Weighted means and an analytic characterization of discs

Weighted means are obtained for solutions of the two-dimensional Helmholtz and modified Helmholtz equations and also for harmonic functions. The presence of a logarithmic weight diminishes the coefficient in the last two mean value identities. A new theorem characterizing analytically discs in the Euclidean plane $\mathbb{R}^2$ is proved. The weighted mean value property of solutions to the modified Helmholtz equation is used for this purpose.