Lemma 5.10.1103676DBFBA61

Proof. Since $v(0)\geq -1>-\infty $, lemma 5.10.5 gives a finite positive measure $\nu $ with support in $D(0,9/10)$ and a harmonic function $h$ on $D(0,4/5)$ such that

at every point of $D(0,4/5)$. Since $|v(0)|\leq 1$, the bounds (5.10.6) give $M=\nu (D)\leq C_0$ and $\sup _{D(0,1/2)}|h|\leq C_0$, where $C_0\geq 1$ is the universal constant of that lemma. Fix $\beta \in (0,1]$ with $\beta C_0\leq 2\pi $.

We show that $\int _{D(0,1/2)}\mathrm e^{-\beta P}\, \mathrm dV\leq 8\pi $. If $M=0$, then $P=0$, and this integral is $\pi /4$. Assume $M>0$ and put $b=\beta M/(2\pi )\in (0,1]$. Fix $z\in D(0,1/2)$. For $N\geq 1$ the Borel function $f_ N(\zeta )=-b\max \{ \log |z-\zeta |,-N\} $ is bounded on $D(0,9/10)$, so Jensen's inequality (lemma 2.2.4), applied to the convex function $\exp $ on $\mathbb R$ and the probability measure $\nu /M$, gives

As $N\to \infty $, the functions $f_ N$ increase to $-b\log |z-\zeta |$, which may take the value $+\infty $, and they are bounded below by the bounded function $f_1$. By monotone convergence, $\int f_ N\, \mathrm d\nu /M$ increases to $\int -b\log |z-\zeta |\, \mathrm d\nu (\zeta )/M=-\beta P(z)$, and therefore

for every $z\in D(0,1/2)$, with the convention $\mathrm e^{\infty }=\infty $. For every $\zeta \in D(0,9/10)$ the translated disc $D(0,1/2)-\zeta $ lies in $D(0,2)$, and $r^{-b}\leq 1+r^{-1}$ for $r>0$ because $0<b\leq 1$; hence

Integrating the bound for $\mathrm e^{-\beta P(z)}$ over $z\in D(0,1/2)$ with Tonelli's theorem, we obtain $\int _{D(0,1/2)}\mathrm e^{-\beta P}\, \mathrm dV\leq 8\pi $. Finally, since $\mathrm e^{-\beta v}=\mathrm e^{-\beta P}\mathrm e^{-\beta h}\leq \mathrm e^{\beta C_0}\mathrm e^{-\beta P}$ on $D(0,1/2)$, the lemma follows with $C=8\pi \mathrm e^{C_0}$. $\square$