View source for 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|Convexity, logarithmic]]). The theorem was established by G.H. Hardy [[#References|[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 [[#References|[1]]]).

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "The mean value of the modulus of an analytic function"  ''Proc. London. Math. Soc. (2)'' , '''14'''  (1915)  pp. 269–277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,   "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Radó,   "Subharmonic functions" , Springer  (1937)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.L. Duren,   "Theory of $H_p$ spaces" , Acad. Press  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1984)  pp. 390</TD></TR></table>