= Zero location bound from Schwarz-Pick
{title2=$|a|\geq|f(0)|/M$}
If a <holomorphic function> $f$ on the <unit disc> satisfies $|f|\leq M$ and $f(a)=0$, the <Schwarz-Pick lemma> applied to $f/M$ gives $|f(0)|/M\leq|a|$. Equality for a nonzero central value forces a scaled <automorphism of the unit disk>. If $f$ extends past the unit circle and $|f(0)|$ is smaller than its minimum boundary modulus, the <maximum modulus principle> applied to $1/f$ proves that a zero occurs inside the circle.
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