If 
and 
are bounded domains, 
, 
are Jordan curves,
and 
is a conformal mapping, then 
(respectively, 
) extends one-to-one and continuously to 
(respectively, 
).
Carathéodory's Theorem
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References
Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 152, 1999.Referenced on Wolfram|Alpha
Carathéodory's TheoremCite this as:
Weisstein, Eric W. "Carathéodory's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CaratheodorysTheorem.html