On the frequency of multiple values of a meromorphic function of small order
Let f be meromorphic in the plane, of (lower) order \(\mu\), and let \(N_ 1(r)\) be the usual branching term. The main result gives a lower asymptotic bound for the ratio \(N_ 1(r)/T(r,F)\) when \(\mu <\). As an application, it is shown that if g and h are linearly independent transcendental functions of order \(<\), then the Wronskian \(W(g,h)=gh'- hg'\) must always have approximately as many zeros as allowed by considering the growths of f and g. Such behavior is conjectured to persist whenever \(2\mu\) is not an integer \(\geq 2\), but this has proved intractible. The proof here uses the hypotheses to construct regions which surround the origin, in which \(| f|\) is large, and then uses the relation between f' and W(g,h) when \(f=g/h\). There is also significant use of a theory of Pólya peaks due to him with \textit{A. E. Eremenko} and \textit{M. L. Sodin} [Minimum modulus theorems for entire functions at Pólya peaks, Teor. Funkts., Funkts. Anal. Prilozh. (to appear)].
scientific article; zbMATH DE number 3914678