Analog of $(a+b)^2 \leq 2(a^2 + b^2)$

Is there an inequality such as $$(a+b)^2 \leq 2(a^2 + b^2)$$ for higher powers of $k$ $$(a+b)^k \leq C(a^k + b^k)?$$

Solution 1:

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.

Solution 2:

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 $C=2$.