If $a+b+c=3$, then $ \frac{a}{5b+c^3}+\frac{b}{5c+a^3}+\frac{c}{5a+b^3} \geq \frac{1}{2}$

Let $a,b,c$ be nonnegative real numbers, no two of which are zero such that $a+b+c=3.$ Prove that $$ \dfrac{a}{5b+c^3}+\dfrac{b}{5c+a^3}+\dfrac{c}{5a+b^3} \geq \dfrac{1}{2}$$

I think this inequality must use the Cauchy-Schwarz inequality $$\sum_{cyc}\dfrac{a^2}{a(5b+c^3)}\sum_{cyc} a(5b+c^3)\ge (a+b+c)^2=9$$

then it suffices to prove that $$\dfrac{9}{5(ab+bc+ac)+(ac^3+ba^3+cb^3)}\ge \dfrac{1}{2}?$$

Solution 1:

This is very ugly proof, but it's a proof. First we homogenize the inequality and then because of the cyclicity it suffices to consider two cases.

1) We have $a \leq b \leq c$, then there exist $u,v \geq 0$ such that $b = a + u$, $c = a + u + v$, then the inequality can be expanded into $$24300 a^7 (u^2 + u v + v^2) + 27 a^6 (3916 u^3 + 8007 u^2 v + 8859 u v^2 + 2384 v^3) + (2 u + v)^2 (29 u^3 + 47 u^2 v + 32 u v^2 + 9 v^3) (13 u^4 + 75 u^3 v + 130 u^2 v^2 + 60 u v^3 + 10 v^4) + 27 a^5 (7609 u^4 + 23750 u^3 v + 33585 u^2 v^2 + 17444 u v^3 + 3013 v^4) + 9 a^4 (25502 u^5 + 107036 u^4 v + 189674 u^3 v^2 + 145480 u^2 v^3 + 51166 u v^4 + 6913 v^5) + a^3 (157534 u^6 + 828219 u^5 v + 1780485 u^4 v^2 + 1807807 u^3 v^3 + 962376 u^2 v^4 + 264369 u v^5 + 29752 v^6) + a (2 u + v) (7620 u^7 + 52167 u^6 v + 138412 u^5 v^2 + 180418 u^4 v^3 + 133318 u^3 v^4 + 57590 u^2 v^5 + 13535 u v^6 + 1355 v^7) + a^2 (65669 u^7 + 413968 u^6 v + 1051107 u^5 v^2 + 1329962 u^4 v^3 + 949111 u^3 v^4 + 394122 u^2 v^5 + 89315 u v^6 + 8575 v^7) \geq 0$$ which is obvious.

2) We have $a \leq c \leq b$, then there exist $u,v \geq 0$ such that $b = a + u + v$, $c = a + u$, again expanding the inequality it becomes $$24300 a^7 (u^2 + u v + v^2) + 27 a^6 (3916 u^3 + 3741 u^2 v + 4593 u v^2 + 2384 v^3) + (2 u + v)^2 (29 u^3 + 40 u^2 v + 25 u v^2 + 5 v^3) (13 u^4 - 23 u^3 v - 17 u^2 v^2 + 27 u v^3 + 18 v^4) + 27 a^5 (7609 u^4 + 6686 u^3 v + 7989 u^2 v^2 + 8912 u v^3 + 3013 v^4) + 9 a^4 (25502 u^5 + 20474 u^4 v + 16550 u^3 v^2 + 36346 u^2 v^3 + 28594 u v^4 + 6913 v^5) + a^3 (157534 u^6 + 116985 u^5 v + 2400 u^4 v^2 + 182623 u^3 v^3 + 302685 u^2 v^4 + 163011 u v^5 + 29752 v^6) + a (2 u + v) (7620 u^7 + 1173 u^6 v - 14570 u^5 v^2 - 4163 u^4 v^3 + 19126 u^3 v^4 + 21491 u^2 v^5 + 9035 u v^6 + 1355 v^7) + a^2 (65669 u^7 + 45715 u^6 v - 53652 u^5 v^2 + 14468 u^4 v^3 + 159388 u^3 v^4 + 154038 u^2 v^5 + 59966 u v^6 + 8575 v^7)$$ where the inequalities $$13 u^4 - 23 u^3 v - 17 u^2 v^2 + 27 u v^3 + 18 v^4 \geq 0$$ $$7620 u^7 + 1173 u^6 v - 14570 u^5 v^2 - 4163 u^4 v^3 + 19126 u^3 v^4 + 21491 u^2 v^5 + 9035 u v^6 + 1355 v^7 \geq 0$$ $$65669 u^7 + 45715 u^6 v - 53652 u^5 v^2 + 14468 u^4 v^3 + 159388 u^3 v^4 + 154038 u^2 v^5 + 59966 u v^6 + 8575 v^7 \geq 0$$ are true and easy to prove.