A further note on a result of Bank, Frank, and Laine (Q1412412)

The paper deals with homogeneous linear differential equations of the form \[ f^{(n)}+ P_{n-1}(z) f^{(n-1)}+\cdots+ P_0(z) f= 0, \] where \(P_0(z),P_1(z),\dots, P_{n-1}(z)\) are polynomials with \(P_0(z)\neq 0\). It is shown that under certain conditions on the coefficients \(P_j\), the solutions \(f\) must satisfy \(\lambda(f)= \rho(f)\) or \(f\) has only finitely many zeros, which generalizes a result of Bank, Frank, and Laine, and a result of Wang. Here, \(\rho(f)\) denotes the order growth of \(f\) and \(\lambda(f)\) the exponent of convergence of the sequence of zeros of \(f\).