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Is this inequality true?

$a^2+b^2+c^2 +2abc+1\ge2(ab+bc+ca)$, where $a,b,c\gt0$.

Can you find a counterexample if not?

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I believe it is true. Try Cauchy. –  Ahaan S. Rungta Oct 29 '13 at 14:53
    
Equality holds for $a=b=c=1$ –  Axel Kemper Oct 29 '13 at 15:08
    
Use AMGM on left hand side: $2abc+1 \ge 3(abc)^{2/3}$. The resulting inequality is weaker than Schur's inequality. –  user27126 Oct 30 '13 at 8:34

2 Answers 2

up vote 4 down vote accepted

EDIT: I found a much better solution: WLOG assume that $b,c$ are on the same side of $1$. Bringing the RHS over to the LHS, it suffices to show that $$(a-1)^2+2a(b-1)(c-1)+(b-c)^2\ge 0$$ But since $b,c$ are on the same side of $1$, this is true.


Old solution:

We prove the inequality for $a,b,c\ge 0$. The inequality is equivalently $$ a^2+2(bc-b-c)a+b^2+c^2+1-2bc\ge 0 $$ If $bc>b+c$, then the expression is minimized when $a=0$. But in this case it is only left to prove $b^2+c^2-2bc+1=(b-c)^2+1\ge 0$, which is clear.

If $bc\le b+c$, then then expression is minimized when $a = b+c-bc$. We must show $$ -(bc-b-c)^2+b^2+c^2+1\ge 2bc\Leftrightarrow 2(b+c)bc+1\ge 4bc+(bc)^2 $$ But letting $bc=k$, we have $$ 2(b+c)bc+1\ge 4k^{3/2}+1 $$ And so we want to prove $$ k^2-4k^{3/2}+4k-1=(\sqrt{k}-1)^2(k-2\sqrt{k}-1)\le 0 $$ given $bc\le b+c\le 2\sqrt{bc}\Rightarrow 0\le k\le 4$. But then $k-2\sqrt{k}-1=\sqrt{k}(\sqrt{k}-2)-1\le -1<0$, so the inequality holds.

Equality holds when $a=b=c=1$.

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The following substitution transforms the constrained inequality into an unconstrained one:

$$a = t^2 \gt 0$$ $$b = u^2 \gt 0$$ $$c = v^2 \gt 0$$

The original inequality becomes:

$$F(t, u, v) = t^4+u^4+v^4 +2t^2u^2v^2+1 - 2*(t^2u^2 + t^2v^2 + u^2v^2) \ge 0$$

The new expression is fully symmetric in $t, u $ and $v$. Therefore, the minimum must be symmetric as well, which means $t=u=v$.

$$F(t,t,t) = -3t^4 + 2t^6 + 1$$ $$\frac{\delta F}{\delta t} = -12t^33 + 12 t^5 = 12t^3(t^2-1)$$

For $t=1$ we get an extremum $F(1,1,1)=0$ which confirms our original inequality.


The following steps should convince us that the solution is in fact symmetric:

To be at the minimum of the left-hand-side, the partial derivatives have to be zero.

$$\frac{\delta F}{\delta t} = 4t^3 + 4tu^2v^2 - 4tu^2 - 4tv^2 = 0$$ $$\frac{\delta F}{\delta u} = 4u^3 + 4t^2uv^2 - 4t^2u - 4uv^2 = 0$$ $$\frac{\delta F}{\delta v} = 4v^3 + 4t^2u^2v - 4t^2v - 4u^2v = 0$$

this leads to:

$$t(t^2 - v^2 + u^2(v^2 - 1)) = 0$$ $$u(u^2 - v^2 + t^2(v^2 - 1)) = 0$$ $$v(v^2 - u^2 + t^2(u^2 - 1)) = 0$$

All variables have to be greater than zero.

$$t^2 - v^2 + u^2(v^2 - 1) = 0$$ $$u^2 - v^2 + t^2(v^2 - 1) = 0$$ $$v^2 - u^2 + t^2(u^2 - 1) = 0$$

The first two of these equations combine to:

$$ (t^2 - u^2)v^2 = 0$$

The last two equations give us:

$$ t^2(u^2+v^2-2) = 0$$

The first and the third equation yield:

$$t^2 - u^2 +u^2(v^2-1) + t^2(u^2-1)$$

Assuming $t=u$ we get:

$$t^2(t^2+v^2-2) = 0$$

or

$$v^2 = 2 - t^2$$

Inserting this we get

$$- t^4 + 3t^2 -2 = 0$$

This can be factored into

$$(t-1)(t+1)(t^2-2)=0$$

For $t^2-2=0$ we get $a=2$, $b=2$ and $c=0$ which violates the constraints.

Therefore, $a=b=c=1$ is the global minmum.

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