x n ) be a strictly convex function of x for all x; then f(x) = max [f(y) + (x - y, ep (y))] , (3) ,. 30 1. The Fundamental Inequalities and Related Matters where 1>(Y) = (of/oYl' of/oY2' ... ), the gradient of f(y). The unique maximum occurs at Y = x. This type of quasi linearization has been extensively used by BELLMAN [1], [2] and KALABA [3] in eonneetion with the analytic and eomputational treatment of nonlinear funetional equations.

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