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
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 .
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 ).
Hardy's theorem generalizes to subharmonic functions (cf. Subharmonic function) in a ball of $\mathbf R^n$, $n\geq2$ (see also ).
|||G.H. Hardy, "The mean value of the modulus of an analytic function" Proc. London. Math. Soc. (2) , 14 (1915) pp. 269–277|
|||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|
Hardy theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_theorem&oldid=34155