Shifted rate roots for quantitative expansion

PDF p. 36: Proposition 6.9, equations (6.7), (6.8)

Let $h(x)=(1+x)\log (1+x)-x$ for $x>-1$, where $\log$ is the natural logarithm. For real parameters $0<\epsilon <1$ and $1+\epsilon <\gamma <1+{\epsilon }^{-2}$, define $c^{-}=c^{-}(\gamma ,\epsilon )$ to be the unique real $c<0$ satisfying $1+c-\epsilon >0$ and $\gamma h(c-\epsilon )=1+\epsilon$. Define $c^{+}=c^{+}(\gamma ,\epsilon )$ to be the unique real $c>0$ satisfying $1+c+\epsilon >0$ and $\gamma h(c+\epsilon )=1+\epsilon$.

Equivalently, the respective equations are $\gamma ((1+c-\epsilon )\log (1+c-\epsilon )-(c-\epsilon ))=1+\epsilon$ and $\gamma ((1+c+\epsilon )\log (1+c+\epsilon )-(c+\epsilon ))=1+\epsilon$.

Degree large-deviation rate function and roots

Entropy rates, inverse branches and shifted root contracts

The rate-function definition gives $h(x)=I(1+x)$, where $I(a)=a\log a-a+1$ for $a>0$ (Degree large-deviation rate function and roots, Entropy rates, inverse branches and shifted root contracts). Put $t=(1+\epsilon )\mathrm{/}\gamma$. The hypotheses give $0<t<1$. The entropy-root proposition supplies exactly two positive solutions of $I(a)=t$: $a_{-}(t)\in (0,1)$ and $a_{+}(t)>1$. It also gives $a_{-}(t)-1+\epsilon <0$ and $a_{+}(t)-1-\epsilon >0$ under these parameter hypotheses.

For the first equation set $a=1+c-\epsilon$. Its two admissible roots are $c=a_{-}(t)-1+\epsilon$ and $c=a_{+}(t)-1+\epsilon$. The first is negative by the supplied sign bound, while the second is positive because $a_{+}(t)>1$ and $\epsilon >0$. Thus exactly one root is negative. For the second equation set $a=1+c+\epsilon$. Its roots are $c=a_{-}(t)-1-\epsilon$ and $c=a_{+}(t)-1-\epsilon$. The first is negative because $a_{-}(t)<1$ and $\epsilon >0$, while the second is positive by the supplied sign bound. Thus exactly one root is positive. The substitutions ensure that the logarithm arguments are positive.

The notation and equations are attributed to Proposition 6.9, equations (6.7) and (6.8), of the cited article; the justification above uses the separate mathematical suppliers.