Quasiconformal modifications and Bank-Laine functions
Let \(A\) be an entire function, and \(f_1\), \(f_2\) be linearly independent solutions of \(w''+A(z)w=0\), normalized by \(W(f_1,f_2)=f_1f_2'-f_1'f_2\equiv 1\). As observed by \textit{S. Bank} and \textit{I. Laine} [Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026)], the product \(E=f_1f_2\), which is trivially entire, satisfies \(E'(z)=\pm 1\) at every zero~\(z\) of~\(E\). Conversely, if \(E\) is an entire function satisfying the above condition, then \(A=\frac 14((E'/E)^2-2E''/E-1/E^2)\) is entire and \(E\) is the product of linearly independent normalized solutions of \(w''+A(z)w=0\). A long-standing open conjecture is that \(\rho(A)\) is a positive integer whenever \(A\) is transcendental entire of finite order \(\rho(A)\), and \(\max(\lambda(f_1),\lambda(f_2))<\infty\), \(\lambda(f)\) being the exponent of convergence for the zero-sequence of~\(f\). Since \(E\) is of finite order under the assumptions of this conjecture, an idea to attack this problem is trying to construct entire functions of the above type, called Bank-Laine functions. Such functions of finite order are associated with a polynomial~\(A\) if and only if \(m(r,\frac 1E)=O(\log r)\), as proved in (loc. cit.). Up to now, all known examples of Bank-Laine functions associated with a transcendental~\(A\) have been of the form \(E(z)=P(\exp(\alpha z))\exp(Q(z))\), where \(P\), \(Q\) are polynomials and \(\alpha\) a constant. The main result in this article is to construct, by using quasiconformal modifications of \[ (\sin(\pi z))\exp(2\pi iz^2-\log\pi), \] a Bank-Laine function \(F(z)\) of finite order, associated with a transcendental \(A\), and not being representable as \(F(z)=P(\exp(\alpha z))\exp(Q(z))\). Moreover, \(F(z)\) has the same zeros as \(\sin(\pi z)\) which is a Bank-Laine function itself, associated with \(A(z)=\pi^2/4\). Finally, it will be proved that zero-sequences cannot be shared in this way for non-constant polynomials \(A\).
scientific article; zbMATH DE number 1214005