View source for 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|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Privalov,   "Subharmonic functions" , Moscow-Leningrad  (1937)  pp. Chapt. 3  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.K. Hayman,   P.B. Kennedy,   "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>