Then we can find a chart from a neighborhood of p _ 0 to the unit disk \ mathbb { D } such that f ( \ phi ^ {-1 } ( z ) ) is holomorphic on the unit disk and has a maximum at \ phi ( p _ 0 ) \ in \ mathbb { D }, so it is constant, by the maximum modulus principle.
12.
If the growth rate of | g | is guaranteed to not be " too fast, " as specified by an appropriate growth condition, the " Phragm�n Lindel�f principle " can be applied to show that boundedness of | g | on the region's boundary implies that | g | is in fact bounded in the whole region, effectively extending the maximum modulus principle to unbounded regions.
13.
This is known as the " maximum modulus principle . " ( In fact, since \ overline { \ Omega } is compact and | f | is continuous, there actually exists some w _ 0 \ in \ partial \ Omega such that | f ( w _ 0 ) | = \ sup _ { z \ in \ partial \ Omega } | f ( z ) | . ) The maximum modulus principle is generally used to conclude that a holomorphic function is bounded in a region after showing that it is bounded on the boundary of that region.
14.
This is known as the " maximum modulus principle . " ( In fact, since \ overline { \ Omega } is compact and | f | is continuous, there actually exists some w _ 0 \ in \ partial \ Omega such that | f ( w _ 0 ) | = \ sup _ { z \ in \ partial \ Omega } | f ( z ) | . ) The maximum modulus principle is generally used to conclude that a holomorphic function is bounded in a region after showing that it is bounded on the boundary of that region.
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