On the value distribution of an entire function of order at most one (Q750671)

This article is written to prove the following theorem: Let E be transcendental entire of order \(\rho\leq 1\), \(Q\neq 0\) be rational, and assume that \(E'-Q\) vanihes at every zero of E with finitely many exceptions. Then \(\rho =1\), E is of regular growth and \(A=(E'-Q)/E\) is either a rational function with a non-zero limit at infinity or transcendental of regular growth with order one and with finitely many poles. This theorem is closely related with some recent results on second order homogeneous linear differential equations in the complex plane.