Brück’s conjecture on shared values between an entire function and its derivative

Prove that for every nonconstant entire function f(z) with finite hyper-order s(f) such that s(f) is not a positive integer, if f(z) and f′(z) share one complex value a counting multiplicities (meaning that f(z)−a and f′(z)−a have the same zeros with the same multiplicities), then there exists a constant c ∈ C such that f′(z)−a = c(f(z)−a) holds identically.

Background

The paper studies meromorphic solutions of the first-order differential equation f′(z) = h(z)f(z) + 1 where h(z) = S(z)e^{P(z)} with P a polynomial and S a nonzero rational function. A key outcome is a lower bound on the modulus of solutions along suitable curves and the identification that the hyper-order s(f) of such solutions equals the degree of P, which informs uniqueness theory.

Within this context, Brück’s conjecture concerns entire functions of finite hyper-order (not an integer) that share a value with their derivative. Translating the sharing condition to a differential relation leads to the equation g′(z) = e^{p(z)}g(z) + 1 for g(z) = (f(z)−a)/a, linking the conjecture to the structural analysis performed in the paper. The paper provides partial progress (Theorem 3.1), but the full conjecture remains a central open problem in uniqueness theory of meromorphic functions.