On the Restricted Mean Value Property
P. C. Fenton · Proceedings of the American Mathematical Society · 1987
Suppose that $u$ is continuous in the open unit disc and has the restricted mean value property. It is shown that if $u$ has finite boundary limits almost everywhere, and if $u$ possesses a harmonic majorant and minorant, the difference between which has finite radial upper limits everywhere, then $u$ is harmonic.