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On the restricted mean value property. (English) Zbl 0634.31002
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.

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
Full Text: DOI
[1] P. C. Fenton, Functions having the restricted mean value property, J. London Math. Soc. (2) 14 (1976), no. 3, 451 – 458. · Zbl 0344.31002 · doi:10.1112/jlms/s2-14.3.451 · doi.org
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