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Difference between revisions of "Hardy theorem"

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''in the theory of functions of a complex variable''
 
''in the theory of functions of a complex variable''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463801.png" /> is a regular analytic function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463803.png" /> is a positive number, and if
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If $f(z)$ is a regular analytic function in the disc $|z|<R$, $\alpha$ is a positive number, and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463804.png" /></td> </tr></table>
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$$M_\alpha(r)=\left\lbrace\frac{1}{2\pi}\int\limits_0^{2\pi}|f(re^{i\theta})|^\alpha d\theta\right\rbrace^{1/\alpha},\quad0<r<R,$$
  
is the average value, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463805.png" /> is a non-decreasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463806.png" /> that is logarithmically convex relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463807.png" /> (cf. [[Convexity, logarithmic|Convexity, logarithmic]]). The theorem was established by G.H. Hardy [[#References|[1]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463808.png" /> that is regular in an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h0463809.png" /> (see [[#References|[1]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h04638010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h04638011.png" /> (see also [[#References|[2]]]).
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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====
 
====References====
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====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046380/h04638012.png" /> 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>
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<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>

Revision as of 15:29, 2 August 2014

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=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