Composite Bank-Laine functions and a question of Rubel (Q2759060)

This excellent paper continues previous studies by the same author about determining Bank-Laine functions, see \textit{J. K. Langley} [Arch. Math. 71, 233-239 (1998; Zbl 0930.30028) and Proc. Am. Math. Soc. 129, No. 7, 1969-1978 (2001; Zbl 0969.30015)]. These are entire functions \(E\) such that \(E'(z)=\pm 1\) at every zero \(z\) of \(E\), arising as products \(E=w_1w_2\) of linearly independent solutions \(w_1,w_2\) of \(w''+A(z)w=0\), \(A(z)\) entire, normalized by \(W(w_1,w_2)=w_1w_2'-w_1'w_2\equiv 1\). The essential contents of this paper is to determine all Bank-Laine functions of the form \(E=f\circ g\) with \(f,g\) entire. More precisely, the following theorem will be proved: Let \(f,g\) be non-constant with \(f\) meromorphic and \(g\) entire such that \(E=fg\) is a Bank-Laine function. Then one of the following four alternatives holds: (1) \(f\) is zero-free, (2) \(f\) has one zero \(a\) only, and either \(a\) is a Picard value of \(g\) or \(f'(a)(rg(z)-a)\) is a Bank-Laine function, (3) \(f\) has at least two zeros and \(g(z)=Az^2+Bz+C\) is a quadratic polynomial such that \(f(a)=0\) implies \(f'(a)^2(B^2-4A(C-a))=1\), (4) \(f\) has at least two zeros and \(g(z)=Ae^{2bz}+B+Ce^{-2bz}\), such that whenever \(f(a)=0\), then either \(a\) is a Picard value of \(g\), or \(4b^2f'(a)^2((B-a)^2-4AC)=1\). In addition to this main result, it will be proved that all Bank-Laine functions of finite order are pseudoprime and, if associated with a transcendental coefficient \(A(z)\) in \(w''+A(z)w=0\), zero is an asymptotic value of these functions. The proofs, which are clearly written throughout, need to combine value distribution theory including the Ahlfors geometric version, Wiman-Valiron theory, and some Tsuji type harmonic measure estimates in particular. As a side result, it will be proved that for any transcendental entire function \(f\), there exists a path tending to infinity on which \(f\) and all its derivatives tend to infinity, solving a conjecture of L. Rubel.