Editing Hadamard three-circle theorem

In [[complex analysis]], a branch of [[mathematics]], the
'''Hadamard three-circle theorem''' is a result about the behavior of [[holomorphic function]]s.

== Statement ==

<blockquote>'''Hadamard three-circle theorem:''' Let <math>f(z)</math> be a holomorphic function on the [[annulus (mathematics)|annulus]] <math>r_1\leq\left| z\right| \leq r_3</math>. Let <math>M(r)</math> be the [[maxima and minima|maximum]] of <math>|f(z)|</math> on the [[circle]] <math>|z|=r.</math> Then, <math>\log M(r)</math> is a [[convex function]] of the [[logarithm]] <math>\log (r).</math> Moreover, if <math>f(z)</math> is not of the form <math>cz^\lambda</math> for some [[Coefficient|constants]] <math>\lambda</math> and <math>c</math>, then <math>\log M(r)</math> is  strictly convex as a function of <math>\log (r).</math>
</blockquote>

The conclusion of the [[theorem]] can be restated as

:<math>\log\left(\frac{r_3}{r_1}\right)\log M(r_2)\leq
\log\left(\frac{r_3}{r_2}\right)\log M(r_1)
+\log\left(\frac{r_2}{r_1}\right)\log M(r_3)</math>
	
for any three [[concentric circles]] of radii <math>r_1<r_2<r_3.</math>

==Proof==
The three circles theorem follows from the fact that for any real ''a'', the function Re log(''z''<sup>''a''</sup>''f''(''z'')) is harmonic between two circles, and therefore takes its maximum value on one of the circles. The theorem follows by choosing the constant ''a'' so that this [[harmonic function]] has the same maximum value on both circles.

The theorem can also be deduced  directly from [[Hadamard three-line theorem|Hadamard's three-line theorem]].<ref>{{harvnb|Ullrich|2008}}</ref>

==History==
A statement and proof for the theorem was given by [[J.E. Littlewood]] in 1912, but he attributes it to no one in particular, stating it as a known theorem. [[Harald Bohr]] and [[Edmund Landau]] attribute the theorem to [[Jacques Hadamard]], writing in 1896; Hadamard  published no proof.<ref>{{harvnb|Edwards|1974|loc=Section 9.3}}</ref>

==See also==
*[[Maximum principle]]
*[[Logarithmically convex function]]
*[[Hardy's theorem]]
*[[Hadamard three-line theorem ]]
*[[Borel–Carathéodory theorem]]
*[[Phragmén–Lindelöf principle]]

==Notes==
{{reflist}}

==References==
* {{citation|first=H.M.|last=Edwards|authorlink=Harold Edwards (mathematician)|title=Riemann's Zeta Function|year=1974|publisher=Dover Publications|isbn=0-486-41740-9}}
* {{Citation | last1=Littlewood | first1=J. E. | title=Quelques consequences de l'hypothese que la function ζ(s) de Riemann n'a pas de zeros dans le demi-plan Re(s) > 1/2. | year=1912 | journal=[[Les Comptes rendus de l'Académie des sciences]] | volume=154 | pages=263–266}}
* [[E. C. Titchmarsh]], ''The theory of the Riemann Zeta-Function'', (1951) Oxford at the Clarendon Press, Oxford. ''(See chapter 14)''
* {{citation|title=Complex made simple|volume= 97|series= [[Graduate Studies in Mathematics]]|first=David C.|last= Ullrich|publisher=[[American Mathematical Society]]|year= 2008|isbn=978-0821844793|pages=386–387}}

{{PlanetMath attribution|id=5605|title=Hadamard three-circle theorem}}

== External links ==
* [https://planetmath.org/proofofhadamardthreecircletheorem "proof of Hadamard three-circle theorem"]

[[Category:Inequalities (mathematics)]]
[[Category:Theorems in complex analysis]]