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In [[mathematics]], '''Nevanlinna's criterion''' in [[complex analysis]], proved in 1920 by the Finnish mathematician [[Rolf Nevanlinna]], characterizes [[holomorphic]] [[univalent functions]] on the [[unit disk]] which are [[star domain|starlike]]. Nevanlinna used this criterion to prove the [[Bieberbach conjecture]] for starlike univalent functions

==Statement of criterion==
A univalent function ''h'' on the unit disk satisfying ''h''(0)&nbsp;=&nbsp;0 and ''h'''(0)&nbsp;=&nbsp;1 is starlike, i.e. has image invariant under multilpication by real numbers in [0,1], if and only if <math>z h^\prime(z)/h(z)</math> has positive real part for |''z''|&nbsp;<&nbsp;1 and takes the value 1 at&nbsp;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<ref>{{harvnb|Hayman|1994|p=14}}</ref>

:<math>f_t(z)=h^{-1}(e^{-t}h(z)), \, </math>

a semigroup of holomorphic mappinga of ''D'' into itself fixing 0.

Moreover ''h'' is the [[Koenigs function]] for the semigroup ''f''<sub>''t''</sub>.

By the [[Schwarz lemma]], |''f''<sub>''t''</sub>(''z'')| decreases as ''t'' increases.

Hence

:<math>\partial_t |f_t(z)|^2 \le 0.</math>

But, setting ''w'' = ''f''<sub>''t''</sub>(''z''),

:<math> \partial_t |f_t(z)|^2 =2\Re\, \overline{f_t(z)} \partial_t f_t(z) = 2 \Re\, \overline{w} v(w),</math>

where

:<math>v(w)= -{h(w)\over h^\prime(w)}.</math>

Hence

:<math> \Re\, \overline{w} {h(w)\over h^\prime (w)} \ge 0.</math>

and so, dividing by |''w''|<sup>2</sup>,

:<math> \Re\, {h(w)\over w h^\prime (w)} \ge 0.</math>

Taking reciprocals and letting ''t'' go to 0 gives

:<math> \Re\, z {h^\prime(z)\over h(z)} \ge  0</math>

for all |''z''| < 1.  Since the left hand side is a [[harmonic function]], the [[maximum principle]] implies the inequality is strict.

Conversely if

:<math> g(z) =z {h^\prime(z)\over h(z)}</math>

has positive real part and ''g''(0)&nbsp;=&nbsp;1, then ''h'' can vanish only at 0, where it must have a simple zero.

Now

:<math>\partial_\theta \arg h(re^{i\theta})=\partial_\theta \Im\, \log h(z) = \Im\, \partial_\theta \log h(z)=\Im\, {\partial z\over \partial\theta} \cdot \partial_z \log h(z) =\Re\, z {h^\prime(z)\over h(z)}.</math>

Thus as ''z'' traces the circle <math> z=re^{i\theta}</math>, the argument of the image <math>h(re^{i\theta})</math> increases strictly. By the [[argument principle]], since <math>h</math> 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

:<math> N(a) ={1\over 2\pi i} \int_{|z|=r} {h^\prime(z) \over h(z)-a}\, dz.</math>

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]] proved in 1907 that if

:<math> g(z)= 1 +b_1 z + b_2 z^2 + \cdots.</math>

is a holomorphic function on the unit disk  ''D'' with  positive real part, then<ref>{{harvnb|Duren|1982|p=41}}</ref><ref>{{harvnb|Pommerenke|1975|p=40}}</ref>

:<math> |b_n|\le 2.</math>
 
In fact it suffices to show the result with ''g'' 
replaced by ''g''<sub>''r''</sub>(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]]

:<math> g(z) = {1\over 2\pi} \int_0^{2\pi} { e^{i\theta}+  z\over e^{i\theta} -z} \Re g(e^{i\theta})\, d\theta.</math>

Using the identity

:<math>  { e^{i\theta}+  z\over e^{i\theta} -z} = 1 +2 \sum_{n\ge 1} e^{-in\theta} z^n,</math>

it follows that

:<math>\int_0^{2\pi}  \Re g(e^{i\theta}) \,d\theta =1</math>,

so defines a probability measure, and

:<math>b_n =2\int_0^{2\pi} e^{-int} \Re g(e^{i\theta}) \,d\theta.</math>

Hence

:<math> |b_n| \le 2 \int_0^{2\pi} \Re g(e^{i\theta}) \,d\theta =2. </math>

===Proof for starlike functions===
Let

:<math> f(z) = z + a_2 z^2 + a_3 z^3 + \cdots </math>

be a univalent starlike function in |''z''| < 1. {{harvtxt|Nevanlinna|1921}} proved that

:<math>|a_n|\le n.</math>

In fact by Nevanlinna's criterion

:<math> g(z) = z{f^\prime(z)\over f(z)} = 1 + b_1 z + b_2 z^2 + \cdots</math>

has positive real part for |''z''|<1. So by Carathéodory's lemma

:<math> |b_n|\le 2.</math>

On the other hand

:<math> z f^\prime(z) = g(z) f(z)</math>

gives the recurrence relation

:<math> (n-1) a_n = \sum_{k=1}^{n-1} b_{n-k}a_k.</math>

where ''a''<sub>1</sub> = 1. Thus

:<math> |a_n|\le {2\over n-1} \sum_{k=1}^{n-1} |a_k|,</math>

so it follows by induction that

:<math>|a_n|\le n.</math>

==Notes==
{{reflist}}

==References==
*{{citation|first=C.|last=Carathéodory|title=Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen|journal= Math. Ann.|volume= 64|pages= 95–115|year=1907}}
*{{citation|last=Duren|first=P. L.|title=
Univalent functions|series=Grundlehren der Mathematischen Wissenschaften|volume= 259|publisher= Springer-Verlag|year= 1983|isbn= 0-387-90795-5|pages=41–42}}
*{{citation|last=Hayman|first= W. K.|title=Multivalent functions|edition=2nd|series=Cambridge Tracts in Mathematics|volume=110|publisher= Cambridge University Press|year= 1994|isbn= 0-521-46026-3}}
*{{citation|last=Nevanlinna|first= R.|title=Über die konforme Abbildung von Sterngebieten|journal=Ofvers. Finska Vet. Soc. Forh. |volume=53 |year=1921|pages=1–21}}
*{{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}}

[[Category:Analytic functions]]