Is $L^p$ linear for $0

The following is an exercise from Bruckner's Real Analysis:

Show that for all $0 <p< \infty$ the collections $L^p$ of measurablefunctions defined on a measure space $(X, \mathcal{M},μ)$ such that $ \int_X |f|^p d\mu < \infty$ are linear spaces. [Hint : Use the inequality $(a + b)^p ≤ 2^p(a^p + b^p)$.]

The spaces is closed for all $p$ under scalar multiplication but for closed under sum we use the hint : The case $x=0$ is obvious so letting $a>0$ and considering $x=a/b$, we have $f(x)=(x + 1)^p - 2^p(x^p + 1) \le 0$ holds for all $0\le x$ when $1 \le p$ but doesn't hold for $0<p<1$always. So does really the claim fails for $0<p<1$ or I make mistakes?

So the question is about validity of $f(x)=(x + 1)^p - 2^p(x^p + 1) \le 0$ for $0<p<1$.