Prove an inequality using Jensen's

Question

$\forall (x_n),(y_n)\ge0$,pf:

$$(x_1\cdot...\cdot x_n)^{1/n}+(y_1\cdot...\cdot y_n)^{1/n}\le((x_1+y_1)\cdot...\cdot (x_n+y_n))^{1/n}$$

I've turned it into: $$log(1+(z_1\cdot...\cdot z_n)^{1/n})\le\dfrac1 n\sum_{k=1}^{n} log(1+z_k)$$

Look like the average of a function with the function over an average now, but it's not convex...

Could someone give a simple proof to this inequality(even without Jensen's) :)

Thanks~

Answer 1

It's just Holder: $$(x_1+y_1)(x_2+y_2)...(x_n+y_n)\geq\left(\sqrt[n]{x_1x_2...x_n}+\sqrt[n]{y_1y_2...y_n}\right)^n$$