MHB Challenge Problem #8: 3Σ(1/(√(a^3+1))≥2Σ(√(a+b))

[Olinguito]

Let $a,b,c$ be positive real numbers such that $a+b+c=2$. Prove that
$$3\left(\frac1{\sqrt{a^3+1}}+\frac1{\sqrt{b^3+1}}+\frac1{\sqrt{c^3+1}}\right)\ \geqslant\ 2\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right).$$

 

[Olinguito]

Solution:

Spoiler

We have
$$\frac3{a^3+1}\ =\ \frac{2-a}{a^2-a+1}+\frac1{a+1}\ \geqslant\ 2\sqrt{\dfrac{2-a}{a^3+1}}$$
putting into partial fractions and applying AM–GM (noting that all terms are positive).

Hence
$$\frac3{\sqrt{a^3+1}}\ \geqslant\ 2\sqrt{2-a}\ =\ 2\sqrt{b+c}.$$
Similarly
$$\frac3{\sqrt{b^3+1}}\ \geqslant\ 2\sqrt{c+a}$$
and
$$\frac3{\sqrt{c^3+1}}\ \geqslant\ 2\sqrt{a+b};$$
summing gives the required inequality.

 

[kaliprasad]

Solution:

Spoiler

We have
$$\frac3{a^3+1}\ =\ \frac{2-a}{a^2-a+1}+\frac1{a+1}\ \leqslant\ 2\sqrt{\dfrac{2-a}{a^3+1}}$$
putting into partial fractions and applying AM–GM (noting that all terms are positive).

Hence
$$\frac3{\sqrt{a^3+1}}\ \leqslant\ 2\sqrt{2-a}\ =\ 2\sqrt{b+c}.$$
Similarly
$$\frac3{\sqrt{b^3+1}}\ \leqslant\ 2\sqrt{c+a}$$
and
$$\frac3{\sqrt{c^3+1}}\ \leqslant\ 2\sqrt{a+b};$$
summing gives the required inequality.

Spoiler

the question says it is $>=$ but answer says it is $<=$

 

[Olinguito]

I’ve fixed the typo in my solution.