How prove this inequality $\sum\limits_{cyc}\frac{a^3}{b+c+d}\ge \dfrac{1}{3}$,if $\sum\limits_{cyc}a\sqrt{bc}\ge 1$

Question

Let $a$, $b$, $c$ and $d$ be non-negative numbers such that $$a\sqrt{bc}+b\sqrt{cd}+c\sqrt{da}+d\sqrt{ab}\ge 1.$$ Show that $$\dfrac{a^3}{b+c+d}+\dfrac{b^3}{a+c+d}+\dfrac{c^3}{a+b+d}+\dfrac{d^3}{a+b+c}\ge \dfrac{1}{3}$$

My try:use Cauchy-Schwarz inequality we have $$\sum_{cyc}\dfrac{a^3}{b+c+d}\left(\sum_{cyc}(b+c+d)\right)\ge \left(\sum_{cyc}a^{3/2}\right)^2$$

then I can't contious, and I found this simaler problem IMO 1990 problem:Let $a,b,c,d\ge 0$,and such $$ab+bc+cd+ad= 1$$

show that $$\dfrac{a^3}{b+c+d}+\dfrac{b^3}{a+c+d}+\dfrac{c^3}{a+b+d}+\dfrac{d^3}{a+b+c}\ge \dfrac{1}{3}$$ solution can see http://www.artofproblemsolving.com/Forum/viewtopic.php?p=360441&sid=a8e8d6ea18dc9e7121bccaedbd15f41f#p360441

Answer 1

WLOG, to simplify expressions, we can consider the squares of $a, b, c, d$ instead and let $s=a^2+b^2+c^2+d^2$, so the problem can be rephrased to show that for $a, b, c, d \ge 0$, $$\color{blue}{\sum_{cyc} a^2bc\ge 1 \implies \sum_{cyc} \frac{a^6}{s-a^2} \geqslant \frac13}$$

Convexity of $t \mapsto \dfrac{t^3}{s-t}$ and Jensen's inequality gives us: $$\sum_{cyc} \frac{a^6}{s-a^2} \ge 4\frac{\left(\frac{s}4\right)^3}{s-\frac{s}4} = \frac{s^2}{12}$$

So it is enough to show that $\displaystyle \sum_{cyc} a^2bc \ge 1 \implies s^2 \ge 4$

We have $$1\le \sum_{cyc} a^2bc = ac(ab+cd)+bd(ad+bc)$$

If $ad+bc \le ab+cd$, then we get from the above, \begin{align} 1 &\le (ac+bd)(ab+cd) \\ &\color{red}{\le} \left(\frac{ac+bd+ab+cd}2\right)^2 =\left(\frac{(a+d)(b+c)}2\right)^2 \\ &\color{red}{\le} 4 \left(\frac{a+b+c+d}4\right)^4 \\ &\color{green}{\le} 4 \left(\frac{a^2+b^2+c^2+d^2}4\right)^2 = \frac{s^2}4 \\ \end{align} where we have used $\color{red}{AM-GM}$ and $\color{green}{AM-QM}$ inequalities.

OTOH, if $ab+cd \le ad+bc$, proceeding along similar lines, by symmetry we get the same result. Hence proved.

Answer 2

By Holder $$\sum_{cyc}\frac{a^3}{b+c+d}\geq\frac{(a+b+c+d)^3}{4\sum\limits_{cyc}(b+c+d)}=\frac{(a+b+c+d)^2}{12}.$$ Thus, it remains to prove that $$(a+b+c+d)^2\geq4(a\sqrt{bc}+b\sqrt{cd}+c\sqrt{da}+d\sqrt{ab}).$$ Let $\{a,b,c,d\}=\{x,y,z,t\}$, where $x\geq y\geq z\geq t$.

Hence, by Rearrangement, AM-GM and C-S we obtain: $$4(a\sqrt{bc}+b\sqrt{cd}+c\sqrt{da}+d\sqrt{ab})=4(\sqrt{a}\sqrt{abc}+\sqrt{b}\sqrt{bcd}+\sqrt{c}\sqrt{cda}+\sqrt{d}\sqrt{dab})\leq$$ $$\leq4(\sqrt{x}\sqrt{xyz}+\sqrt{y}\sqrt{xyt}+\sqrt{z}\sqrt{xzt}+\sqrt{t}\sqrt{yzt})=4\left(\sqrt{xz}+\sqrt{yt}\right)\left(\sqrt{xy}+\sqrt{zt}\right)\leq$$ $$\leq4\left(\frac{\sqrt{xz}+\sqrt{yt}+\sqrt{xy}+\sqrt{zt}}{2}\right)^2=\left((\sqrt{x}+\sqrt{t})(\sqrt{y}+\sqrt{z})\right)^2\leq$$ $$\leq\left(\frac{\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}}{2}\right)^4\leq\frac{1}{16}\left(\sqrt{4(x+y+z+t)}\right)^4=$$ $$=(x+y+z+t)^2=(a+b+c+d)^2.$$ Done!