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New York Journal of Mathematics
Volume 28 (2022), 1230-1255

  

Haley K. Bambico, Mehmet Celik, Sarah T. Gross, and Francis Hall

Generalization of the excess area and its geometric interpretation

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Published: August 26, 2022.
Keywords: Holomorphic functions, Blaschke product, Harmonic functions, Area of an Image.
Subject [2010]: Primary 30H05; Secondary 30J10.

Abstract
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.

Acknowledgements

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


Author information

Haley K. Bambico:
Texas A & M University-Commerce
Department of Mathematics
Commerce, TX 75429, USA

hbambico@leomail.tamuc.edu

Mehmet Celik:
Texas A & M University-Commerce
Department of Mathematics
Commerce, TX 75429, USA

mehmet.celik@tamuc.edu

Sarah T. Gross:
Texas A & M University-Commerce
Department of Mathematics
Commerce, TX 75429, USA

sgross2@leomail.tamuc.edu

Francis Hall:
Texas A & M University-Commerce
Department of Mathematics
Commerce, TX 75429, USA

fhall2@leomail.tamuc.edu