Two-constants theorem

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 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 ) $[1], [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 [3]. Hadamard's three-circles theorem (cf. Hadamard theorem) is obtained from it as a special case. For possible analogues of the two-constants theorem for harmonic functions in space see [4], [5].

References

[1]	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)
[2]	A. Ostrowski, "Über allgemeine Konvergenzsätze der komplexen Funktionentheorie" Jahresber. Deutsch. Math.-Ver. , 32 : 9–12 (1923) pp. 185–194
[3]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[4]	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)
[5]	E.D. Solomentsev, "Three-spheres theorem for harmonic functions" Dokl. Akad. Nauk ArmSSR , 42 : 5 (1966) pp. 274–278 (In Russian)

Comments

There is a more general $ n $- constants theorem, [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

[a1]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. 210–214 (Translated from Russian)
[a2]	E. Hille, "Analytic function theory" , 2 , Chelsea, reprint (1987) pp. 409–410