Nevanlinna's criterion

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 [0,1], if and only if z h^\prime(z)/h(z) has positive real part for |*z''|

Note that, by applying the result to a•h(rz), the criterion applies on any disc |z|

Proof of criterion

Let h(z) be a starlike univalent function on |z|

For t

:f_t(z)=h^{-1}(e^{-t}h(z)), ,

a semigroup of holomorphic mappings of D into itself fixing 0.

Moreover h is the Koenigs function for the semigroup f**t.

By the Schwarz lemma, |f**t(z)| decreases as t increases.

Hence

:\partial_t |f_t(z)|^2 \le 0.

But, setting w = f**t(z),

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

where

:v(w)= -{h(w)\over h^\prime(w)}.

Hence

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

and so, dividing by |w|2,

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

Taking reciprocals and letting t go to 0 gives

: \Re, z {h^\prime(z)\over h(z)} \ge 0

for all |z|

Conversely if

: g(z) =z {h^\prime(z)\over h(z)}

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

Now

:\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)}.

Thus as z traces the circle z=re^{i\theta}, the argument of the image h(re^{i\theta}) increases strictly. By the argument principle, since h 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|

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

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|

Application to Bieberbach conjecture

Carathéodory's lemma

Constantin Carathéodory proved in 1907 that if

: g(z)= 1 +b_1 z + b_2 z^2 + \cdots.

is a holomorphic function on the unit disk D with positive real part, then

: |b_n|\le 2.

In fact it suffices to show the result with g replaced by g**r(z) = g(rz) for any r In that case g extends to a continuous function on the closed disc with positive real part and by Schwarz formula

: 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.

Using the identity

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

it follows that

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

so defines a probability measure, and

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

Hence

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

Proof for starlike functions

Let

: f(z) = z + a_2 z^2 + a_3 z^3 + \cdots

be a univalent starlike function in |z|

:|a_n|\le n.

In fact by Nevanlinna's criterion

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

has positive real part for |z|

: |b_n|\le 2.

On the other hand

: z f^\prime(z) = g(z) f(z)

gives the recurrence relation

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

where a1 = 1. Thus

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

so it follows by induction that

:|a_n|\le n.

Notes

References

References

1. {{harvnb. Hayman. 1994