Logarithmically-subharmonic function

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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. 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.


[1] I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) pp. Chapt. 3 (In Russian)



[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:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article