Characterizations of discs via weighted means

New theorems characterizing analytically discs in the Euclidean plane ${\text{RR}}^2$ are proved. Weighted mean value properties of solutions to the modified Helmholtz equation and harmonic functions are used for this purpose. The presence of a logarithmic weight diminish coefficients in the mean value identities. A weighted mean is also valid for solutions of the Helmholtz equation.