Does this inequality hold true for all $\alpha\in\mathbb{R}$? Let $x$ and $y$ be two real and positive numbers. Let $\alpha\in\mathbb{R}$. I am trying to understand if the inequality
$$ x^{\alpha} + y^{\alpha} \leq (x+y)^{\alpha}$$
holds true. By attemps, I found that it holds true only if $\alpha \geq1$. Could anyone please tell me if it is true?
Moreover, there is a simple way to justify it instead of proceeding by trial?
Thank you in advance!

We have that
$$x^{\alpha} + y^{\alpha} \leq (x+y)^{\alpha} \iff \left(\frac x{x+y}\right)^\alpha+\left(\frac y{x+y}\right)^\alpha \le 1$$
and assuming wlog $x+y=1$
$$x^{\alpha} + y^{\alpha} \leq 1$$
which is true by convexity for $\alpha \ge 1$.
