Meromorphic functions in the class \(S\) and the zeros of the second derivative
Let \(S\) be a class of transcendental meromorphic functions in the plane that have finitely many critical and asymptotic values. The author has proved that if \(f \in S\) has finite order (respectively finite lower order), \(f''/f'\) is non-constant and \(f^{-1}\) has finitely many singular values. Then \[ \log^+ M(r, f''/f') = o (T(r, f''/f')) \] as \(r \to \infty\) in a set of logarithmic density 1 (respectively upper logarithmic density 1). Here \(M\) and \(T\) are the standard notions of the Nevanlina theory (Theorem 4). Moreover in these conditions the Nevanlinna deficiency \(\delta (0, f''/f') = 0\) (Theorem 5).
scientific article; zbMATH DE number 5305521
scientific article