Just skimming it, it looks like (a) is just the statement of the mean value property for the harmonic functions Re(f(z)) and Im(f(z)). I think that's proven in the "integral over a bounding circle" form by citing Cauchy's Theorem that the counter clockwise integral of an holomorphic function (throughout the interior too) around a closed curve of f(z) / ( f(z)-f(a) ) = 2 pi i f(a). Then put that "mean value property:boundary circle" into "mean value property:disk" form by applying Stokes theorem. Something like that - I think that's how it can be done. Maybe I'll try to work it out later. Good luck.