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Inequality 6(Christos Patilas)

Problem:

Let \displaystyle x,y,z\in (0,+\infty)\wedge n\in \mathbb{N}. Prove that

\displaystyle (x^{n+3}-x^n+3)(y^{n+3}-y^n+3)(z^{n+3}-z^n+3)\geq (x+y+z)^3.

Solution:

We will prove that

\displaystyle \sum_{cyc}(x^{n+3}-x^n+3)\geq \sum_{cyc}(x^3+2).

Indeed, if we bring everything in the left hand side we get that

\displaystyle \sum_{cyc}(x^n-1)(x^3-1)\geq 0,

which is obviously true.

So, it remains to prove that

\displaystyle \prod_{cyc}(x^3+1+1)\geq (x+y+z)^3,

which is a direct conclusion of Holder’s inequality, Q.E.D.

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