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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]].
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| 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.
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| ''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.
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| ==Formal definition==
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| 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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| :<math>\varphi \colon G \to {\mathbb{R}} \cup \{ - \infty \}</math>
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| 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(x) \leq h(x)</math> for all <math>x</math> on the [[boundary (topology)|boundary]] <math>\partial B(x,r)</math> of <math>B(x,r)</math> we have <math>\varphi(x) \leq h(x)</math> for all <math>x \in B(x,r).</math>
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| 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 <math>u</math> is called ''superharmonic'' if <math>-u</math> is subharmonic.
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| ==Properties==
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| * A function is [[harmonic function|harmonic]] [[if and only if]] it is both subharmonic and superharmonic.
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| * If <math>\phi\,</math> is ''C''<sup>2</sup> ([[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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| :<math> \Delta \phi \ge 0</math> on <math>G</math>
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| :where <math>\Delta</math> is the [[Laplacian]].
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| * 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, this is the so-called [[maximum principle]]. However, the [[minimum]] of a subharmonic function can be achieved in the interior of its domain.
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| * Subharmonic functions make a [[convex cone]], that is a linear combination of subharmonic functions
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| with positive coefficients is also subharmonic. | |
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| *Pointwise maximum of two subharmonic functions is subharmonic.
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| *The limit of a decreasing sequence of subharmonic functions is subharmonic (or identicaly equal to
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| <math>-\infty</math>).
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| ==Subharmonic functions in the complex plane==
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| Subharmonic functions are of a particular importance in [[complex analysis]], where they are intimately connected to [[holomorphic function]]s.
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| 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
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| :<math> \varphi(z) \leq \frac{1}{2\pi} \int_0^{2\pi} \varphi(z+ r \mathrm{e}^{i\theta}) \, d\theta. </math>
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| 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]].
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| If <math>f</math> is a holomorphic function, then
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| :<math>\varphi(z) = \log \left| f(z) \right|</math>
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| is a subharmonic function if we define the value of <math>\varphi(z)</math> at the zeros of <math>f</math> to be −∞. It follows that
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| :<math>\psi_\alpha(z) = \left| f(z) \right|^\alpha</math>
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| is subharmonic for every ''α'' > 0. This observation plays a role in the theory of [[Hardy spaces]], especially for the study of ''H<sup>p</sup>'' when 0 < ''p'' < 1.
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| 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}</math> that is constant in the imaginary direction is convex in the real direction and vice versa.
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| ===Harmonic majorants of subharmonic functions===
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| 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>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>
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| === Subharmonic functions in the unit disc. Radial maximal function ===
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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, 1). The ''radial maximal function'' for the function ''φ'' (restricted to the unit disc) is defined on the unit circle by
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| :<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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| If ''P''<sub>''r''</sup> denotes the [[Poisson kernel]], it follows from the subharmonicity that
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| :<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>
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| It can be shown that the last integral is less than the value at e<sup> i''θ''</sup> of the [[Hardy–Littlewood maximal function]] ''φ''<sup>∗</sup> of the restriction of ''φ'' to the unit circle '''T''',
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| :<math> \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} t}\right) \, \mathrm{d}t,</math>
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| so that 0 ≤ ''M'' ''φ'' ≤ ''φ''<sup>∗</sup>. It is known that the Hardy–Littlewood operator is bounded on [[Lp space|''L''<sup>''p''</sup>('''T''')]] when 1 < ''p'' < ∞.
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| It follows that for some universal constant ''C'',
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| ::<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>
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| If ''f'' is a function holomorphic in ''Ω'' and 0 < ''p'' < ∞, then the preceding inequality applies to ''φ'' = |''f''<sup> </sup>|<sup> ''p''/2</sup>. It can be deduced from these facts that any function ''F'' in the classical Hardy space ''H<sup>p</sup>'' satisfies
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| ::<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>
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| With more work, it can be shown that ''F'' has radial limits ''F''(e<sup> i''θ''</sup>) almost everywhere on the unit circle, and (by the [[dominated convergence theorem]]) that ''F<sub>r</sub>'', defined by ''F<sub>r</sub>''(e<sup> i''θ''</sup>) = ''F''(''r''<sup> </sup>e<sup> i''θ''</sup>) tends to ''F'' in ''L''<sup>''p''</sup>('''T''').
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| == Subharmonic functions on Riemannian manifolds ==
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| Subharmonic functions can be defined on an arbitrary [[Riemannian manifold]].
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| ''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<sub>1</sub>'' 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''.
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| 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 operator|Laplacian]].<ref>{{Cite journal
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| | author = Greene, R. E.
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| | year = 1974
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| | title = Integrals of subharmonic functions on manifolds of nonnegative curvature
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| | journal = Inventiones Mathematicae
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| | volume = 27
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| | pages = 265–298
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| | doi = 10.1007/BF01425500
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| | last2 = Wu
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| | first2 = H.
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| | postscript = <!--None-->
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| | issue = 4
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| }}, {{MathSciNet | id = 0382723}}</ref>
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| ==See also==
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| * [[Plurisubharmonic function]] — generalization to [[several complex variables]]
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| * [[Classical fine topology]]
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| == Notes ==
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| <references />
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| == References ==
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| * {{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 }}
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| * {{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 }}
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| *{{cite book
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| | last = Doob
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| | first = Joseph Leo
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| | authorlink = Joseph Leo Doob
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| | title = Classical Potential Theory and Its Probabilistic Counterpart
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| | publisher = [[Springer-Verlag]]
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| | location = Berlin Heidelberg New York
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| | year = 1984
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| | isbn = 3-540-41206-9 }}
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| *{{cite book
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| |last1 = Rosenblum
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| |first1 = Marvin
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| |last2 = Rovnyak
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| |first2 = James
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| |title = Topics in Hardy classes and univalent functions
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| |series = Birkhauser Advanced Texts: Basel Textbooks
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| |publisher = Birkhauser Verlag
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| |address = Basel
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| |year = 1994
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| }}
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| {{PlanetMath attribution|id=5796|title=Subharmonic and superharmonic functions}}
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| [[Category:Subharmonic functions]]
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| [[Category:Potential theory]]
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| [[Category:Complex analysis]]
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| [[Category:Types of functions]]
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