View source for Two-constants theorem

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Let  $  D $
be a finitely-connected Jordan domain in the  $  z $-
plane and let  $  w ( z) $
be a regular analytic function in  $  D $
satisfying the inequality  $  | w ( z) | \leq  M $, 
as well as the relation

$$ 
\lim\limits _ {z \rightarrow \zeta }  \sup  | w ( z) |  \leq  \ 
m  <  M ,\  z \in D ,\  \zeta \in \alpha ,
$$

on some arc  $  \alpha $
of the boundary  $  \partial  D $. 
Then, at each point  $  z $
of the set

$$ 
\{ {z \in D } : {0 < \lambda \leq  \omega ( z ;  \alpha , D ) < 1 } \}
,
$$

where  $  \omega ( z ;  \alpha , D ) $
is the [[Harmonic measure|harmonic measure]] of the arc  $  \alpha $
with respect to  $  D $
at  $  z $, 
the inequality

$$ 
| w ( z) |  \leq   m  ^  \lambda  \cdot M ^ {1- \lambda }
$$

is satisfied. If for some  $  z $(
satisfying the condition  $  \omega ( z ;  \alpha , D ) = \lambda $) 
equality is attained, equality will hold for all  $  z \in D $
and for all  $  \lambda $, 
$  0 \leq  \lambda \leq  1 $, 
while the function  $  w ( z) $
in this case has the form

$$ 
w ( z)  =  e ^ {ia } m ^ {\phi ( z) } M ^ {1 - \phi ( z) } ,
$$

where  $  a $
is a real number and  $  \phi ( z) $
is an analytic function in  $  D $
for which  $   \mathop{\rm Re}  \phi ( z) = \omega ( z ;  \alpha , D ) $[[#References|[1]]], [[#References|[2]]].

The two-constants theorem gives a quantitative expression of the unique determination of analytic functions by their boundary values and has important applications in the theory of functions [[#References|[3]]]. Hadamard's three-circles theorem (cf. [[Hadamard theorem|Hadamard theorem]]) is obtained from it as a special case. For possible analogues of the two-constants theorem for harmonic functions in space see [[#References|[4]]], [[#References|[5]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Ostrowski,   "Über allgemeine Konvergenzsätze der komplexen Funktionentheorie"  ''Jahresber. Deutsch. Math.-Ver.'' , '''32''' :  9–12  (1923)  pp. 185–194</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.N. Mergelyan,   "Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation"  ''Uspekhi Mat. Nauk'' , '''11''' :  5  (1956)  pp. 3–26  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.D. Solomentsev,   "Three-spheres theorem for harmonic functions"  ''Dokl. Akad. Nauk ArmSSR'' , '''42''' :  5  (1966)  pp. 274–278  (In Russian)</TD></TR></table>

====Comments====
There is a more general  $  n $-
constants theorem, [[#References|[a2]]]: Let  $  f( z) $
be holomorphic in a domain  $  D $
whose boundary is the union of  $  n $
distinct rectifiable arcs  $  \alpha _ {1} \dots \alpha _ {n} $; 
suppose that for each  $  j $
there is a constant  $  M _ {j} $
such that if  $  z $
approaches any point of  $  \alpha _ {j} $, 
then the limits of  $  f ( z) $
do not exceed  $  M _ {j} $
in absolute value. Then for each  $  z \in D $,

$$ 
 \mathop{\rm log}  | f( z) |  \leq   \sum _ { j= } 1 ^ { m } 
\omega ( z, \alpha _ {j} ;  D)   \mathop{\rm log}  M _ {j} .
$$

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''2''' , Chelsea  (1977)  pp. 210–214  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hille,   "Analytic function theory" , '''2''' , Chelsea, reprint  (1987)  pp. 409–410</TD></TR></table>