TITLE: Analog of $(a+b)^2 \leq 2(a^2 + b^2)$
QUESTION [6 upvotes]: Is there an inequality such as
$$(a+b)^2 \leq 2(a^2 + b^2)$$
for higher powers of $k$
$$(a+b)^k \leq C(a^k + b^k)?$$

REPLY [14 votes]: The generalized mean inequality states that
$$\dfrac{a+b}2\leq \left(\dfrac{a^k+b^k}2\right)^{1/k},$$
with equality if and only if $a=b$, from which it follows that
$$(a+b)^k\leq 2^{k-1}(a^k+b^k),$$
with equality if and only if $a=b$. Thus $C=2^{k-1}$ works and no smaller $C$ works.

REPLY [10 votes]: Take the ratio $$f(a, b)=\frac{(a+b)^k}{a^k+b^k}$$and observe that it is homogeneous of degree zero, that is $f(a, b)=f(\lambda a, \lambda b)$ for all $\lambda >0$. So $f$ is constant along rays in $\mathbb{R}^2$ and so, in particular, 
$$f(a, b)\le \max_{(x, y)\in \mathbb{S}^1} f(x, y),$$
where $\mathbb{S}^1$ is the unit circle. This means that the sought inequality is true with $C=\max_{\mathbb{S}^1} f(x,y)$. You can check that with $k=2$ you recover $C=2$.