If a, b, c are positive real numbers such that $a^2+ b^2+ c^2 = 1$ $( \frac{1}{a} +\frac{1}{b} +\frac{1}{c}) +a +b +c \geq 4\sqrt{3}$

Question

If a, b, c are positive real numbers such that $a^2+ b^2+ c^2 = 1$
Show that: $$\frac{1}{a} +\frac{1}{b} +\frac{1}{c}+a +b +c \geq 4\sqrt{3}.$$

My attempt:

First , I used Holder's : $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{27}{3(a+b+c)}$$

similarly:

$$ a+b+c \geq \frac{(a+b+c)^3}{3(a^2+b^2+c^2)} = \frac{(a+b+c)^3}{3}$$

Which gives us:

$$LHS \geq \frac{27}{3(a+b+c)}+ \frac{(a+b+c)^3}{3}$$

By AM-GM:

$$LHS \geq 2\sqrt{3} \times (a+b+c)$$

So all we need to do is prove that $$2\sqrt{3} \times (a+b+c)\geq 4\sqrt{3}$$

And this means we need to prove $$a+b+c \geq 2 $$

Which I can't.

I am not looking for a solution , only hints that would guide me through solving it .

Thanks in advance for any communicated help !

Answer 1

Alternative approach using the point of incidence technique.

We will first employ the AM-GM inequality, followed by QM-AM: \begin{align*} a+b+c+\frac1a+\frac1b+\frac1c&=a+b+c+\frac13\left(\frac1a+\frac1b+\frac1c\right) + \frac23\left(\frac1a+\frac1b+\frac1c\right)\\ &\geqslant \frac{6}{\sqrt{3}}\sqrt[6]{abc\cdot\frac1{abc}}+\frac{2}{\sqrt[3]{abc}}\\ &\geqslant 2\sqrt{3}+\frac2{\sqrt{\frac{a^2+b^2+c^2}3}}\\ &= 2\sqrt{3}+\frac2{\sqrt{\frac13}}=4\sqrt{3} \end{align*}

Equality holds iff $a=b=c=\frac1{\sqrt{3}}$.

---

Observation. This problem is a stronger version of an inequality that appeared in Macedonia's National Olympiad back in $1999$. Notwithstanding, a similar idea to that of Macedonia yields a solution for this problem.

Answer 2

By your argument, we already have $$\frac1a+\frac1b+\frac1c+a+b+c\geq\frac9{a+b+c}+\frac{(a+b+c)^3}3=\frac9u+\frac{u^3}3\equiv f(u)$$ where $u=a+b+c>0$. Try to show that $f'(x)=0$ only has one root $x^*=\sqrt3$ when $x>0$ and $f''(x^*)>0$, therefore $f(u)\geq\min_{x>0}f(x)=f(x^*)=4\sqrt3$.