# Two-constants theorem

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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 , .

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=12855
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article