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Re: which one is greater than or equal?

• To: mathgroup at smc.vnet.net
• Subject: [mg41307] Re: [mg41285] which one is greater than or equal?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 14 May 2003 08:10:43 -0400 (EDT)
• References: <200305130818.EAA18227@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Son Bui wrote:
> 
> can any one tell me which one is greater than? Please give your proof.
> 
> sqrt((a1+a2+...+an)/n)  ?   (sqrt(a1)+sqrt(a2)+...+sqrt(an))/n
> 
> Bui


For a1,...,an>0, Sqrt[(a1+...+an)/n] >= (Sqrt[a1]+...+Sqrt[an])/n

with equality only when a1==a2==...==an.

For the start of a proof see

http://mathworld.wolfram.com/ConvexFunction.html

and apply to f[x_] := -Sqrt[x]

Another proof uses the Cauchy-Schwarz inequality with vectors

a = {a1,...,an} and b = {1,...,1}.


Daniel Lichtblau
Wolfram Research

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  • which one is greater than or equal?
    • From: shbui@yahoo.com (Son Bui)