[不等式] 最近比较流行的几个Vasc不等式

1.If$a,b,c$   are nonnegative real numbers such that$ab+bc+ca=4$ , then
\[ \sqrt{a^2+9bc}+\sqrt{b^2+9ca}+\sqrt{c^2+9ab}\ge 10. \]



2. If$a,b,c$   are nonnegative real numbers such that$ab+bc+ca=3$ , then
\[ \sqrt{a^2+4bc}+\sqrt{b^2+4ca}+\sqrt{c^2+4ab}\ge\sqrt{a^2+b^2+c^2+42}. \]

3.
This inequality inspires other similar nice inequalities for $a,b,c\ge 0$.
$(A) \ \ \ \  \sum\sqrt{(ab+c^2)(ac+b^2)}\le \frac 3{4}(a+b+c)^2;$

$(B) \ \ \ \  \sum\sqrt{(a^2+ab+b^2)(a^2+ac+c^2)}\ge (a+b+c)^2;$

$(C) \ \ \ \  \sum\sqrt{(9ab+c^2)(9ac+b^2)}\ge 7(ab+bc+ca);$

$(D) \ \ \ \  \sum\sqrt{(a^2+7ab+b^2)(a^2+7ac+c^2)}\ge 7(ab+bc+ca).$

Also, the cyclic inequality holds:
$(B1) \ \ \ \  \sum\sqrt{(a^2+b^2+bc)(b^2+c^2+ca)}\ge (a+b+c)^2;$

有何流行之处？