# 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 $\mathop{\rm Re} \phi ( z) = \omega ( z ; \alpha , D )$, .

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 . 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 , .

How to Cite This Entry:
Two-constants theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-constants_theorem&oldid=49049
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article