prove that when : $(a+b)(c+d)(x+y)≥(\sqrt[3]{acx}+\sqrt[3]{bdy})^3$

Question

How is proved ? when $a,b,c,d,x,y \in R^+$:

$$(a+b)(c+d)(x+y)≥(\sqrt[3]{acx}+\sqrt[3]{b d y})^3$$

I want a simple way.thank you very much !and What is the name of this unequal? my try :

$(a+b)(c+d)(x+y)=a c x + b c x + a d x + b d x + a c y + b c y + a d y + b d y$

now ??

Answer 1

It's just Holder for three series.

After your expanding we can use AM-GM: $$bcx+adx+acy\geq3\sqrt[3]{a^2c^2x^2bdy}$$ The second is similar.