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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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Formally, the definition can be stated as follows. Let <math>G</math> be a subset of the [[Euclidean space]] <math>\R^n</math> and let
<math display="block">\varphi \colon G \to \R \cup \{ - \infty \}</math>
be an [[semi-continuity|upper semi-continuous function]]. Then, <math>\varphi </math> is called ''subharmonic'' if for any [[closed ball]] <math>\overline{B(x,r)}</math> of center <math>x</math> and radius <math>r</math> contained in <math>G</math> and every [[real number|real]]-valued [[continuous function]] <math>h</math> on <math>\overline{B(x,r)}</math> that is [[harmonic function|harmonic]] in <math>B(x,r)</math> and satisfies <math>\varphi(y) \leq h(y)</math> for all <math>y</math> on the [[boundary (topology)|boundary]] <math>\partial B(x,r)</math> of <math>B(x,r)</math>, we have <math>\varphi(y) \leq h(y)</math> for all <math>y \in B(x,r).</math>
Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.
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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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