Subharmonic function: Difference between revisions

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

Intuitively, subharmonic functions are related to [[convex function]]s of one variable as follows. If the [[graph of a function|graph]] of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a [[harmonic function]] on the ''boundary'' of a [[ball (mathematics)|ball]], then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball.

''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the [[additive inverse|negative]] of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions.

==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

:<math>\varphi \colon G \to {\mathbb{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>

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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.

One can show that a real-valued, continuous function <math>\varphi</math> of a complex variable (that is, of two real variables) defined on a set <math>G\subset \mathbb{C}</math> is subharmonic if and only if for any closed disc <math>D(z,r) \subset G</math> of center <math>z</math> and radius <math>r</math> one has

:<math> \varphi(z) \leq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r \mathrm{e}^{i\theta}) \, d\theta. </math>

Intuitively, this means that a subharmonic function is at any point no greater than the [[arithmetic mean|average]] of the values in a circle around that point, a fact which can be used to derive the [[maximum principle]].

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

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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>

If ''P''<sub>''r''</supsub> denotes the [[Poisson kernel]], it follows from the subharmonicity that

:<math> 0 \le \varphi(r \mathrm{e}^{\mathrm{i} \theta}) \le \frac{1}{2\pi} \int_0^{2\pi} P_r\left(\theta- t\right) \varphi\left(\mathrm{e}^{\mathrm{i} t}\right) \, \mathrm{d} t, \ \ \ r < 1.</math>

It can be shown that the last integral is less than the value at e^{&nbsp;i''θ''} of the [[Hardy–Littlewood maximal function]] ''φ''^∗ of the restriction of ''φ'' to the unit circle '''T''',