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In complex analysis, a branch of mathematics, theHadamard three-circle theorem is a result about the behavior of holomorphic functions.
Let f {\displaystyle f} be a holomorphic function on the annulus
Let M {\displaystyle M} be the maximum of | f | {\displaystyle |f|} on the circle | z | = r . {\displaystyle |z|=r.} Then, log M {\displaystyle \log M} is a convex function of the logarithm log . {\displaystyle \log.} Moreover, if f {\displaystyle f} is not of the form c z λ {\displaystyle cz^{\lambda }} for some constants λ {\displaystyle \lambda } and c {\displaystyle c} , then log M {\displaystyle \log M} is strictly convex as a function of log . {\displaystyle \log.}
The conclusion of the theorem can be restated as
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