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

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