Least harmonic majorant by expanding disk lifts ($h(z)={\sup }_{|z|<r<1}{H}_r(z)$)

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 $H_s\ge H_r\ge u$ 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.