I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.

Under the standard hypotheses for the Riemann mapping theorem, if $\Omega = \mathbb C\setminus \mathcal K$ is a simply connected domain, then the Green's potential $G(z)$ of $\Omega$ (i.e. $G(z)=0$ on $\mathcal K$ and $G(z) \simeq \ln |z|$ at infinity) satisfies $$ G(z) \leq \sqrt{ \frac{\operatorname{dist}(z,\mathcal K)}{\operatorname{Cap}(\mathcal K)}} $$ where $\operatorname{Cap}(\mathcal K)$ is the capacity. I tried to follow the proof but it invokes a Theorem 1 Ch. IV of Goluzin's book, which seemingly has nothing to do with the property where it is used.

Question: is such an inequality proven somewhere else (or provable)? I am thinking of Köbe 1/4 theorem….

Note that the statement should not rely on any regularity whatsoever of $\Omega$. It would seem to me that this should be some very classical result (and if true, a quite nice one at that).

I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.

Under the standard hypotheses for the Riemann mapping theorem, if $\Omega =\widehat{ \mathbb C}\setminus \mathcal K$ is a simply connected domain (of the Riemann sphere), then the Green's potential $G(z)$ of $\Omega$ (i.e. $G(z)=0$ on $\mathcal K$ and $G(z) \simeq \ln |z|$ at infinity) satisfies $$ G(z) \leq \sqrt{ \frac{\operatorname{dist}(z,\mathcal K)}{\operatorname{Cap}(\mathcal K)}} $$ where $\operatorname{Cap}(\mathcal K)$ is the capacity. I tried to follow the proof but it invokes a Theorem 1 Ch. IV of Goluzin's book, which seemingly has nothing to do with the property where it is used.

Question: is such an inequality proven somewhere else (or provable)? I am thinking of Köbe 1/4 theorem….

Note that the statement should not rely on any regularity whatsoever of $\Omega$. It would seem to me that this should be some very classical result (and if true, a quite nice one at that).

I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.

Under the standard hypotheses for the Riemann mapping theorem, if $\Omega = \mathbb C\setminus \mathcal K$ is a simply connected domain, then the Green's potential $G(z)$ of $\Omega$ (i.e. $G(z)=0$ on $\mathcal K$ and $G(z) \simeq \ln |z|$ at infinity) satisfies $$ G(z) \leq \sqrt{ \frac{{\rm dist}(z,\mathcal K)}{Cap(\mathcal K)}} $$ where $Cap(\mathcal K)$ is the capacity. I tried to follow the proof but it invokes a Theorem 1 Ch. IV of Goluzin's book, which seemingly has nothing to do with the property where it is used.

Question: is such an inequality proven somewhere else (or provable)? I am thinking of Köbe 1/4 theorem...

Note that the statement should not rely on any regularity whatsoever of $\Omega$. It would seem to me that this should be some very classical result (and if true, a quite nice at that).

I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.

Under the standard hypotheses for the Riemann mapping theorem, if $\Omega = \mathbb C\setminus \mathcal K$ is a simply connected domain, then the Green's potential $G(z)$ of $\Omega$ (i.e. $G(z)=0$ on $\mathcal K$ and $G(z) \simeq \ln |z|$ at infinity) satisfies $$ G(z) \leq \sqrt{ \frac{\operatorname{dist}(z,\mathcal K)}{\operatorname{Cap}(\mathcal K)}} $$ where $\operatorname{Cap}(\mathcal K)$ is the capacity. I tried to follow the proof but it invokes a Theorem 1 Ch. IV of Goluzin's book, which seemingly has nothing to do with the property where it is used.

Question: is such an inequality proven somewhere else (or provable)? I am thinking of Köbe 1/4 theorem….

Note that the statement should not rely on any regularity whatsoever of $\Omega$. It would seem to me that this should be some very classical result (and if true, a quite nice one at that).

I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.

Under the standard hypotheses for the Riemann mapping theorem, if $\Omega = \mathbb C\setminus \mathcal K$ is a simply connected domain, then the Green's potential $G(z)$ of $\Omega$ (i.e. $G(z)=0$ on $\mathcal K$ and $G(z) \simeq \ln |z|$ at infinity) satisfies $$ G(z) \leq \sqrt{ \frac{{\rm dist}(z,\mathcal K)}{Cap(\mathcal K)}} $$ where $Cap(\mathcal K)$ is the capacity. I tried to follow the proof but it invokes a Theorem 1 Ch. IV of Goluzin's book, which seemingly has nothing to do with the property where it is used.

Question: is such an inequality proven somewhere else (or provable)? I am thinking of Köbe 1/4 theorem...

Note that the statement should not rely on any regularity whatsoever of $\Omega$. It would seem to me that this should be some very classical result (and if true, a quite nice at that).

I read in a paper by E. A. Rakhmanov "Orthogonal Polynomials and S-curves" the following statement, in Lemma 9.9 therein, which I state here in my words.

Under the standard hypotheses for the Riemann mapping theorem, if $\Omega = \mathbb C\setminus \mathcal K$ is a simply connected domain, then the Green's potential $G(z)$ of $\Omega$ (i.e. $G(z)=0$ on $\mathcal K$ and $G(z) \simeq \ln |z|$ at infinity) satisfies $$ G(z) \leq \sqrt{ \frac{{\rm dist}(z,\mathcal K)}{Cap(\mathcal K)}} $$ where $Cap(\mathcal K)$ is the capacity. I tried to follow the proof but it invokes a Theorem 1 Ch. IV of Goluzin's book, which seemingly has nothing to do with the property where it is used.

Question: is such an inequality proven somewhere else (or provable)? I am thinking of Köbe 1/4 theorem...

Note that the statement should not rely on any regularity whatsoever of $\Omega$. It would seem to me that this should be some very classical result (and if true, a quite nice at that).