= Least harmonic majorant by expanding disk lifts
{title2=$h(z)=\sup_{|z|<r<1}H_r(z)$}
For continuous <subharmonic function> $u$ on the <unit disk>, let $H_r$ be the <Poisson integral> of its values on $|z|=r$, evaluated inside that disk. Harmonic comparison gives $H_s\geq H_r\geq u$ for $|z|<r<s<1$. The <Harnack inequality for harmonic functions> makes their limit either finite and harmonic everywhere or everywhere infinite. In the finite case every <harmonic majorant> dominates every $H_r$, so their limit is the least harmonic majorant. The supremum must restrict $r>|z|$; the Poisson expression evaluated outside its circle does not have this interpretation.
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