From: Jim Ferry <jferry@delete_this_field.uiuc.edu>
Newsgroups: sci.math
Subject: Re: Use of AM-GM ?
Date: Fri, 24 Jul 1998 13:54:46 -0500

Peter L. Montgomery wrote:
> 
> In article <35AF9704.3B0D@excite.com> r38matt@excite.com writes:
> >Prove that,
> >Sqrt { 1 + [(X1 + ...+ Xn)/n]^2 }  <= {Sqrt(1+X1^2) + Sqrt(1+X2^2)
> >+...+Sqrt(1+Xn^2) } /n
> 
> >This problem may need induction with Cauchy-Schwartz inequality or
> >Arithmetic-Mean-Geometric-Mean(AM-GM) ineq.
> 
> >What would the easiest way to complete the proof ?
> 
>         You want to prove
> 
>            ( X1 + X2 + ... + Xn )     f(X1) + ... + f(Xn)
> (*)       f( ------------------ ) <=  --------------------
>            (           n        )              n
> 
> where f(x) = sqrt(1 + x^2).

Jensen's inequality states: if f(x) is convex, then (*) is true.
A convex function is defined as an f such that (*) is true for
n=2, for all x1, x2.  (Geometrically, a convex function is one
such that any chord drawn between 2 points on its graph is lies
on or above the intermediate points on the graph.)

f''(x) >= 0 for all x implies f is convex, and hence the above
inequality.

Indeed, since in this case f''(x) = (1+x^2)^-1.5 > 0 for all x,
f is strictly convex (i.e., a chord btwn 2 different points is
strictly above the intermediate points on the graph).  For f
strictly convex, Jensen's inequality states that equality holds
in (*) only when all the Xi are equal.
 
|  Jim Ferry (your imaginary friend) | Center for Simulation  |
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