Subharmonic function: Difference between revisions

Revision as of 08:17, 13 April 2012

Revision as of 03:37, 14 April 2012

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==Formal definition==

Formally, the definition can be stated as follows. Let <math>G</math> be a subset of the [[Euclidean space]] <math>{\mathbb{R}}^n</math> and let

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==Properties==

* A function is [[harmonic function|harmonic]] [[if and only if]] it is both subharmonic and superharmonic.

* If <math>\phi\,</math> is ''C''² ([[smooth function|twice continuously differentiable]]) on an [[open set]] <math>G</math> in <math>{\mathbb{R}}^n</math>, then <math>\phi\,</math> is subharmonic [[if and only if]] one has

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==Subharmonic functions in the complex plane==

Subharmonic functions are of a particular importance in [[complex analysis]], where they are intimately connected to [[holomorphic function]]s.

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=== Subharmonic functions in the unit disc. Radial maximal function ===

Let ''φ'' be subharmonic, continuous and non-negative in an open subset ''Ω'' of the complex plane containing the closed unit disc ''D''(0,&nbsp;1). The ''radial maximal function'' for the function ''φ'' (restricted to the unit disc) is defined on the unit circle by

:<math> (M \varphi)(\mathrm{e}^{\mathrm{i} \theta}) = \sup_{0 \le r < 1} \varphi(r \mathrm{e}^{\mathrm{i} \theta}). </math>

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== Subharmonic functions on Riemannian manifolds ==

Subharmonic functions can be defined on an arbitrary [[Riemannian manifold]].

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|year = 1994

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{{planetmathPlanetMath attribution|id=5796|title=Subharmonic and superharmonic functions}}

[[Category:Subharmonic functions]]