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From: Jim Ferry on 8 Jun 2010 12:25
Let e(x)=x^x. Show
e(a+b)^2 e(b+c)^2 <= e(a) e(b)^2 e(c) e(a+2b+c)
for all a,b,c >= 0, with equality iff b^2 = a c.

This arose in something I'm working on -- that it's true follows
trivially from what I was doing. However, it doesn't seem obvious
when stated in isolation.
From: Gerry Myerson on 8 Jun 2010 18:54
In article
<617b6dba-0d27-4b04-a3e8-7074008faf08(a)z8g2000yqz.googlegroups.com>,
Jim Ferry <corklebath(a)hotmail.com> wrote:

> Let e(x)=x^x. Show
> e(a+b)^2 e(b+c)^2 <= e(a) e(b)^2 e(c) e(a+2b+c)
> for all a,b,c >= 0, with equality iff b^2 = a c.

Does e(b)^2 mean (b^b)^2 or (b^2)^(b^2)?

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Jim Ferry on 9 Jun 2010 00:22
On Jun 8, 6:54 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> In article
> <617b6dba-0d27-4b04-a3e8-7074008fa...(a)z8g2000yqz.googlegroups.com>,
>  Jim Ferry <corkleb...(a)hotmail.com> wrote:
>
> > Let e(x)=x^x.  Show
> > e(a+b)^2 e(b+c)^2 <= e(a) e(b)^2 e(c) e(a+2b+c)
> > for all a,b,c >= 0, with equality iff b^2 = a c.
>
> Does e(b)^2 mean (b^b)^2 or (b^2)^(b^2)?

e(b)^2 = (b^b)^2. I probably should have called the function f(x).
The e(x) looks weird.
From: William Elliot on 9 Jun 2010 00:35
On Tue, 8 Jun 2010, Jim Ferry wrote:

> Let e(x)=x^x. Show
> e(a+b)^2 e(b+c)^2 <= e(a) e(b)^2 e(c) e(a+2b+c)
> for all a,b,c >= 0, with equality iff b^2 = a c.

Let f(x) = x^x, for all x in [0,oo).
What's f(0)? f(0) = lim(x->0) x^x = 0 ?

Show
f(a+b)^2 f(b+c)^2 <= f(a) f(b)^2 f(c) f(a+2b+c)
for all a,b,c >= 0, with equality iff b^2 = a c.

That statement is false for b = 0. For all a,c >= 0,
0 <= f(a)^2 f(c)^2 <= f(a) f(0)^2 f(c) f(a + c) = 0
f(a).f(c) = 0

----
From: Henry on 9 Jun 2010 04:55
On 9 June, 05:35, William Elliot <ma...(a)rdrop.remove.com> wrote:
> On Tue, 8 Jun 2010, Jim Ferry wrote:
> > Let e(x)=x^x.  Show
> > e(a+b)^2 e(b+c)^2 <= e(a) e(b)^2 e(c) e(a+2b+c)
> > for all a,b,c >= 0, with equality iff b^2 = a c.
>
> Let f(x) = x^x, for all x in [0,oo).
> What's f(0)?  f(0) = lim(x->0) x^x = 0 ?
>

lim(x->0) x^x = 1 when the limit is taken from above.
E.g. 0.001^0.001 = 0.993...
0.0000001^0.0000001 = 0.999998...

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