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Schwarz's Lemma

Let f be analytic on the unit disk, and assume that

1. |f(z)|<=1 for all z and

2. f(0)=0.

Then |f(z)|<=|z| and |f^'(0)|<=1.

If either |f(z)|=|z| for some z!=0 or if |f^'(0)|=1, then f is a rotation, i.e., f(z)=az for some complex constant a with |a|=1.

See also

Möbius Transformation, Schwarz-Pick Lemma

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References

Krantz, S. G. "Schwarz's Lemma." §5.5.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 78, 1999.

Referenced on Wolfram|Alpha

Schwarz's Lemma

Cite this as:

Weisstein, Eric W. "Schwarz's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchwarzsLemma.html

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