# maximum-modulus theorem

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## maximum-modulus theorem

[′mak·sə·məm ′mäj·ə·ləs ‚thir·əm]
(mathematics)
For a complex analytic function in a closed bounded simply connected region its modulus assumes its maximum value on the boundary of the region.
References in periodicals archive ?
Hence, by the maximum modulus principle, 1/[f.sub.n] [right arrow] 1/[f.sub.0] on whole D.
Thus by [f.sub.n] [right arrow] 0 on D \ E and the maximum modulus principle, [f.sub.n] [right arrow] 0 uniformly on F.
Furthermore, an analog of the maximum modulus principle holds (see ).
As in the complex case, the quaternionic Phragmen-Lindelof principle generalizes the maximum modulus principle 2.6 to unbounded domains.
As a consequence of points 1-4, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.0] [member of] [[partial derivative].sub.[infinity]][OMEGA]' and, by an easy application of the maximum modulus principle 2.6, [absolute value of [[omega].sup.[delta].sub.r] f] [less than or equal to] max {M, [M.sub.r], N} in [OMEGA]'.
As before, the maximum modulus principle 2.6 yields that [[omega].sup.[delta].sub.r] f must be constant.
of Liverpool) examine necessary and sufficient conditions for validity of the classical maximum modulus principles of strongly elliptical systems of the second order and systems of parabolic equations.
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