How to prove this inequality $\frac{a}{\sqrt{b^2+b+1}}+\frac{b}{\sqrt{c^2+c+1}}+\frac{c}{\sqrt{a^2+a+1}}\ge\sqrt{3}\;?$

Question

Let $a,b,c\ge 0$, and assume $a+b+c=3$.
I'd like to show that $$\frac{a}{\sqrt{b^2+b+1}}+\frac{b}{\sqrt{c^2+c+1}}+\frac{c}{\sqrt{a^2+a+1}}\ge\sqrt{3}$$

My try: Use Hölder inequality $$\left(\sum_{cyc}\dfrac{a}{\sqrt{b^2+b+1}}\right)\left(a\sqrt{b^2+b+1}\right)^2\ge (a+b+c)^3\,,$$ then we only need to prove this $$\sum_{cyc}a(b^2+b+1)=\sum_{cyc}(ab^2+ab+a)\le 9$$ but $$ab^2+bc^2+ca^2+ab+bc+ac\le 6$$ is not true.

So I can't prove it.

Answer 1

I'm going to use two facts. The first one can be obtained as following: $$\begin{align} ab + bc + ca &\leqslant a^2 + b^2 + c^2, \\ 3(ab + bc + ca) &\leqslant (a + b + c)^2 = 9, \\ ab + bc + ca &\leqslant 3. \end{align}$$

And the second one: $$\frac{1}{\sqrt{x^2 + x + 1}} \geqslant \frac{\sqrt{3}}{2} - \frac{x}{2\sqrt{3}}$$ (here's Wolfram Alpha visualization, a bit later I'm going to prove it more rigorously).

So, putting it all together: $$ \begin{align} \frac{a}{\sqrt{b^2 + b + 1}} + \frac{b}{\sqrt{c^2 + c + 1}} + \frac{c}{\sqrt{a^2 + a + 1}} &\geqslant \frac{\sqrt{3}}{2}(a + b + c) - \frac{ab + bc + ca}{2\sqrt{3}} \\ &\geqslant \frac{3\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = \sqrt{3} \end{align}$$

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Lemma. $\displaystyle \frac{1}{\sqrt{x^2 + x + 1}} \geqslant \frac{\sqrt{3}}{2} - \frac{x}{2\sqrt{3}},\ \forall x \geqslant 0$.

Note that $\displaystyle \frac{\sqrt{3}}{2} - \frac{x}{2\sqrt{3}} = \frac{1}{2\sqrt{3}}(3 - x) < 0$ for $x > 3$. So, for these $x$ we get obvious inequality that square root of something is greater than $0$.

For $x \in [0, 3]$: $$ \begin{align} \frac{1}{\sqrt{x^2 + x + 1}} \geqslant \frac{\sqrt{3}}{2} - \frac{x}{2\sqrt{3}} &\Longleftrightarrow 12 \geqslant (x^2 + x + 1)(3 - x)^2 \\ &\Longleftrightarrow x^4 - 5x^3 + 4x^2 +3x - 3 \leqslant 0 \\ &\Longleftrightarrow (x - 1)(x^3 - 4x^2 + 3) \leqslant 0 \\ &\Longleftrightarrow (x - 1)^2(x^2 - 3x - 3) \leqslant 0 \end{align}$$ which is true because roots of $x^2 - 3x - 3$ are: $x_1 = \frac{1}{2}(3 - \sqrt{21}) < 0$, $x_2 = \frac{1}{2}(3 + \sqrt{21}) > 3$. So for $x \in [0, 3]$ holds $x^2 - 3x - 3 < 0.$