In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex...

hold, although other properties of harmonic functions may fail. More generally, a function is subharmonic if and only if, in the interior of any ball...

potential theory, a subharmonic function f satisfies Δf ≥ 0. Any subharmonic function that is bounded above by a harmonic function for all points on the...

discrete Laplace operator. Similar to continuous subharmonic functions one can define subharmonic functions for finite-difference approximations u h {\displaystyle...

condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex domain...

Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian...

of the ball (mean value property). Also subharmonic function and superharmonic function. Elementary function: composition of arithmetic operations, exponentials...

Bloch space, Bergman space and Sobolev space. Subharmonic function – Class of mathematical functions Kellogg's theorem Garabedian, P. R.; Schiffer, M...

whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are...

natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which Δ u ≥ 0 , {\displaystyle...

sets), convex functions (on vector and affine spaces), harmonic and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one...

in which he gave a solution to Plateau's problem, and in 1935, "Subharmonic Functions". His work focused on computer science in the last decade of his...

the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar...

interest in potential theory because Iαμ is then a (continuous) subharmonic function off the support of μ, and is lower semicontinuous on all of Rn. Consideration...

rigidity results for functions on complete Riemannian manifolds. A particularly famous result of his says that a subharmonic function cannot be both positive...

using the Perron method, which relies on the maximum principle for subharmonic functions. This approach is described in many text books. It is not well-suited...

Research Problems in Function Theory, London: Athlone Press, pp. vii+56. Hayman, W. K.; Kennedy, P. B. (1976), Subharmonic functions. Volume 1, London Mathematical...

understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and...

kind of "complex convexity" remains, namely the fact that z → |z|q is subharmonic for every q > 0. As a consequence, if F ( z ) = ∑ n = 0 + ∞ c n z n ...

{\displaystyle n\geq 2} ) is a polar set if there is a non-constant subharmonic function u {\displaystyle u} on R n {\displaystyle \mathbb {R} ^{n}} such...

(\log \left|A-\lambda I\right|),\;\lambda \in \mathbb {C} ,} is a subharmonic function and its Laplacian in the distributional sense is a probability measure...

the function f r ( z ) = f ( r z ) . {\displaystyle f_{r}(z)=f(rz).} The inequalities can also be deduced, following Riesz (1925), using subharmonic functions...

Peter; Schoen, Richard (1984). "Lp and mean value properties of subharmonic functions on Riemannian manifolds". Acta Mathematica. 153 (3–4): 279–301....

harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic...

{R} ^{n}} , the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Finer topology If X is a set,...

a conjecture of Heins, which concerned a conjecture of Heins on subharmonic functions and gives positive results. That same year he was appointed as an...

State University in 1979 (Asymptotic Properties of Meromorphic and Subharmonic Functions), and is currently a distinguished professor at Purdue University...

Times, November 3, 2009. Cunzer, Ela-Chaim (1937). On convex and subharmonic functions (Catalog entry for translated manuscript). Translated by Wlordarski...

1007/978-1-4757-1705-1_1, ISBN 9781475717068 Beckenbach, E. F.; Rado, T. (1933). "Subharmonic Functions and Surfaces of Negative Curvature". Transactions of the American...

Ortega-Cerda', Joaquim; Tilli, Paolo (2022). "A monotonicity theorem for subharmonic functions on manifolds". arXiv:2212.14008 [math.CA]. Video of a lecture by...