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In 2006 Carbery raised a question about an improvement on the na\"ive norminequality $\|f+g\|_p^p \leq 2^{p-1}(\|f\|_p^p + \|g\|_p^p)$ for two functionsin $L^p$ of any measure space. When $f=g$ this is an equality, but when thesupports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed.Carbery's question concerns a proposed interpolation between the two situationsfor $p>2$. The interpolation parameter measuring the overlap is $\|fg\|_{p/2}$.We prove an inequality of this type that is stronger than the one Carberyproposed. Moreover, our stronger inequalities are valid for all $p$.

Inequalities for $L^p$-norms that sharpen the triangle inequality and complement Hanner's Inequality