Hardy theorem
From Encyclopedia of Mathematics
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in the theory of functions of a complex variable
If is a regular analytic function in the disc , is a positive number, and if
is the average value, then is a non-decreasing function of that is logarithmically convex relative to (cf. Convexity, logarithmic). The theorem was established by G.H. Hardy [1].
The assertion on the logarithmic convexity remains valid for a function that is regular in an annulus (see [1]).
Hardy's theorem generalizes to subharmonic functions (cf. Subharmonic function) in a ball of , (see also [2]).
References
[1] | G.H. Hardy, "The mean value of the modulus of an analytic function" Proc. London. Math. Soc. (2) , 14 (1915) pp. 269–277 |
[2] | I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian) |
Comments
References
[a1] | T. Radó, "Subharmonic functions" , Springer (1937) |
[a2] | P.L. Duren, "Theory of spaces" , Acad. Press (1970) |
[a3] | J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 |
How to Cite This Entry:
Hardy theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_theorem&oldid=16176
Hardy theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_theorem&oldid=16176
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article