NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/research/448

Text File | 1992-09-11 | 1.6 KB | 39 lines

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From: [e-mail] (Vincent Broman)
Subject: Re: Better than Cauchy-Schwarz
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Date: Thu, 10 Sep 1992 23:25:35 GMT
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let C = 1/8,
let f be L1 and L2 on [0,\infty),
and let \int_0^\infty f(x) x dx < C \int_0^\infty f(x) dx.

Then
       \int_0^C f(x) (C-x) dx >= \int_C^\infty f(x) (x-C) dx
implies with a slight amount of work
       \int_0^{2 C} f(x) dx >= \int_{2 C}^\infty f(x) dx,
meaning that the median is not more than twice the mean.

Apply Cauchy-Schwartz to f and the characteristic function on [0,2C].
Then,
       \int_0^\infty f^2 >= ( \int_0^{2 C} f )^2 / (2 C)
                         >= ( 1/2 \int_0^\infty f )^2 / (2 C)
                          = ( \int_0^\infty f )^2.

Vincent Broman,  code 572 bayside,                                  =   o
Naval Command Control and Ocean Surveillance Center, RDT&E Div.   =  _ /- _
San Diego, CA  92152-5000,  USA                                  =  (_)> (_)
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