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