A Cyclic Inequality of Degree Four

Problem

A Cyclic Inequality of Degree Four, problem

Solution

Observe that the inequality is equivalent to

$\displaystyle \sum_{cycl}b(a^4-a\sqrt{3}+2)\ge 0.$

But

$\displaystyle\begin{align} a^4-a\sqrt{3}+2&=(a^2-1)^2+2a^2-a\sqrt{3}+1\\ &=(a^2-1)^2+\left(\sqrt{2}a-\sqrt{\frac{3}{8}}\right)^2+\frac{5}{8}\\ &\gt 0. \end{align}$

Extension

Note that the original inequality is weak, for example,

$\displaystyle\begin{align} a^4-a\sqrt{5}+2&=(a^2-1)^2+2a^2-a\sqrt{5}+1\\ &=(a^2-1)^2+\left(\sqrt{2}a-\sqrt{\frac{5}{8}}\right)^2+\frac{3}{8}\\ &\gt 0 \end{align}$

Shows that $\displaystyle \sum_{cycl}a^4b+2\sum_{cycl}a\ge \sqrt{5}\sum_{cycl}ab\,$ is also true which organically leads to the question of finding the maximum $k\,$ such that

$\displaystyle a^4b+b^4c+c^4a+2(a+b+c)\ge k(ab+bc+ca).$

Simply following the derivation above, the maximal $k\,$ satisfies, $k\ge k_0=2\sqrt{2}.$ But we can do better.

Let $f(x)=x^4-xk+2.\,$ Then $f'(x)=4x^3-k=0\,$ leads to $\displaystyle x=\sqrt[3]{\frac{k}{4}}.\,$ It is easy to see that this is a local minimum because $f''\left(\sqrt[3]{\frac{k}{4}}\right)\ge 0.$

Now, $\displaystyle f\left(\sqrt[3]{\frac{k}{4}}\right)=\left(\sqrt[3]{\frac{k}{4}}\right)^4-k\sqrt[3]{\frac{k}{4}}+2=2-3\left(\sqrt[3]{\frac{k}{4}}\right)^4.\,$ From this, $\displaystyle k_1=4\left(\frac{2}{3}\right)^{\frac{3}{4}}.\,$ Now, this is an imporvement since $k_1/k_0\approx 1.0434.\,$

The graph (courtesy wolframalpha) shows that this may not be the last word:

A Cyclic Inequality of Degree Four, graph

However, the graph generated by GeoGebra tells us a different story:

A Cyclic Inequality of Degree Four, graph

Acknowledgment

Dan Sitaru has kindly posted the above problem of his from the Romanian Mathematical Magazine at the CutTheKnotMath facebook page and Leo Giugiuc messaged his solution practically right away.

Jeffrey Samuelson gave an answer to the extended question.

 

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