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.
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
Moreover h is the Koenigs function for the semigroup ft. By the Schwarz lemma, |ft(z)| decreases as t increases.
Since the left hand side is a harmonic function, the maximum principle implies the inequality is strict.
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.
Constantin Carathéodory proved in 1907 that if is a holomorphic function on the unit disk D with positive real part, then[2][3] In fact it suffices to show the result with g replaced by gr(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 Let be a univalent starlike function in |z| < 1.