### On Bank-Laine type functions

Jianming Chang, Yuefei Wang
2013 Annales Academiae Scientiarum Fennicae: Mathematica
We continue our previous study on the Bank-Laine type functions: meromorphic functions f that satisfy f (z) = 0 ⇐⇒ f ′ (z) ∈ {a, b} on the plane, where a, b are two distinct nonzero values. Using quasi-normality, we prove that there is no transcendental meromorphic function with this property when the quotient a/b is a positive integer. Moreover, we prove a quasi-normal criterion for families of such functions. This completes our previous results. In order to prove Theorem 1.2, we first study
more » ... 2, we first study the normality or quasi-normality of the family F a,b (D) which consists of all meromorphic functions f in a plane domain The idea of proving the results in function theory by making use of quasinormality first appears in  where it was proved that the derivative of a transcendental meromorphic function with finitely many simple zeros takes every non-zero values infinitely often. Recall [3, 9] that a family F of meromorphic functions defined in a plane domain D ⊂ C is said to be normal (quasi-normal) on D, in the sense of Montel, if each sequence {f n } ⊂ F contains a subsequence which converges spherically locally uniformly in D (minus a set E that has no accumulation point in D). The set E may depend on the subsequence. If there exists an integer ν ∈ N such that the set E always can be chosen to contain at most ν points, then F is said to be quasi-normal of order ν. Also, we say that the family F is normal (quasi-normal) at a point z 0 ∈ D, if there exists a neighborhood U ⊂ D of z 0 such that F is normal (quasi-normal) on U . An useful fact, which can be proved by making use of the diagonal method, is that F is normal (quasi-normal) on D if and only if F is normal (quasi-normal) at every point in D. Another fact is that if F is not quasi-normal of order ν in D, then there exist a sequence {f n } ⊂ F and ν + 1 points z 1 , z 2 , · · · , z ν+1 ∈ D such that no subsequence of {f n } is normal at each z j . The family F a,b (D) is not quasi-normal in general as showed by the following example.