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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]].

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 display="block">\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>

Note that by the above, the function which is identically &minus;∞−∞ is subharmonic, but some authors exclude this function by definition.

:<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>

Note that by the above, the function which is identically &minus;∞ is subharmonic, but some authors exclude this function by definition.

A function <math>u</math> is called ''superharmonic'' if <math>-u</math> is subharmonic.

==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 <math> \Delta \phi \gegeq 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, thiswhich is the so-called the [[maximum principle]]. However, the [[minimum]] of a subharmonic function can be achieved in the interior of its ___domain.

* 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.

*The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to <math>-\infty</math>).

*Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the [[fine topology (potential theory)|fine topology]] which makes them continuous.

==Examples==

If <math>f</math> is [[analytic function|analytic]] then <math>\log|f|</math> is subharmonic. More examples can be constructed by using the properties listed above,

by taking maxima, convex combinations and limits. In dimension 1, all subharmonic functions can be obtained in this way.

==Riesz Representation Theorem==

If <math>u</math> is subharmonic in a region <math>D</math>, in [[Euclidean space]] of dimension <math>n</math>, <math>v</math> is harmonic in <math>D</math>, and <math>u \leq v</math>, then <math>v</math> is called a harmonic majorant of <math>u</math>. If a harmonic majorant exists, then there exists the least harmonic majorant, and

<math display="block">u(x) = v(x) - \int_D\frac{d\mu(y)}{|x-y|^{n-2}},\quad n\geq 3</math>

while in dimension 2,

<math display="block">u(x) = v(x) + \int_D\log|x-y|d\mu(y),</math>

where <math>v</math> is the least harmonic majorant, and <math>\mu</math> is a [[Borel measure]] in <math>D</math>.

This is called the [[Friedrich Riesz|Riesz]] representation theorem.

==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}Complex</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 display="block"> \varphi(z) \leq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r \mathrm{e}re^{i\theta}) \, d\theta. </math>

:<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

:<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 &minus;∞<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</sup>}}'' when 0&nbsp;< ''p''&nbsp;<&nbsp;1.

In the context of the complex plane, the connection to the [[convex function]]s can be realized as well by the fact that a subharmonic function <math>f</math> on a ___domain <math>G \subset \mathbb{C}Complex</math> that is constant in the imaginary direction is convex in the real direction and vice versa.

===Harmonic majorants of subharmonic functions===

If <math>u</math> is subharmonic in a [[Region (mathematical analysis)|region]] <math>\Omega</math> of the complex plane, and <math>h</math> is [[Harmonic function|harmonic]] on <math>\Omega</math>, then <math>h</math> is a '''harmonic majorant''' of <math>u</math> in <math>\Omega</math> if <math>u</math>≤<math> \leq h</math> in <math>\Omega</math>. Such an inequality can be viewed as a growth condition on <math>u</math>.<ref>Rosenblum, Marvin; Rovnyak, James (1994), p.35 (see References)</ref>

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

If ''P''_''r'' denotes the [[Poisson kernel]], it follows from the subharmonicity that

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

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

:<math display="block"> \varphi^*(\mathrm{e}^{\mathrm{i} \theta}) = \sup_{0 < \alpha \le \pi} \frac{1}{2 \alpha} \int_{\theta - \alpha}^{\theta + \alpha} \varphi\left(\mathrm{e}^{\mathrm{i} tit}\right) \, \mathrm{d}tdt,</math>

so that 0&nbsp;≤ ''M''&nbsp;''φ''&nbsp;≤ ''φ''^∗. It is known that the Hardy–Littlewood operator is bounded on [[Lp space|''L''^''p''('''T''')]] when 1&nbsp;< ''p''&nbsp;< ∞.

It follows that for some universal constant ''C'',

::<math display="block"> \|M \varphi\|_{L^2(\mathbf{T})}^2 \le C^2 \, \int_0^{2\pi} \varphi(\mathrm{e}^{\mathrm{i} \theta})^2 \, \mathrm{d}\theta.</math>

If ''f'' is a function holomorphic in ''Ω'' and 0&nbsp;< ''p''&nbsp;< ∞, then the preceding inequality applies to ''φ''&nbsp;= |''f''<sup>&nbsp;</{{hair space}}|{{i sup>|^{&nbsp;''p''/2}}}. It can be deduced from these facts that any function ''F'' in the classical Hardy space ''H^p'' satisfies

::<math> \|M \varphi\|_{L^2(\mathbf{T})}^2 \le C^2 \, \int_0^{2\pi} \varphi(\mathrm{e}^{\mathrm{i} \theta})^2 \, \mathrm{d}\theta.</math>

::<math display="block"> \int_0^{2\pi} \Biglleft( \sup_{0 \le r < 1} \left|F(r \mathrm{e}^{\mathrm{i} \theta})\right| \Bigrright)^p \, \mathrm{d}\theta \le C^2 \, \sup_{0 \leleq r < 1} \int_0^{2\pi} \left|F(r \mathrm{e}re^{\mathrm{i} \theta})\right|^p \, \mathrm{d}\theta.</math>

With more work, it can be shown that ''F'' has radial limits ''F''(''e<''{{i sup>&nbsp;i|''θiθ''</sup>}}) almost everywhere on the unit circle, and (by the [[dominated convergence theorem]]) that ''F_r'', defined by ''F_r''(''e^{&nbsp;''{{i sup|''θiθ''}}})&nbsp;= ''F''(''r''^&nbsp;{{hair space}}''e<''{{i sup>&nbsp;i|''θiθ''</sup>}}) tends to ''F'' in ''L''^''p''('''T''').

If ''f'' is a function holomorphic in ''Ω'' and 0&nbsp;< ''p''&nbsp;< ∞, then the preceding inequality applies to ''φ''&nbsp;= |''f''^&nbsp;|^{&nbsp;''p''/2}. It can be deduced from these facts that any function ''F'' in the classical Hardy space ''H^p'' satisfies

::<math> \int_0^{2\pi} \Bigl( \sup_{0 \le r < 1} |F(r \mathrm{e}^{\mathrm{i} \theta})| \Bigr)^p \, \mathrm{d}\theta \le C^2 \, \sup_{0 \le r < 1} \int_0^{2\pi} |F(r \mathrm{e}^{\mathrm{i} \theta})|^p \, \mathrm{d}\theta.</math>

With more work, it can be shown that ''F'' has radial limits ''F''(e^{&nbsp;i''θ''}) almost everywhere on the unit circle, and (by the [[dominated convergence theorem]]) that ''F_r'', defined by ''F_r''(e^{&nbsp;i''θ''})&nbsp;= ''F''(''r''^&nbsp;e^{&nbsp;i''θ''}) tends to ''F'' in ''L''^''p''('''T''').

== Subharmonic functions on Riemannian manifolds ==

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

''Definition:'' Let ''M'' be a Riemannian manifold, and <math>f:\; M \to {\Bbb R}</math> an [[upper semicontinuous]] function. Assume that for any open subset <math>U\subset M</math>, and any [[harmonic function]] ''f''₁'' on ''U'', such that <math>f_1 \geq f</math> on the boundary of ''U'', the inequality <math>f_1 \geq f</math> holds on all ''U''. Then ''f'' is called ''subharmonic''.

This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality <math>\Delta f \geq 0</math>, where <math>\Delta</math> is the usual [[Laplace–Beltrami_operator|Laplacian]].<ref>{{Cite journal | author = Greene, R. E. | year = 1974 | title = Integrals of subharmonic functions on manifolds of nonnegative curvature | journal = Inventiones Mathematicae | volume = 27 | pages = 265–298 | doi = 10.1007/BF01425500 | last2 = Wu | first2 = H. | issue = 4| bibcode = 1974InMat..27..265G | s2cid = 122233796 }}, {{MathSciNet | id = 0382723}}</ref>

| author = Greene, R. E.

| year = 1974

| title = Integrals of subharmonic functions on manifolds of nonnegative curvature

| journal = Inventiones Mathematicae

| volume = 27

| pages = 265–298

| doi = 10.1007/BF01425500

| last2 = Wu

| first2 = H.

| postscript = <!--None-->

| issue = 4

}}, {{MathSciNet | id = 0382723}}</ref>

==See also==

* [[Plurisubharmonic function]] &mdash;— generalization to [[several complex variables]]

* [[Classical fine topology]]

Line 89 ⟶ 88:

* {{Cite book | first=John B. | last=Conway | authorlink=John B. Conway | title=Functions of one complex variable | publisher=Springer-Verlag | ___location=New York | year=1978 | isbn=0-387-90328-3 }}

* {{Cite book | first=Steven G. | last=Krantz | title=Function Theory of Several Complex Variables | publisher=AMS Chelsea Publishing | ___location=Providence, Rhode Island | year=1992 | isbn=0-8218-2724-3 }}

*{{cite book | last = Doob | first = Joseph Leo | authorlink = Joseph Leo Doob | title = Classical Potential Theory and Its Probabilistic Counterpart | url = https://archive.org/details/classicalpotenti0000doob | url-access = registration | publisher = [[Springer-Verlag]] | ___location = Berlin Heidelberg New York | year = 1984 | isbn = 3-540-41206-9 }}

*{{cite book

*{{cite book |last1 = Rosenblum |first1 = Marvin |last2 = Rovnyak |first2 = James |title = Topics in Hardy classes and univalent functions |series = Birkhauser Advanced Texts: Basel Textbooks |publisher = Birkhauser Verlag |___location = Basel |year = 1994}}

| last = Doob

| first = Joseph Leo

| authorlink = Joseph Leo Doob

| title = Classical Potential Theory and Its Probabilistic Counterpart

| publisher = [[Springer-Verlag]]

| ___location = Berlin Heidelberg New York

| year = 1984

| isbn = 3-540-41206-9 }}

*{{cite book

|last1 = Rosenblum

|first1 = Marvin

|last2 = Rovnyak

|first2 = James

|title = Topics in Hardy classes and univalent functions

|series = Birkhauser Advanced Texts: Basel Textbooks

|publisher = Birkhauser Verlag

|___location = Basel

|year = 1994

}}

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