# Hardy theorem

*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]).

#### 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 $H_p$ 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: http://encyclopediaofmath.org/index.php?title=Hardy_theorem&oldid=34155