Brück’s Conjecture for Solutions of Second-Order Complex ODE

Abstract

Brück’s conjecture asserts that if a non-constant entire function f(z) with hyper-order \(\rho _{2}(f) \not \in \mathbb {N}\cup \{\infty \}\) shares one finite value a CM (counting multiplicities) with its derivative, then \(f'-a=c(f-a)\), for some non-zero constant c. This conjecture has been affirmed for entire functions with finite order and hyper-order less than one. In this paper, we show that Brück’s conjecture is true for entire functions that satisfy second order differential equations with meromorphic coefficients of finite order.