Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$ Can you help me to prove that
$$(x+y)^n\leq 2^{n-1}(x^n+y^n)$$
for $n\ge1$ and $x,y\ge0$.
I tried by induction, but I didn't get a result.

Look. I also have tried to do it by induction. It is obvious that it holds for $n=1$ and $n=2$. Assume that it also holds for $n$. Let's prove that inequality for $n+1$:
$$
2^n(x^{n+1} + y^{n+1}) - (x+y)^{n+1} = 2^{n}(x^{n+1} + y^{n+1}) -(x+y)^n(x+y)\geq 2^n(x^{n+1} + y^{n+1}) - 2^{n-1}(x^n + y^n)(x+y) = 
2^{n-1}(x^{n+1} + y^{n+1} - x^ny - y^nx) = 2^{n-1}(y(y^n - x^n) + x(x^n - y^n))=
2^{n-1}(y^n-x^n)(y-x) = 2^{n-1}(y-x)^2(y^{n-1}+xy^{n-2}+x^2y^{n-2}+\dots+y^2x^{n-2} + x^{n-1}) \geq 0. 
$$ 
So we have proved that $2^n(x^{n+1} + y^{n+1}) - (x+y)^{n+1} \geq 0$. It is similar to $(x+y)^{n+1} \leq 2^n(x^{n+1} + y^{n+1})$
