Re: On limitations of the Mautsch Principle (was Re: ineqality)

In article <Rb0jf.60045$Dg7.3759222@xxxxxxxxxxxxxxxxxxxxx>,
Dirk Van de moortel <dirkvandemoortel@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

>> So it starts to look like Mautsch's approach may be the more powerful.
>> Naturally, I therefore looked for an example to defeat it :-)
>> Here is one:
>>
>> Problem: Prove that for all real a,b we have
>> 3 a b (2 a^2 - 3 a b + 2 b^2) < ( a^2 + b^2 )^2
>>
>> Solution: RHS - LHS = (a^2+b^2 - 3*a*b)^2 .
>>
>> However, when we substitute b = a + x into RHS-LHS we get a
>> negative coefficient: a^4 + x^4 + 2*a^3*x - a^2*x^2 - 2*a*x^3 .
>> So this inequality cannot be proved using the Mautsch Principle
>> (as useful as it is nonetheless in other cases!).
>
>Sure, but this b = a + x (and if applicable, c = a+x+y etc...)
>should only work when the inequality is to be proven and
>valid for real a and b that are *positive only*, otherwise the
>identity of the sign of the coefficients in the polynomial is
>useless. That is the case with your example where the
>inequality is valid for all real a and b.

I'm not sure I understand what your objection is. Would it help if
I asked you to prove that for _positive_ a and b,

3 a^2 b^2 (2 a^2 - 3 a b + 2 b^2) < a b ( a^2 + b^2 )^2 ?

This one now is false if a and b have opposite signs. I just
multiplied the previous example by ab . Or, perhaps a bit less
transparently, try to prove that

3 a b (a^4 + 2 a^3 b - 8 a^2 b^2 + 2 b^3 a + b^4) < (a^3-b^3)^2

when a and b are positive. This time RHS-LHS = (a*b+(a+b)^2)*P + (a*b)^3
where P is (a^2+b^2 - 3*a*b)^2 as in the previous example, so
(i) the expression is certainly positive if a and b are. It's also
(ii) symmetric in a and b and when you expand with b = a + x you get
(iii) some monomials with negative coefficients. I don't know what other
features you are looking for in an example. Granted, this particular
example lacks the je ne sais quoi of "nice" inequalities, but that's
not a _mathematical_ objection, is it?

dave

.

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  • Re: On limitations of the Mautsch Principle (was Re: ineqality)
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