However, the maximum modulus principle cannot be applied to an unbounded region of the complex plane.
2.
The maximum modulus principle can therefore be applied to " F " " n " in the strip.
3.
There is no need for the Borel Carath�odory theorem, but you do need to consider the directions and the Maximum modulus principle.
4.
Alternatively, the maximum modulus principle can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets.
5.
Reshetnyak's theorem implies that all pure topological results about analytic functions ( such that the Maximum Modulus Principle, Rouch?s theorem etc . ) extend to quasiregular maps.
6.
He is known for the Picard Lindel�f theorem on differential equations and the Phragm�n Lindel�f principle, one of several refinements of the maximum modulus principle that he proved in complex function theory.
7.
Because S _ { x _ 0 } is a bounded region, the maximum modulus principle is applicable and implies that | gh _ \ epsilon | \ leq 1 for all z \ in S _ { x _ 0 }.
8.
:I'll close my eyes and point to Maximum modulus principle and Borel Carath�odory theorem ( and then to sci . math when you find I wasn't helpful . ) iames 19 : 22, 13 June 2007 ( UTC)
9.
Since the complex semigroup has as Shilov boundary the symplectic group, the fact that this representation has a well-defined contractive extension to the semigroup follows from the maximum modulus principle and the fact that the semigroup operators are closed under adjoints.
10.
In a typical Phragm�n Lindel�f argument, we introduce a multiplicative factor h _ \ epsilon to " subdue " the growth of g, such that | gh _ \ epsilon | is bounded on the boundary of a bounded subregion of S and we can apply the maximum modulus principle to gh _ \ epsilon.
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