From: "denis-feldmann" <denis-feldmann@wanadoo.fr>
Subject: Re: Little bugger of an inequality
Date: Fri, 28 Apr 2000 09:03:31 +0200
Newsgroups: sci.math
Summary: [missing]

Dave Hughes <davidh@cse.unsw.EDU.AU> a écrit dans le message :
Pine.GSO.3.95.1000428141440.19225C-100000@tuba00.orchestra.cse.unsw.EDU.AU..
.
> Hi there,
>
> recently doing a real analysis assignment I did one of the questions the
> wrong way and came up against this inequality
>
> for 0 < p < 1
>
> 2^(1/p) + 1 < (2^p + 1)^(1/p)
>
> it's true, and I reckon you can substitute 2 for n >= 1.
>
> Can anyone think of a proof? (even without the n bit)
>
> Cheers,

Those are called (here at least) Holder's inequality. They  are a special
case of what is known as convexity inequalities: if f is convex (i.e.
f''>0), and 0<k<1, whe have f(kx+(1-k)y)<kf(x)+(1-k)f(y). Take the right
function f (for instance X->X^p), the right x,y and k, and you will get your
result (and many others)



>
> Dave Hughes
> 3rd year pure maths, UNSW
>
>
> -------------------------------------------------------------------------
> |                               David Hughes                            |
> |                                   UNSW                                |
> |                                                                       |
> |  "Communism is man against man. Socialism is the other way around."  |
> |                                                                       |
> -------------------------------------------------------------------------
>