TITLE: $(x+y)^a \geq x^a + y^a$
QUESTION [5 upvotes]: Take $x,y \in (0,\infty), a \in (1, \infty)$. For $a \in \mathbb N $ by binomial theorem it holds $$(x+y)^a \geq x^a + y^a.$$
Trying with some numbers it seems true also for $a \in (1, \infty)$. How can I prove it?

REPLY [3 votes]: Assume $y<x$. Then we have
$(x+y)^\alpha = x^\alpha(1+\frac{y}{x})^\alpha > x^\alpha (1+\frac{y}{x})$.
Last inequality since $(1+\frac{y}{x})>1$. But now $\frac{y}{x}<1$, hence $\frac{y}{x} > (\frac{y}{x})^\alpha$, thus
$x^\alpha (1+\frac{y}{x}) > x^\alpha (1+(\frac{y}{x})^\alpha) = x^\alpha+y^\alpha$.