A treatise on differential equations by George Boole

By George Boole

This Elibron Classics e-book is a facsimile reprint of a 1877 version by means of Macmillan and Co., London.

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1). 3 of Chapter 1. 1). Proof. P u t /»oo /-oo J' f2{x)dx and V= x2f2(x)dx. o Jo We may assume that both U and V are finite. Let a and /? be positive 47 48 Multiplicative Inequalities of Carlson Type and Interpolation numbers. )*)"- ( f ^Lpv^VM«x) IT 1 2 v¥ (aU + pV) With a = V and /3 = U, the rightmost expression is TrVt^V, which, upon squaring both sides, yields the desired inequality. Equality in the application of the Schwarz inequality is obtained precisely when for almost all x f(x) = A a + fix1 for some constant A or as claimed.

Our second proof is based on Calculus of Variations. 1). Proof by Calculus of Variations. 2) Jo Jo where a and b are some non-zero numbers. 2). The Lagrange function for this problem is given by L(A0, Ai, A2) f Jo Xof(x) + X1f2(x) + X2x2f2(x) dx. 3). 2). By dividing through by Ao we may therefore assume that AQ = 1. Thus, let us consider h(u) = — u + X\u + X2x u . 1 The main advantage of the method used in the above proof is that it automatically gives the sharp constant and extremizing functions.

Sivo^-'SitTuTi)". The Holder-Rogers inequality with exponents l-6> and 6 yields the desired inequality. • We can now prove our extensions of the Landau and Levin-Steckin results. 3. 5 if we put p = 1. V8 2 l + \ l = (l-9)(l-q) + l - \ l + v(^ + q), V— 1 Then, since J and « i 6=—I 7 i \ —. 3 and the case q = »/|, that 5(0,1) 4 < C-s(l - y 1 - z W l + y ^ , 2 < C - 5 ( l - g , 2 ) 1 - " 5 ( l + g,2)" S(l-q,2)1-eS0. ». 7). Finally, for the case g > 1, let 9=q-±± 2? and „=«fl. , 2)e] = n2S(l-q,2)S(l+q,2).

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