inverse Minkowski inequality for $p \in (0,1)$

Question

I'm trying to find functions $f,g \in L^p(\mathbb{R})$ with $$||f+g||_p > ||f||_p+||g||_p$$ where $p \in (0,1)$. All my ideas failed so fair, any help and hints appreciated!

Answer 1

Take $f$ supported on $[0,1]$ and equal to $1$, $g$ supported on $[1,2]$ and equal to $a>0$.

Then $\|f+g\|_p=(1+a^p)^{1/p}$, $\|f\|_p=1$, $\|g\|_p=a$.

We want $1+a^p > (1+a)^p$. By binomial expansion, the RHS is only $1+a/p+O(a^2)$ which is less than $1+a^p$ for $a>0$ small enough.

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For who doesn't like big $O$ estimates: it suffices to prove that $$\lim_{a \to 0+}\frac{(1+a)^p-1}{a^p} < 1$$ We have by definition of derivative: $$\lim_{a \to 0+}\frac{(1+a)^p-1}{a} = p$$ and so for $p<1$: $$\lim_{a \to 0+}\frac{(1+a)^p-1}{a^p} = p \cdot \lim_{a \to 0+}a^{1-p}=0$$

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The two are essentially the same, the only difference being that Taylor expansion gives the error $O(a^2)$ while the definition of derivative gives $o(a)$, which is sufficient.