= Equality in the Cauchy coefficient estimate
For an <entire function> $f(z)=\sum_jc_jz^j$, equality $|c_n|=\max_{|z|=r}|f(z)|/r^n$ at a fixed $r>0$ holds exactly when $f(z)=cz^n$. The coefficient is the average of $r^{-n}f(re^{i\theta})e^{-in\theta}$. Equality in its modulus bound forces the continuous integrand to have constant modulus and argument, hence to be constant. The <identity theorem> then extends the monomial identity from the circle to the whole plane. The zero function and the constant case $n=0$ are included.
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