Green kernel of a simply connected plane domain from a Riemann map

Because $\phi$ is an injective holomorphic map on the domain $\mathrm{\Omega }$, [F2] gives ${\phi }^{\mathrm{\prime }}(a)\mathrm{\ne }0$; hence $w\mapsto \phi (w)$ has a zero of order one at $a$, and [F2] provides a disc $D(a,\rho )\subseteq \mathrm{\Omega }$ with $\phi (z)=(z-a)q(z)$ for a holomorphic $q$ that is nowhere zero on $D(a,\rho )$. By [F3] there is a holomorphic $L$ on $D(a,\rho )$ with $\exp L=q$, and $\mathrm{Re}L$ is harmonic on $D(a,\rho )$ $-\log \mathrm{\mid }\phi (z)\mathrm{\mid }+\log \mathrm{\mid }z-a\mathrm{\mid }=-\mathrm{Re}L(z)(0<\mathrm{\mid }z-a\mathrm{\mid }<\rho ),$ because $\mathrm{\mid }\phi (z)\mathrm{\mid }=\mathrm{\mid }z-a\mathrm{\mid }\mathrm{\mid }q(z)\mathrm{\mid }=\mathrm{\mid }z-a\mathrm{\mid }{e}^{\mathrm{Re}L(z)}$. The inverse ${\phi }^{-1}:\mathbb{D}\to \mathrm{\Omega }$ is holomorphic by [F2].

On the unit disc the least logarithmic-pole candidate at $0$ is $-\log \mathrm{\mid }w\mathrm{\mid }$. Indeed $-\log \mathrm{\mid }w\mathrm{\mid }$ is positive on $\mathbb{D}\setminus \{0\}$, harmonic there by [F3], and $-\log \mathrm{\mid }w\mathrm{\mid }+\log \mathrm{\mid }w\mathrm{\mid }=0$ extends harmonically across $0$, so it is a candidate. If $U$ is any candidate at $0$, then $D:=U+\log \mathrm{\mid }w\mathrm{\mid }$ agrees on $\mathbb{D}\setminus \{0\}$ with a function harmonic on $\mathbb{D}$, hence is harmonic on $\mathbb{D}$ by [F3]; on the circle $\mathrm{\mid }w\mathrm{\mid }=r$ it satisfies $D\ge \log r$ because $U\ge 0$, so the minimum principle [F4] applied on $\{\mathrm{\mid }w\mathrm{\mid }\le r\}$ gives $D\ge \log r$ on that disc, and letting $r\uparrow 1$ yields $D\ge 0$, that is $U\ge -\log \mathrm{\mid }w\mathrm{\mid }$.

If $\psi :\mathrm{\Omega }\to \mathbb{D}$ is another biholomorphism with $\psi (a)=0$, then $h:=\psi \circ {\phi }^{-1}$ is a biholomorphic self-map of $\mathbb{D}$ fixing $0$, so $\mathrm{\mid }h(w)\mathrm{\mid }\le \mathrm{\mid }w\mathrm{\mid }$ for all $w$; the same bound applied to $h^{-1}$ gives $\mathrm{\mid }h(w)\mathrm{\mid }=\mathrm{\mid }w\mathrm{\mid }$, so $h$ is a rotation by [F4] and therefore $\mathrm{\mid }\psi (z)\mathrm{\mid }=\mathrm{\mid }h(\phi (z))\mathrm{\mid }=\mathrm{\mid }\phi (z)\mathrm{\mid }$ for every $z\in \mathrm{\Omega }$. Hence $-\log \mathrm{\mid }\psi (z)\mathrm{\mid }=-\log \mathrm{\mid }\phi (z)\mathrm{\mid }$ on $\mathrm{\Omega }\setminus \{a\}$.

The function $z\mapsto -\log \mathrm{\mid }\phi (z)\mathrm{\mid }$ is a logarithmic-pole candidate at $a$ on $\mathrm{\Omega }$: it is positive because $\mathrm{\mid }\phi (z)\mathrm{\mid }<1$ on $\mathrm{\Omega }$, it is harmonic on $\mathrm{\Omega }\setminus \{a\}$ because it is the composite of the harmonic function $-\log \mathrm{\mid }\cdot \mathrm{\mid }$ on $\mathbb{C}\setminus \{0\}$ with the holomorphic $\phi$ by [F3], and its corrector across $a$ is the harmonic function $-\mathrm{Re}L$ of step 1.1.

Let $u$ be an arbitrary logarithmic-pole candidate at $a$ on $\mathrm{\Omega }$ and put $U(w):=u({\phi }^{-1}(w))$ for $w\in \mathbb{D}\setminus \{0\}$. Then $U$ is a candidate at $0$ on $\mathbb{D}$: it is nonnegative, harmonic by [F3] because ${\phi }^{-1}$ is holomorphic by step 1.1, and $U(w)+\log \mathrm{\mid }w\mathrm{\mid }=u({\phi }^{-1}(w))+\log \mathrm{\mid }{\phi }^{-1}(w)-a\mathrm{\mid }-\log \frac{\mathrm{\mid }{\phi }^{-1}(w)-a\mathrm{\mid }}{\mathrm{\mid }w\mathrm{\mid }}$ extends harmonically across $0$, because the first two terms are the harmonic corrector of $u$ composed with ${\phi }^{-1}$ and the last term is $-\log \mathrm{\mid }g(w)\mathrm{\mid }$ for the holomorphic function $g(w):=({\phi }^{-1}(w)-a)\mathrm{/}w$, which satisfies $g(0)=({\phi }^{-1}{)}^{\mathrm{\prime }}(0)\mathrm{\ne }0$, so that it has a holomorphic logarithm near $0$ by [F3].

By disc leastness, step 1.2 applied to the candidate $U$ of step 2.2 gives $u({\phi }^{-1}(w))\ge -\log \mathrm{\mid }w\mathrm{\mid }$ for every $w\in \mathbb{D}\setminus \{0\}$; writing $w=\phi (z)$ yields $u(z)\ge -\log \mathrm{\mid }\phi (z)\mathrm{\mid }$ on $\mathrm{\Omega }\setminus \{a\}$. So $-\log \mathrm{\mid }\phi \mathrm{\mid }$ is the pointwise least candidate and hence $g_{\mathrm{\Omega }}(z,a)=-\log \mathrm{\mid }\phi (z)\mathrm{\mid }$ by [F1]; by step 1.3 the same formula holds for every biholomorphism sending $a$ to $0$. The supplied biholomorphism is the only place where a choice principle could enter, and by [F5] its existence is exactly what the Axiom of Choice is assumed for.