The image area of the unit disk under the function zh(z) exceeds the image area under the holomorphic function h(z). In his book, Hermitian Analysis, J. D'Angelo precisely determines how this excess image area of the unit disk grows. In our work, we replace the multiplier z with a finite Blaschke product and observe that the excess area growth is a solution for the Dirichlet problem on the unit disk. We replace holomorphic functions with harmonic ones in the formulation and observe a new identity. Furthermore, we show that the excess area growth idea can also be implemented to some other domains conformal to the unit disk.

The MAA's grant number 3-8-710-890 (NSF grant number DMS-1652506) partly supported the research.