# 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)

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