Two-constants theorem
Let be a finitely-connected Jordan domain in the
-plane and let
be a regular analytic function in
satisfying the inequality
, as well as the relation
![]() |
on some arc of the boundary
. Then, at each point
of the set
![]() |
where is the harmonic measure of the arc
with respect to
at
, the inequality
![]() |
is satisfied. If for some (satisfying the condition
) equality is attained, equality will hold for all
and for all
,
, while the function
in this case has the form
![]() |
where is a real number and
is an analytic function in
for which
[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 -constants theorem, [a2]: Let
be holomorphic in a domain
whose boundary is the union of
distinct rectifiable arcs
; suppose that for each
there is a constant
such that if
approaches any point of
, then the limits of
do not exceed
in absolute value. Then for each
,
![]() |
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 |
Two-constants theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-constants_theorem&oldid=49049