Degree large-deviation rate function and roots

PDF p. 32: Equation (6.3) and following paragraph; Figure 3

Let $\log$ be the natural logarithm, the inverse of the exponential function on the positive reals, and write $e=\exp (1)$. For real $c>-1$, define $h(c)=(1+c)\log (1+c)-c$.

For real $\gamma >1$, let $c^{-}(\gamma )$ be the unique root of $\gamma h(c)=1$ in $(-1,0)$ and let $c^{+}(\gamma )$ be its unique root in $(0,\mathrm{\infty })$. Extend the negative branch by $c^{-}(\gamma )=-1$ for $0<\gamma \le 1$, and set $c^{+}(1)=e-1$.

The natural logarithm is defined on positive reals by The natural logarithm as the inverse of the exponential function, so the formula for $h$ is defined for every $c>-1$. In Entropy rates, inverse branches and shifted root contracts, $I(a)=a\log a-a+1$ has a lower inverse branch $a_{-}:(0,1)\to (0,1)$ and an upper inverse branch $a_{+}:(0,\mathrm{\infty })\to (1,\mathrm{\infty })$. Since $h(c)=I(1+c)$ and $1\mathrm{/}\gamma \in (0,1)$ when $\gamma >1$, the two designated roots exist uniquely and equal $a_{-}(1\mathrm{/}\gamma )-1$ and $a_{+}(1\mathrm{/}\gamma )-1$.

As $\gamma \downarrow 1$, the lower inverse tends to $0$: for every $0<\delta <1$, strict decrease of $I$ on $(0,1)$ and $I(\delta )<1$ place $a_{-}(t)$ below $\delta$ once $t<1$ is sufficiently close to $1$. Thus the assigned value $c^{-}(1)=-1$ makes the negative extension continuous at $1$; it is constant for $0<\gamma \le 1$. The upper inverse is continuous at $1$, and $I(e)=e\log e-e+1=1$ because $\log (\exp 1)=1$. Its unique upper-branch value at $1$ is therefore $e$, so $c^{+}(1)=e-1$ is the continuous endpoint value.

The notation and endpoint assignments are attributed to the article, PDF p. 32, equation (6.3), its following paragraph and Figure 3. The article is not used as the justification for the existence or uniqueness of the roots.