Difference between revisions of "Logarithmically-subharmonic function"
From Encyclopedia of Mathematics
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| − | A positive function | + | {{TEX|done}} |
| + | A positive function $u(x)$ in a domain of the Euclidean space $\mathbf R^n$, $n\geq2$, whose logarithm $\log u(x)$ is a [[Subharmonic function|subharmonic function]]. For example, the modulus $|f(z)|$ of an analytic function $f(z)$ of a complex variable is a logarithmically-subharmonic function, but there are continuous logarithmically-subharmonic functions in planar domains that cannot be represented as the modulus of any analytic function. The logarithmically-subharmonic functions constitute a subclass of the strongly-subharmonic functions (cf. [[Subharmonic function|Subharmonic function]]). For $n=1$ they correspond to logarithmically-convex functions. | ||
The main property of logarithmically-subharmonic functions is that not only the product, but also a positive linear combination, of several logarithmically-subharmonic functions is a logarithmically-subharmonic function. | The main property of logarithmically-subharmonic functions is that not only the product, but also a positive linear combination, of several logarithmically-subharmonic functions is a logarithmically-subharmonic function. | ||
Latest revision as of 15:22, 29 December 2018
A positive function $u(x)$ in a domain of the Euclidean space $\mathbf R^n$, $n\geq2$, whose logarithm $\log u(x)$ is a subharmonic function. For example, the modulus $|f(z)|$ of an analytic function $f(z)$ of a complex variable is a logarithmically-subharmonic function, but there are continuous logarithmically-subharmonic functions in planar domains that cannot be represented as the modulus of any analytic function. The logarithmically-subharmonic functions constitute a subclass of the strongly-subharmonic functions (cf. Subharmonic function). For $n=1$ they correspond to logarithmically-convex functions.
The main property of logarithmically-subharmonic functions is that not only the product, but also a positive linear combination, of several logarithmically-subharmonic functions is a logarithmically-subharmonic function.
References
| [1] | I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) pp. Chapt. 3 (In Russian) |
Comments
References
| [a1] | L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) pp. 36 (Translated from Russian) |
| [a2] | W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976) |
How to Cite This Entry:
Logarithmically-subharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmically-subharmonic_function&oldid=43560
Logarithmically-subharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmically-subharmonic_function&oldid=43560
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article