Positive harmonic function: Difference between revisions

Revision as of 09:51, 23 October 2022

Revision as of 15:03, 8 August 2025

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Then

:<math> f_n(re^{i\theta}) = {1\over 2\pi}\int_0^{2\pi} {1-r^2\over 1-2r\cos(\theta-\varphi) + r^2 }\, f_n(\varphi)\,d\phivarphi =\int_0^{2\pi}{1-r^2\over 1-2r\cos(\theta-\varphi) + r^2 } d\mu_n(\varphi)</math>

where

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Let

:<math> f(z)=1 + a_1b_1 z + a_2b_2 z^2 + \cdots</math>

be a holomorphic function on the unit disk. Then ''f''(''z'') has positive real part on the disk