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Difference between revisions of "Logarithmically-subharmonic function"

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A positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606701.png" /> in a domain of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606703.png" />, whose logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606704.png" /> is a [[Subharmonic function|subharmonic function]]. For example, the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606705.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606706.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060670/l0606707.png" /> they correspond to logarithmically-convex functions.
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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
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article