let $a,b,c,d&gt;0$,show that $$(a+b)^3+(b+c)^3+(c+d)^3+(d+a)^3\ge 8(a^2b+b^2c+c^2d+d^2a)$$ I want use AM-GM inequality to solve it,such $$(a+b)^3=a^3+b^3+3a^2b+3ab^2$$ so we must to prove $$2(a^3+b^3+c^3+d^3)+3(ab^2+bc^2+cd^2+da^2)\ge 5(a^2b+b^2c+c^2d+d^2a)$$ Then I can't
let $x,y,z,w&gt;0$,show that $$(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$$  it seem use AM-GM inequality to solve it,But I can't it,Thanks
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