On sufficient conditions for harmonicity

Abstract

Suppose that u is continuous in the plane and that given any complex number z there is a number ρ = ρ ( z ) > 0 \rho = \rho (z) > 0 such that u ( z ) = 1 2 π ∫ 0 2 π u ( z + ρ e i θ ) d θ \begin{equation} u(z) = \frac {1} {{2\pi }}\int _0^{2\pi } {u(z + \rho {e^{i\theta }})} d\theta \end{equation} The main result is: if u possesses a harmonic majorant and ρ ( z ) \rho (z) is continuous and satisfies a further condition (which may not be omitted) then u is harmonic. Another result in the same vein is proved.