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.
|||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)|
Logarithmically-subharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logarithmically-subharmonic_function&oldid=43560