In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk |z| < r onto a starlike domain for r ⤠tanh ÃÂ/4.
The radius of starlikeness of an univalent function f satisfying f(0) = 0 is the largest radius r for which the function f maps the open disk |z| < r into a starlike domain with respect to the origin.
Let f be a univalent holomorphic function on the unit disc D such that f(0) = 0. Then for all r ⤠tanh ÃÂ/4, the image of the disc |z| < r is starlike with respect to 0, i.e. it is closed under multiplication by real numbers in (0,1).
If f(z) is univalent on D with f(0) = 0, then
Taking the real and imaginary parts of the logarithm, this implies the two inequalities
and
For fixed z, both these equalities are attained by suitable Koebe functions
where |w| = 1.
originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in , relied on the Loewner equation. More elementary proofs were subsequently given based on Goluzin's inequalities, an equivalent form of Grunsky's inequalities (1939) for the Grunsky matrix.
For a univalent function g in z > 1 with an expansion
Goluzin's inequalities state that
where the z<sub>i</sub> are distinct points with |z<sub>i</sub>| > 1 and û<sub>i</sub> are arbitrary complex numbers.
Taking n = 2. with û<sub>1</sub> = â û<sub>2</sub> = û, the inequality implies
If g is an odd function and ÷ = â ö, this yields
Finally if f is any normalized univalent function in D, the required inequality for f follows by taking
with
Let f be a univalent function on D with f(0) = 0. By Nevanlinna's criterion, f is starlike on |z| < r if and only if
for |z| < r. Equivalently
On the other hand, by the inequality of Grunsky above,
Thus if
the inequality holds at z. This condition is equivalent to
and hence f is starlike on any disk |z| < r with r ⤠tanh ÃÂ/4.