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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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* A function is [[harmonic function|harmonic]] [[if and only if]] it is both subharmonic and superharmonic.
* If <math>\phi</math> is ''C''<sup>2</sup> ([[smooth function|twice continuously differentiable]]) on an [[open set]] <math>G</math> in <math>\R^n</math>, then <math>\phi</math> is subharmonic [[if and only if]] one has <math> \Delta \phi \geq 0</math> on <math>G</math>, where <math>\Delta</math> is the [[Laplacian]].
* The [[maxima and minima|maximum]] of a subharmonic function cannot be achieved in the [[interior (topology)|interior]] of its ___domain unless the function is constant,
* Subharmonic functions make a [[convex cone]], that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic.
*The [[pointwise maximum]] of two subharmonic functions is subharmonic. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic.
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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 display="block">\psi_\alpha(z) = \left| f(z) \right|^\alpha</math>
is subharmonic for every ''α'' > 0. This observation plays a role in the theory of [[Hardy spaces]], especially for the study of ''H{{i sup|p}}'' when 0 < ''p'' < 1.
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