How prove this inequality $9c^2(a+c)(a+2b+c)\le 2(a+b+c)(ab+bc+ac)(4c+a+b)$

Question

let $a\ge b\ge c\ge0$,show that $$9c^2(a+c)(a+2b+c)\le 2(a+b+c)(ab+bc+ac)(4c+a+b)$$

I have try: WLOG: $a+b+c=1$,then $$RHS-LHS=2(ab+bc+ac)(4c+a+b)-9c^2(a+c)(a+2b+c)=2(ac+b(a+c))(1+3c)-9c^2(1-b)(1+b) =2(ac+b(1-b))(1+3c)-9c^2(1-b^2)$$ then I can't it

Answer 1

A brute-force approach . . .

If $a=0$, then $b=c=0$, and the inequality holds.

Thus we can assume $a > 0$.

Since the inequality is homogeneous, we can scale $a,b,c$ by an arbitrary positive constant, hence it suffices to consider the case $a=1$.

Since $b\ge c$, we can write $b=c+x$ for some $x\ge 0$.

Replacing $a$ by $1$, $b$ by $c+x$, and then expanding, we get $$ \text{RHS}-\text{LHS}=P+Q \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\; $$ where \begin{align*} P&=21c^2+4c-7c^4+18c^3\\[4pt] Q&=24cx+2x+4x^2+48c^2x+22cx^2+16c^3x+16c^2x^2+2x^3+2cx^3\\[4pt] \end{align*} Note that all terms of $Q$ are nonnegative, hence $Q\ge 0$.

Also, since $0\le c\le 1$, we have $0\le c^4\le c^3$, so $$P=21c^2+4c-7c^4+18c^3\ge 21c^2+4c-7c^3+18c^3=21c^2+4c+11c^3\ge 0$$ hence $P+Q\ge 0$, and we're done.

Answer 2

We'll prove a stronger inequality:

Let $a\geq b\geq c\geq 0$. Show that: $$27c^2(a+c)(a+2b+c)\le 4(a+b+c)(ab+bc+ac)(4c+a+b).$$

Indeed, let $b=c+u$ and $a=c+u+v$.

Thus, $u$ and $v$ are non-negatives and $$4(a+b+c)(ab+bc+ac)(4c+a+b)-27c^2(a+c)(a+2b+c)=$$ $$=18(13u+5v)c^3+(327u^2+300uv+57v^2)c^2+$$ $$+4(34u^3+51u^2v+21uv^2+2v^3)c+4u(u+v)(2u+v)^2\geq0.$$

Answer 3

You have $$a + 2b + c \leq 2(a + b + c)$$ $$6c \leq 4c + a + b$$ $${3 \over 2}(c(a + c)) \leq ab + bc + ac$$ The first two inequalities follow immediately from $a \geq b \geq c \geq 0$. For the third one, the idea is that since $a \geq b \geq c$, one will have $ac + c^2 \leq {2 \over 3}(ab + bc + ac)$ since the smaller numbers are weighted more in $ac + c^2$ than they are in $ab + bc + ac$. It can be verified directly by doing some simple algebra which I'll omit. At any rate, multiplying these three equations gives your inequality.