Maximum-modulus principle

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A theorem expressing one of the basic properties of the modulus of an analytic function. Let be a regular analytic, or holomorphic, function of complex variables , , defined on an (open) domain of the complex space , which is not a constant, . The local formulation of the maximum-modulus principle asserts that the modulus of does not have a local maximum at a point , that is, there is no neighbourhood of for which , . If in addition , then also cannot be a local minimum point of the modulus of . An equivalent global formulation of the maximum-modulus principle is that, under the same conditions as above, the modulus of does not attain its least upper bound

at any . Consequently, if is continuous in a finite closed domain , then can only be attained on the boundary of . These formulations of the maximum-modulus principle still hold when is a holomorphic function on a connected complex (analytic) manifold, in particular, on a Riemann surface or a Riemann domain (cf. Riemannian domain) .

The maximum-modulus principle has generalizations in several directions. First, instead of being holomorphic, it is sufficient to assume that is a (complex) harmonic function. Another generalization is connected with the fact that for a holomorphic function the modulus is a logarithmically-subharmonic function. If is a bounded holomorphic function in a finite domain and if

holds for all , except at some set of outer capacity zero (in ), then everywhere in . See also Two-constants theorem; Phragmén–Lindelöf theorem.

The maximum-modulus principle can also be generalized to holomorphic mappings. Let be a holomorphic mapping of an (open) domain , , into , that is, , , where , , are holomorphic functions on , and is the Euclidean norm. Then does not attain a local maximum at any . The maximum-modulus principle is valid whenever the principle of preservation of domain is satisfied (cf. Preservation of domain, principle of).


[1] S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian)
[2] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)


This principle is also called the maximum principle, cf. [a2].


[a1] R.B. Burchel, "An introduction to classical complex analysis" , 1 , Acad. Press (1979)
[a2] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241
How to Cite This Entry:
Maximum-modulus principle. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article