Subharmonic function: Difference between revisions

Revision as of 18:35, 14 July 2021

Revision as of 18:37, 14 July 2021

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If <math>f</math> is a holomorphic function, then

<math display="block">\varphi(z) = \log \left| f(z) \right|</math>

is a subharmonic function if we define the value of <math>\varphi(z)</math> at the zeros of <math>f</math> to be &minus;∞−∞. It follows that

<math display="block">\psi_\alpha(z) = \left| f(z) \right|^\alpha</math>

is subharmonic for every ''α''&nbsp;> 0. This observation plays a role in the theory of [[Hardy spaces]], especially for the study of ''H<{{i sup>|p</sup>}}'' when 0&nbsp;< ''p''&nbsp;<&nbsp;1.

In the context of the complex plane, the connection to the [[convex function]]s can be realized as well by the fact that a subharmonic function <math>f</math> on a ___domain <math>G \subset \Complex</math> that is constant in the imaginary direction is convex in the real direction and vice versa.