I'm looking for the name and some references for the proof of the inequality below. I founded that is due to T. Carleman but no reference was given.


Let $f(z)$ be an analytic function on a subdomain $D$ of $\mathbb{C}$ containing a segment $I$ of the real axis.
We assume that $|f(z)|<M$ in $D$ and $|f(x)|<m$ in $I$.
Consider the regular function $h(z)$, harmonic in $D$, such that $h(z)=0$ on $I$, $h(z)=1$ on the boundary $\partial D$.
Then we have 
$$ |f(z)|\leq m^{1-h(z)} M^{h(z)},$$
for all $z \in D$.



An inequality of T. Carleman
This is the **two-constants theorem** :


Let $D$ be a domain of $\mathbb{C}$ with (non-polar) boundary $\partial D$, and let $B$ a Borel subset of $\partial D$. If $u$ is subharmonic on $D$ and satsifies
$$
u(z)\leq M,~z\in D\quad\text{ and }\quad\limsup\_{z\to\zeta}u(z)\leq m,~\zeta\in B,
$$
then
$$
u(z)\leq m\omega\_D(z,B)+M(1-\omega\_D(z,B)),
$$
where $\omega\_D$ denotes the harmonic measure for $D$. When the relative boundary of $B$ in $\partial D$ is polar, the function $\omega\_D(z,B)$ is exactly the solution of the Dirichlet problem with boundary data $1$ on $B$ and 0 on $\partial D\setminus B$. For a reference, see Theorem 4.3.7 in



> 
> T. Ransford, Potential Theory in the Complex Plane, Cambridge
>  University Press, Cambridge, 1995.
> 
> 
> 


The Two-constants (actually the $n$-constants) theorem is due to the Nevanlinna brothers and to A. Ostrowski :



> 
> F. Nevanlinna, R. Nevanlinna, Über die Eigenschaften einer
>  analytischen Funktion in der Umgebung einer singulären Stelle oder
>  Linie, Acta Soc. Sci. Fennica, 5 : 5 (1922)
> 
> 
> A. Ostrowski, Über allgemeine Konvergenzsätze der komplexen
>  Funktionentheorie, Jahresber. Deutsch. Math.-Ver., 32 : 9–12 (1923)
>  pp. 185–194
> 
> 
> 


