- Source: Hadamard three-circle theorem
In complex analysis, a branch of mathematics, the
Hadamard three-circle theorem is a result about the behavior of holomorphic functions.
Let
f
(
z
)
{\displaystyle f(z)}
be a holomorphic function on the annulus
r
1
≤
|
z
|
≤
r
3
.
{\displaystyle r_{1}\leq \left|z\right|\leq r_{3}.}
Let
M
(
r
)
{\displaystyle M(r)}
be the maximum of
|
f
(
z
)
|
{\displaystyle |f(z)|}
on the circle
|
z
|
=
r
.
{\displaystyle |z|=r.}
Then,
log
M
(
r
)
{\displaystyle \log M(r)}
is a convex function of the logarithm
log
(
r
)
.
{\displaystyle \log(r).}
Moreover, if
f
(
z
)
{\displaystyle f(z)}
is not of the form
c
z
λ
{\displaystyle cz^{\lambda }}
for some constants
λ
{\displaystyle \lambda }
and
c
{\displaystyle c}
, then
log
M
(
r
)
{\displaystyle \log M(r)}
is strictly convex as a function of
log
(
r
)
.
{\displaystyle \log(r).}
The conclusion of the theorem can be restated as
log
(
r
3
r
1
)
log
M
(
r
2
)
≤
log
(
r
3
r
2
)
log
M
(
r
1
)
+
log
(
r
2
r
1
)
log
M
(
r
3
)
{\displaystyle \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})}
for any three concentric circles of radii
r
1
<
r
2
<
r
3
.
{\displaystyle r_{1}
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.
Proof
The three circles theorem follows from the fact that for any real a, the function Re log(zaf(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's three-line theorem.
See also
Maximum principle
Logarithmically convex function
Hardy's theorem
Hadamard three-line theorem
Borel–Carathéodory theorem
Phragmén–Lindelöf principle
Notes
References
Edwards, H.M. (1974), Riemann's Zeta Function, Dover Publications, ISBN 0-486-41740-9
Littlewood, J. E. (1912), "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.", Les Comptes rendus de l'Académie des sciences, 154: 263–266
E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)
Ullrich, David C. (2008), Complex made simple, Graduate Studies in Mathematics, vol. 97, American Mathematical Society, pp. 386–387, ISBN 0821844792
This article incorporates material from Hadamard three-circle theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
External links
"proof of Hadamard three-circle theorem"
Kata Kunci Pencarian:
- Hadamard three-circle theorem
- Hadamard three-lines theorem
- Hardy's theorem
- List of circle topics
- List of complex analysis topics
- List of things named after Jacques Hadamard
- Hadamard factorization theorem
- Hadamard (disambiguation)
- List of theorems
- Brouwer fixed-point theorem
