Logarithmically-subharmonic function

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A positive function in a domain of the Euclidean space , , whose logarithm is a subharmonic function. For example, the modulus of an analytic function 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 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. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098