{"history":[],"article":{"id":"0074929a-28d4-4975-a0d3-fd0a71e3b478","title":"Subharmonic function","slug":"subharmonic-function","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **subharmonic** and **superharmonic** [functions](https://bliptext.com/articles/function-mathematics) are important classes of [functions](https://bliptext.com/articles/function-mathematics) used extensively in [partial differential equations](https://bliptext.com/articles/partial-differential-equations), [complex analysis](https://bliptext.com/articles/complex-analysis) and [potential theory](https://bliptext.com/articles/potential-theory).\nIntuitively, subharmonic functions are related to [convex functions](https://bliptext.com/articles/convex-function) of one variable as follows.\nIf the [graph](https://bliptext.com/articles/graph-of-a-function) of a convex function and a line intersect at two points, then the [graph](https://bliptext.com/articles/graph-of-a-function) of the convex function is below the line between those points.\nIn the same way, if the values of a sub[harmonic function](https://bliptext.com/articles/harmonic-function) are no larger than the values of a [harmonic function](https://bliptext.com/articles/harmonic-function) on the boundary of a [ball](https://bliptext.com/articles/ball-mathematics), then the values of the sub[harmonic function](https://bliptext.com/articles/harmonic-function) are no larger than the values of the [harmonic function](https://bliptext.com/articles/harmonic-function) also inside the [ball](https://bliptext.com/articles/ball-mathematics).\nSuperharmonic functions can be defined by the same description, only replacing \"no larger\" with \"no smaller\".\nAlternatively, a superharmonic function is just the [negative](https://bliptext.com/articles/additive-inverse) of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions.\n\n# Formal definition\nFormally, the definition can be stated as follows.\nLet G be a subset of the [Euclidean space](https://bliptext.com/articles/euclidean-space) \\R^n and let\nbe an [upper semi-continuous function](https://bliptext.com/articles/semi-continuity).\nThen, \\varphi is called sub[harmonic](https://bliptext.com/articles/harmonic-function) if for any [closed ball](https://bliptext.com/articles/closed-ball) of center x and radius r contained in G and every [real](https://bliptext.com/articles/real-number)-valued [continuous function](https://bliptext.com/articles/continuous-function) h on that is [harmonic](https://bliptext.com/articles/harmonic-function) in B(x,r) and satisfies for all y on the [boundary](https://bliptext.com/articles/boundary-topology) of B(x,r), we have for all\nNote that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.\nA function u is called superharmonic if -u is subharmonic.\n\n# Properties\n\n# Examples\nIf f is [analytic](https://bliptext.com/articles/analytic-function) then \\log|f| is subharmonic.\nMore examples can be constructed by using the properties listed above,\nby taking maxima, convex combinations and limits.\nIn dimension 1, all subharmonic functions can be obtained in this way.\n\n# Riesz Representation Theorem\nIf u is subharmonic in a region D, in [Euclidean space](https://bliptext.com/articles/euclidean-space) of dimension n, v is harmonic in D, and u \\leq v, then v is called a harmonic majorant of u. If a harmonic majorant exists, then there exists the least harmonic majorant, and\nwhile in dimension 2,\nwhere v is the least harmonic majorant, and \\mu is a [Borel measure](https://bliptext.com/articles/borel-measure) in D. This is called the [Riesz](https://bliptext.com/articles/friedrich-riesz) representation theorem.\n\n# Subharmonic functions in the complex plane\nSubharmonic functions are of a particular importance in [complex analysis](https://bliptext.com/articles/complex-analysis), where they are intimately connected to [holomorphic functions](https://bliptext.com/articles/holomorphic-function).\nOne can show that a real-valued, continuous function \\varphi of a complex variable (that is, of two real variables) defined on a set is subharmonic if and only if for any closed disc of center z and radius r one has\nIntuitively, this means that a subharmonic function is at any point no greater than the [average](https://bliptext.com/articles/arithmetic-mean) of the values in a circle around that point, a fact which can be used to derive the [maximum principle](https://bliptext.com/articles/maximum-principle).\nIf f is a holomorphic function, then\nis a subharmonic function if we define the value of \\varphi(z) at the zeros of f to be -\\infty.\nIt follows that\nis subharmonic for every α > 0.\nThis observation plays a role in the theory of [Hardy spaces](https://bliptext.com/articles/hardy-spaces), especially for the study of H when 0 < p < 1.\nIn the context of the complex plane, the connection to the [convex functions](https://bliptext.com/articles/convex-function) can be realized as well by the fact that a subharmonic function f on a domain that is constant in the imaginary direction is convex in the real direction and vice versa.\n\n## Harmonic majorants of subharmonic functions\nIf u is sub[harmonic](https://bliptext.com/articles/harmonic-function) in a [region](https://bliptext.com/articles/region-mathematical-analysis) \\Omega of the complex plane, and h is [harmonic](https://bliptext.com/articles/harmonic-function) on \\Omega, then h is a **[harmonic](https://bliptext.com/articles/harmonic-function) majorant** of u in \\Omega if u \\leq h in \\Omega.\nSuch an inequality can be viewed as a growth condition on u.\n\n## Subharmonic functions in the unit disc. Radial maximal function\nLet φ be subharmonic, continuous and non-negative in an open subset Ω of the complex plane containing the closed unit disc D(0, 1).\nThe radial maximal function for the function φ (restricted to the unit disc) is defined on the unit circle by\nIf Pr denotes the [Poisson kernel](https://bliptext.com/articles/poisson-kernel), it follows from the subharmonicity that\nIt can be shown that the last integral is less than the value at e of the [Hardy–Littlewood maximal function](https://bliptext.com/articles/hardy-littlewood-maximal-function) φ∗ of the restriction of φ to the unit circle **T**,\nso that 0 ≤ M φ ≤ φ∗.\nIt is known that the Hardy–Littlewood operator is bounded on Lp(**T**) when 1 < p < ∞.\nIt follows that for some universal constant C,\nIf f is a function holomorphic in Ω and 0 < p < ∞, then the preceding inequality applies to φ = |f |.\nIt can be deduced from these facts that any function F in the classical Hardy space Hp satisfies\nWith more work, it can be shown that F has radial limits F(e) almost everywhere on the unit circle, and (by the [dominated convergence theorem](https://bliptext.com/articles/dominated-convergence-theorem)) that Fr, defined by Fr(e) = F(r e) tends to F in Lp(**T**).\n\n# Subharmonic functions on Riemannian manifolds\nSubharmonic functions can be defined on an arbitrary [Riemannian manifold](https://bliptext.com/articles/riemannian-manifold).\nDefinition: Let M be a Riemannian manifold, and an [upper semicontinuous](https://bliptext.com/articles/upper-semicontinuous) function.\nAssume that for any open subset U\\subset M, and any [harmonic function](https://bliptext.com/articles/harmonic-function) f1 on U, such that f_1 \\geq f on the boundary of U, the inequality f_1 \\geq f holds on all U.\nThen f is called subharmonic.\nThis definition is equivalent to one given above.\nAlso, for twice differentiable functions, subharmonicity is equivalent to the inequality, where \\Delta is the usual [Laplacian](https://bliptext.com/articles/laplace-beltrami-operator).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T06:53:28.953Z","updated_at":"2025-01-26T06:53:28.953Z","search_vector":"'function':2 'subharmon':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **subharmonic** and **superharmonic** [functions](https://bliptext.com/articles/function-mathematics) are important classes of [functions](https://bliptext.com/articles/function-mathematics) used extensively in [partial differential equations](https://bliptext.com/articles/partial-differential-equations), [complex analysis](https://bliptext.com/articles/complex-analysis) and [potential theory](https://bliptext.com/articles/potential-theory).\nIntuitively, subharmonic functions are related to [convex functions](https://bliptext.com/articles/convex-function) of one variable as follows.\nIf the [graph](https://bliptext.com/articles/graph-of-a-function) of a convex function and a line intersect at two points, then the [graph](https://bliptext.com/articles/graph-of-a-function) of the convex function is below the line between those points.\nIn the same way, if the values of a sub[harmonic function](https://bliptext.com/articles/harmonic-function) are no larger than the values of a [harmonic function](https://bliptext.com/articles/harmonic-function) on the boundary of a [ball](https://bliptext.com/articles/ball-mathematics), then the values of the sub[harmonic function](https://bliptext.com/articles/harmonic-function) are no larger than the values of the [harmonic function](https://bliptext.com/articles/harmonic-function) also inside the [ball](https://bliptext.com/articles/ball-mathematics).\nSuperharmonic functions can be defined by the same description, only replacing \"no larger\" with \"no smaller\".\nAlternatively, a superharmonic function is just the [negative](https://bliptext.com/articles/additive-inverse) of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions.\n\n# Formal definition\nFormally, the definition can be stated as follows.\nLet G be a subset of the [Euclidean space](https://bliptext.com/articles/euclidean-space) \\R^n and let\nbe an [upper semi-continuous function](https://bliptext.com/articles/semi-continuity).\nThen, \\varphi is called sub[harmonic](https://bliptext.com/articles/harmonic-function) if for any [closed ball](https://bliptext.com/articles/closed-ball) of center x and radius r contained in G and every [real](https://bliptext.com/articles/real-number)-valued [continuous function](https://bliptext.com/articles/continuous-function) h on that is [harmonic](https://bliptext.com/articles/harmonic-function) in B(x,r) and satisfies for all y on the [boundary](https://bliptext.com/articles/boundary-topology) of B(x,r), we have for all\nNote that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.\nA function u is called superharmonic if -u is subharmonic.\n\n# Properties\n\n# Examples\nIf f is [analytic](https://bliptext.com/articles/analytic-function) then \\log|f| is subharmonic.\nMore examples can be constructed by using the properties listed above,\nby taking maxima, convex combinations and limits.\nIn dimension 1, all subharmonic functions can be obtained in this way.\n\n# Riesz Representation Theorem\nIf u is subharmonic in a region D, in [Euclidean space](https://bliptext.com/articles/euclidean-space) of dimension n, v is harmonic in D, and u \\leq v, then v is called a harmonic majorant of u. If a harmonic majorant exists, then there exists the least harmonic majorant, and\nwhile in dimension 2,\nwhere v is the least harmonic majorant, and \\mu is a [Borel measure](https://bliptext.com/articles/borel-measure) in D. This is called the [Riesz](https://bliptext.com/articles/friedrich-riesz) representation theorem.\n\n# Subharmonic functions in the complex plane\nSubharmonic functions are of a particular importance in [complex analysis](https://bliptext.com/articles/complex-analysis), where they are intimately connected to [holomorphic functions](https://bliptext.com/articles/holomorphic-function).\nOne can show that a real-valued, continuous function \\varphi of a complex variable (that is, of two real variables) defined on a set is subharmonic if and only if for any closed disc of center z and radius r one has\nIntuitively, this means that a subharmonic function is at any point no greater than the [average](https://bliptext.com/articles/arithmetic-mean) of the values in a circle around that point, a fact which can be used to derive the [maximum principle](https://bliptext.com/articles/maximum-principle).\nIf f is a holomorphic function, then\nis a subharmonic function if we define the value of \\varphi(z) at the zeros of f to be -\\infty.\nIt follows that\nis subharmonic for every α > 0.\nThis observation plays a role in the theory of [Hardy spaces](https://bliptext.com/articles/hardy-spaces), especially for the study of H when 0 < p < 1.\nIn the context of the complex plane, the connection to the [convex functions](https://bliptext.com/articles/convex-function) can be realized as well by the fact that a subharmonic function f on a domain that is constant in the imaginary direction is convex in the real direction and vice versa.\n\n## Harmonic majorants of subharmonic functions\nIf u is sub[harmonic](https://bliptext.com/articles/harmonic-function) in a [region](https://bliptext.com/articles/region-mathematical-analysis) \\Omega of the complex plane, and h is [harmonic](https://bliptext.com/articles/harmonic-function) on \\Omega, then h is a **[harmonic](https://bliptext.com/articles/harmonic-function) majorant** of u in \\Omega if u \\leq h in \\Omega.\nSuch an inequality can be viewed as a growth condition on u.\n\n## Subharmonic functions in the unit disc. Radial maximal function\nLet φ be subharmonic, continuous and non-negative in an open subset Ω of the complex plane containing the closed unit disc D(0, 1).\nThe radial maximal function for the function φ (restricted to the unit disc) is defined on the unit circle by\nIf Pr denotes the [Poisson kernel](https://bliptext.com/articles/poisson-kernel), it follows from the subharmonicity that\nIt can be shown that the last integral is less than the value at e of the [Hardy–Littlewood maximal function](https://bliptext.com/articles/hardy-littlewood-maximal-function) φ∗ of the restriction of φ to the unit circle **T**,\nso that 0 ≤ M φ ≤ φ∗.\nIt is known that the Hardy–Littlewood operator is bounded on Lp(**T**) when 1 < p < ∞.\nIt follows that for some universal constant C,\nIf f is a function holomorphic in Ω and 0 < p < ∞, then the preceding inequality applies to φ = |f |.\nIt can be deduced from these facts that any function F in the classical Hardy space Hp satisfies\nWith more work, it can be shown that F has radial limits F(e) almost everywhere on the unit circle, and (by the [dominated convergence theorem](https://bliptext.com/articles/dominated-convergence-theorem)) that Fr, defined by Fr(e) = F(r e) tends to F in Lp(**T**).\n\n# Subharmonic functions on Riemannian manifolds\nSubharmonic functions can be defined on an arbitrary [Riemannian manifold](https://bliptext.com/articles/riemannian-manifold).\nDefinition: Let M be a Riemannian manifold, and an [upper semicontinuous](https://bliptext.com/articles/upper-semicontinuous) function.\nAssume that for any open subset U\\subset M, and any [harmonic function](https://bliptext.com/articles/harmonic-function) f1 on U, such that f_1 \\geq f on the boundary of U, the inequality f_1 \\geq f holds on all U.\nThen f is called subharmonic.\nThis definition is equivalent to one given above.\nAlso, for twice differentiable functions, subharmonicity is equivalent to the inequality, where \\Delta is the usual [Laplacian](https://bliptext.com/articles/laplace-beltrami-operator).","revisionCount":0},"hasMore":false,"nextCursor":null}

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