Subharmonic function: Difference between revisions

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{{Short description|Class of mathematical functions}}{{Moreinline|date=June 2025}}

In [[mathematics]], '''subharmonic''' and '''superharmonic''' functions are important classes of [[function (mathematics)|functions]] used extensively in [[partial differential equations]], [[complex analysis]] and [[potential theory]].

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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 −∞<math>-\infty</math>. 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}}'' when 0&nbsp;< ''p''&nbsp;<&nbsp;1.