Lower bound for $(x + y)^k $?

Question

I'm wondering, is there a lower bound for $(x + y)^k $? For example, if $x,y,k > 0$, can we say that $(x + y)^k \geq x^k + y^k$? If anyone has a source/reference for this, that would be great.

Answer 1

Assuming that $k \in \Bbb N$. $$(x+y)^k = \sum_{i=0}^k \binom k i x^iy^{k-i}$$ When $x, y > 0$, all the $\binom k i x^iy^{k-i} > 0$, so indead $$(x+y)^k \ge x^k+y^k$$