Positive harmonic function

In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a positive measure on the circle. This result, the Herglotz representation theorem, was proved by Gustav Herglotz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.

References

• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN 0-387-90795-5 {{citation}}: Text "Duren, Peter L." ignored (help)
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht