A Generalisation of a Malmquist's theorem
Kôsaku Yosida · Japanese journal of mathematics · 1932
J. Malmquist (1) has proved the following important theorem :is not a Riccati's equation( 2), all its one-valued solutions must be rational.In the present paper, I intend to give a generalisation of it (see the Theorem 1 and 2 below), as an application of the Nevanlinna's (3) theory of meromorphic functions.1. Let R(x, z) be an irreducible rational function whose degree in z is d.If we denote by z(x) any transcendental meromorphic function we shall have, in Nevanlinna's notation :Lemma 1. (of G. Valiron)(4),(1) T(r,R(x,z(x)))=dT(r, z(x))+O(logr), and