In
mathematics, Nevanlinna's criterion in
complex analysis, proved in 1920 by the Finnish mathematician
Rolf Nevanlinna
Rolf Herman Nevanlinna (né Neovius; 22 October 1895 – 28 May 1980) was a Finnish mathematician who made significant contributions to complex analysis.
Background
Nevanlinna was born Rolf Herman Neovius, becoming Nevanlinna in 1906 when his fa ...
, characterizes
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
univalent functions on the
unit disk which are
starlike. Nevanlinna used this criterion to prove the
Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It ...
for starlike univalent functions.
Statement of criterion
A univalent function ''h'' on the unit disk satisfying ''h''(0) = 0 and ''h(0) = 1 is starlike, i.e. has image invariant under multiplication by real numbers in
,1 if and only if
has positive real part for , ''z'', < 1 and takes the value 1 at 0.
Note that, by applying the result to ''a''•''h''(''rz''), the criterion applies on any disc , ''z'', < r with only the requirement that ''f''(0) = 0 and ''f(0) ≠ 0.
Proof of criterion
Let ''h''(''z'') be a starlike univalent function on , ''z'', < 1 with ''h''(0) = 0 and ''h(0) = 1.
For ''t'' < 0, define
:
a semigroup of holomorphic mappinga of ''D'' into itself fixing 0.
Moreover ''h'' is the
Koenigs function for the semigroup ''f''
''t''.
By the
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapp ...
, , ''f''
''t''(''z''), decreases as ''t'' increases.
Hence
:
But, setting ''w'' = ''f''
''t''(''z''),
:
where
:
Hence
:
and so, dividing by , ''w'',
2,
:
Taking reciprocals and letting ''t'' go to 0 gives
:
for all , ''z'', < 1. Since the left hand side is a
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
, the
maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
implies the inequality is strict.
Conversely if
:
has positive real part and ''g''(0) = 1, then ''h'' can vanish only at 0, where it must have a simple zero.
Now
:
Thus as ''z'' traces the circle
, the argument of the image
increases strictly. By the
argument principle
In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
Specifically, i ...
, since
has a simple zero at 0,
it circles the origin just once. The interior of the region bounded by the curve it traces is therefore starlike. If ''a'' is a point in the interior then the number of solutions ''N''(''a'') of ''h(z)'' = ''a'' with , ''z'', < ''r'' is given by
:
Since this is an integer, depends continuously on ''a'' and ''N''(0) = 1, it is identically 1. So ''h'' is univalent and starlike in each disk , ''z'', < ''r'' and hence everywhere.
Application to Bieberbach conjecture
Carathéodory's lemma
Constantin Carathéodory
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
proved in 1907 that if
:
is a holomorphic function on the unit disk ''D'' with positive real part, then
:
In fact it suffices to show the result with ''g''
replaced by ''g''
''r''(z) = ''g''(''rz'') for any ''r'' < 1 and then pass to the limit ''r'' = 1.
In that case ''g'' extends to a continuous function on the closed disc with positive real part and by
Schwarz formula
:
Using the identity
:
it follows that
:
,
so defines a probability measure, and
:
Hence
:
Proof for starlike functions
Let
:
be a univalent starlike function in , ''z'', < 1. proved that
:
In fact by Nevanlinna's criterion
:
has positive real part for , ''z'', <1. So by Carathéodory's lemma
:
On the other hand
:
gives the recurrence relation
:
where ''a''
1 = 1. Thus
:
so it follows by induction that
:
Notes
References
*
*
*
*
*{{citation, last=Pommerenke, first= C., authorlink=Christian Pommerenke, title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, series= Studia Mathematica/Mathematische Lehrbücher, volume=15, publisher= Vandenhoeck & Ruprecht, year= 1975
Analytic functions