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New theorems characterizing analytically discs in the Euclidean plane $\RR^2$are proved. Weighted mean value properties of solutions to the modifiedHelmholtz equation and harmonic functions are used for this purpose. Thepresence of a logarithmic weight diminish coefficients in the mean valueidentities. A weighted mean is also valid for solutions of the Helmholtzequation.

Characterizations of discs via weighted means