Re: Proof for an inequality

In news:<45420c48@xxxxxxxxxxxxx> schrieb Thomas Mautsch <mautsch@xxxxxxx>:
In news:<1161908998.746740.196850@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
schrieb david petry <david_lawrence_petry@xxxxxxxxx>:
This leads to another question:

for what values of 't' is the following always true?

9(1-t)(a^3+b^3+c^3) + 27 t abc >= (a+b+c)^3

The inequality holds for all t <= 5/9.
Proof by my favourite principle:

Assume w.l.o.g. 0 <= a <= b <= c, so that you can write

b = a + x
c = a + x + y

with nonnegative x and y; then expand

9(1-t)(a^3+b^3+c^3) + 27 t abc - (a+b+c)^3

into monomials in the variables a, x and y. -
All coefficients of these monomials will be non-negative.

I just received a request to explain
which monimials I am referring to in this sentence.

Well, monomials in a, x, and y are expressions
a^m * x^n * y^q
with non-negative integers m, n, and q;
and the coefficients of these monomials in the expanded version of

9(1-t)(a^3+(a+x)^3+(a+x+y)^3) + 27 t a(a+x)(a+x+y) - (a+(a+x)+(a+x+y))^3

will be linear functions of t,
which will be non-negative for all t <= 5/9.

This proves the inequality for t <= 5/9.
.

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