Brück conjecture in one complex variable (single shared value with derivative)

Prove that if f is a non-constant entire function on the complex plane C with hyper-order ρ1(f) not belonging to N ∪ {∞}, and a ∈ C, and if f and its first derivative f^{(1)} share the value a counting multiplicities, then f^{(1)}(z) − a = c (f(z) − a) for some non-zero constant c.

Background

The classical uniqueness results of Rubel–Yang and Mues–Steinmetz show that an entire function sharing two values with its derivative must equal its derivative. Motivated by whether a single shared value could force a similarly rigid conclusion, Brück proposed a conjecture giving a precise form for f when f and f′ share one value counting multiplicities, provided the hyper-order avoids the integers and infinity.

This conjecture is known to hold in special cases (e.g., a = 0), and under additional growth hypotheses, but it fails when the hyper-order is an integer or infinite, as indicated by explicit counterexamples. The paper recalls this conjecture as the foundational open problem that motivates both its restatement and the several-variable extension studied here.