Entropy summand ($f(x)=-x{\log }_{2}x$)

The contribution of a symbol with probability $x$ to Shannon entropy, with $f(0)=0$. It is a concave function because $f^{\prime \prime }(x)=-1/(x\ln 2)<0$ on $(0,1)$ and its endpoint extension is continuous. Its sharp bound for pairs exactly distance $v$ apart in $[0,1]$ isTo prove this, the increment $f(t+v)-f(t)$ decreases with $t$, from $f(v)$ to $-f(1-v)$. For $v\le 1/2$, the maximum is $f(v)$: the difference $f(v)-f(1-v)$ vanishes at zero and one half and has second derivative $(2v-1)/(v(1-v)\ln 2)\le 0$ between them. Hence it is nonnegative by concavity.