Hadamard's theorem of three circles

The Three circles set of Hadamard , even hadamardscher Three circles set called English Hadamard's three-circle theorem is a proposition on the mathematical branch of function theory . The sentence goes back to the French mathematician Jacques Hadamard (1865–1963). It can be derived from the maximum principle of function theory and entails a number of further theorems of function theory, in particular Liouville's theorem .

Formulation of the sentence

The three-circle theorem can be given as follows:

Given a region and a holomorphic function defined on it , which is not the null function . ${\displaystyle G\subseteq \mathbb {C} }$ ${\displaystyle f\colon G\to \mathbb {C} \;;\;z\mapsto f(z)}$

If further be two real numbers plus one in contained annulus . ${\displaystyle a,b\;(0<a<b)}$ ${\displaystyle G}$ ${\displaystyle A=\{z\in \mathbb {C} \colon a\leq |z|\leq b\}}$

Then applies to the associated real-valued function

${\displaystyle M\colon [a,b]\to \mathbb {R} ^{>0}\;;\;r\mapsto M(r):=\max _{|z|=r}\;{|f(z)|}}$

always the inequality

${\displaystyle M(r)\leq {M(a)}^{\frac {\ln(b)-\ln(r)}{\ln(b)-\ln(a)}}\cdot {M(b)}^{\frac {\ln(r)-\ln(a)}{\ln(b)-\ln(a)}}}$.

In other words:

The real-valued function is an in- convex function and therefore always satisfies the inequality ${\displaystyle r\mapsto \ln {\bigl (}M(r){\bigr )}}$ ${\displaystyle \ln(r)}$

${\displaystyle \ln {\bigl (}(M(r){\bigr )}\leq {{\frac {\ln(b)-\ln(r)}{\ln(b)-\ln(a)}}\cdot \ln {\bigl (}(M(a){\bigr )}+{{\frac {\ln(r)-\ln(a)}{\ln(b)-\ln(a)}}\cdot \ln {\bigl (}(M(b){\bigr )}}\;(a\leq r\leq b)}}$.

Application: Jentzsch's theorem

As Edmund Landau showed, another well-known result of the theory of functions can be derived by applying the three-circle theorem, namely the Jentzsch theorem . This goes back to the inaugural dissertation by Robert Jentzsch from 1914. The sentence was then published by Jentzsch in the Acta Mathematica of 1916 and gave rise to many further functional theoretical investigations. It can be formulated as follows:

A power series developed around the point of development is given ${\displaystyle \mathbb {C} }$ ${\displaystyle z_{0}=0}$ ${\displaystyle f(z)=\sum _{n=0}^{\infty }{a_{n}z^{n}}=a_{0}+a_{1}z^{1}+a_{2}z^{2}+\ldots +a_{n}z^{n}+a_{n+1}z^{n+1}\ldots }$

with finite radius of convergence and circle of convergence . ${\displaystyle r\in (0\;,\;\infty )}$ ${\displaystyle D=\{z\in \mathbb {C} \colon |z|<r\}}$

The associated complex-valued function ${\displaystyle f\colon \,D\to \mathbb {C} ,\;z\mapsto f(z)}$

is not constant and it applies . ${\displaystyle a_{0}\neq 0}$

Be further

${\displaystyle s_{k}(z)=\sum _{n=0}^{k}{a_{n}z^{n}}=a_{0}+a_{1}z+a_{2}z^{2}+\ldots +a_{k}z^{k}\;(k\in \mathbb {N} _{0})}$

the created section functions .

Then:

In every arbitrarily small open neighborhood of every edge point of the convergence circle, an infinite number of section functions each have at least one zero .

literature

Monographs

• Robert B. Burckel: An introduction to classical complex analysis . tape 1 . Birkhäuser Verlag, Basel / Stuttgart 1979, ISBN 3-7643-0989-X ( MR0555733 ).
• GM Golusin : Geometrical Function Theory (=  university books for mathematics . Volume 31 ). VEB Deutscher Verlag der Wissenschaften, Berlin 1957 ( MR0089896 ).
• Adolf Hurwitz , Richard Courant : Lectures on general function theory and elliptic functions (=  The basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Volume 3 ). 4th, increased and improved edition. Springer-Verlag, Berlin ( inter alia) 1964 ( MR0173749 ).
• Edmund Landau, Dieter Gaier : Presentation and justification of some recent results of the function theory . 3rd, expanded edition. Springer-Verlag, Berlin ( inter alia ) 1986, ISBN 3-540-16886-9 ( MR0869998 ).
• Rolf Nevanlinna : Unambiguous analytical functions (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 46 ). Springer-Verlag, Berlin (inter alia), ISBN 3-540-06233-5 .
• Walter Rudin : Real and Complex Analysis . 2nd improved edition. Oldenbourg Wissenschaftsverlag, Munich / Vienna 1999, ISBN 3-486-24789-1 ( MR1736644 ).
• Fritz Rühs : Function Theory (=  university books for mathematics . Volume 56 ). 3rd, corrected edition. VEB Deutscher Verlag der Wissenschaften, Berlin 1976 ( MR0486433 ).
• EC Titchmarsh : The Theory of Functions . Oxford University Press , Oxford / London (among others) 1978.

Original work

• Aryeh Dvoretzky : On the theorem of Jentzsch . In: Proc. Nat. Acad. Sci. USA band 35 , 1949, pp. 246-252 ( MR0029995 ).
• Robert Jentzsch: Investigations on the theory of the consequences of analytical functions . In: Acta Mathematica . tape 41 , 1916, pp. 219-251 ( link.springer.com ). MR1555151
• Raphael M. Robinson : Hadamard's three circles theorem . In: Bulletin of the American Mathematical Society . tape 50 , 1944, pp. 795-802 ( projecteuclid.org ). MR0011120

Individual evidence

1. ↑ There is also the spelling "Drei-Kreise-Satz" instead of "Dreikreisesatz" in German-language sources, as well as the spelling "three circles theorem" instead of "three-circle theorem" in English sources.
2. ↑ Robert B. Burckel: An introduction to classical complex analysis. Vol.1. 1979, pp. 147, 187
3. ↑ GM Golusin: Geometrical Function Theory . 1957, pp. 299-300
4. ^ Adolf Hurwitz, Richard Courant: Lectures on general function theory ..... 1964, pp. 429-430
5. ^ ^{A b} Edmund Landau, Dieter Gaier: Presentation and justification of some recent results of the theory of functions. 1986, pp. 88-95, pp. 145-148
6. ↑ Rolf Nevanlinna: Unique analytical functions. 1974, p. 43
7. ↑ Fritz Rühs: Function theory. 1976, pp. 117-119, 145-146
8. ^ Walter Rudin: Real and Complex Analysis. 1999, p. 316
9. ^ EC Titchmarsh: The Theory of Functions. 1978, pp. 172-173