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	~~I woke~~ [~~http~~://~~www.museodecarruajes.org cheap luke bryan tickets~~] up [~~http:~~//~~Photo~~.~~net~~/~~gallery~~/~~tag-search~~/~~search?query_string=yesterday yesterday~~] ~~and realised - Today I've been solitary for~~ a ~~little while and after much intimidation from pals I today locate myself signed up for web dating~~. ~~They promised me that there are lots of entertaining, sweet and regular folks to meet, so the pitch is gone by here!~~<br>~~I make an effort to stay as physically healthy as possible being at~~ the ~~gymnasium luke bryan live in concert;~~ [~~http~~://~~www~~.~~banburycrossonline~~.~~com banburycrossonline.com~~], ~~several times weekly. I appreciate my athletics and attempt~~ to ~~perform or see since many~~ a ~~possible~~. ~~I am going~~ to ~~regularly at Hawthorn matches being winter. Notice: Supposing~~ [~~http://lukebryantickets~~.~~pyhgy.com luke bryan on tour]~~ that ~~you would contemplated buying~~ a ~~sport I don~~'~~t mind~~, ~~I have seen~~ the ~~carnage of fumbling suits at stocktake revenue.~~<br>~~My family~~ and ~~friends are wonderful and spending time with them at bar gigabytes or meals is always critical~~. ~~I have never been in to nightclubs as~~ [~~http:~~/~~/www~~.~~guardian~~.Co.uk/~~search?q~~=~~I+discover I discover~~] ~~that one may not get a nice conversation against~~ the ~~sound~~. ~~In addition, I have two very adorable and unquestionably cheeky puppies who are constantly excited to meet fresh folks~~.~~<br><br>My homepage ::~~ [~~http~~:~~//www.senatorwonderling.com cheap luke bryan concert tickets~~]		In [[complex analysis]], a branch of [[mathematics]], the
			'''Hadamard three-circle theorem''' is a result about the behavior of [[holomorphic function]]s.

			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>

			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>

			==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\|Ed74}}

			==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-lines theorem\|Hadamard's three-lines theorem]].<ref>{{harvnb\|Ullrich\|2008}}</ref>

			==See also==
			*[[maximum principle]]
			*[[logarithmically convex function]]
			*[[Hardy's theorem]]
			*[[Hadamard three-lines theorem]]
			*[[Borel–Carathéodory theorem]]
			*[[Phragmén–Lindelöf principle]]

			==Notes==
			{{reflist}}
			* {{note\|Ed74}} [[Harold Edwards (mathematician)\|H.M. Edwards]], ''Riemann's Zeta Function'', (1974) Dover Publications, ISBN 0-486-41740-9 ''(See section 9.3.)''

			==References==
			*{{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\|id=ISBN 0821844792\|pages=386–387}}

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

			==External links==
			* {{PlanetMath reference\|7943\|proof of Hadamard three-circle theorem}}

			[[Category:Theorems in complex analysis]]

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