Hardy theorem

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in the theory of functions of a complex variable

If $f(z)$ is a regular analytic function in the disc $|z|<R$, $\alpha$ is a positive number, and if

$$M_\alpha(r)=\left\lbrace\frac{1}{2\pi}\int\limits_0^{2\pi}|f(re^{i\theta})|^\alpha d\theta\right\rbrace^{1/\alpha},\quad0<r<R,$$

is the average value, then $M_\alpha(r)$ is a non-decreasing function of $r$ that is logarithmically convex relative to $\ln r$ (cf. Convexity, logarithmic). The theorem was established by G.H. Hardy [1].

The assertion on the logarithmic convexity remains valid for a function $f(z)$ that is regular in an annulus $0\leq\rho<|z|<R$ (see [1]).

Hardy's theorem generalizes to subharmonic functions (cf. Subharmonic function) in a ball of $\mathbf R^n$, $n\geq2$ (see also [2]).


[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)



[a1] T. Radó, "Subharmonic functions" , Springer (1937)
[a2] P.L. Duren, "Theory of $H_p$ 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:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article