Nevanlinna's criterion

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 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) = 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| < 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^[1]

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

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, |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| < 1. Since the left hand side is a harmonic function, the maximum principle implies the inequality is strict.

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| < r is given by

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| < r and hence everywhere.

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^[2]^[3]

|b_n|\le 2.

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

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| < 1. Nevanlinna (1921) proved that

|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|<1. So by Carathéodory's lemma

|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

Carathéodory, C. (1907), "Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen", Math. Ann. 64: 95–115, doi:10.1007/bf01449883
Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften 259, Springer-Verlag, pp. 41–42, ISBN 0-387-90795-5
Hayman, W. K. (1994), Multivalent functions, Cambridge Tracts in Mathematics 110 (2nd ed.), Cambridge University Press, ISBN 0-521-46026-3
Nevanlinna, R. (1921), "Über die konforme Abbildung von Sterngebieten", Ofvers. Finska Vet. Soc. Forh. 53: 1–21
Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher 15, Vandenhoeck & Ruprecht

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