Reformulation of Brück conjecture via exponential ratio

Prove that if f is a non-constant entire function on C with hyper-order ρ1(f) not in N ∪ {∞} and a ∈ C, and if f^{(1)}(z) − a = e^{α(z)} (f(z) − a) for an entire function α, then α is constant and consequently f(z) = c1 e^{cz} + a − a/c, where c = e^{d} for the constant d = α and c1 ≠ 0.

Background

Because (f{(1)} − a)/(f − a) is entire and zero-free when f and f′ share a counting multiplicities, it can be written as e{α} for some entire α. Brück’s conjecture is therefore equivalent to proving that α must be constant under the specified hyper-order restriction, yielding an exponential form for f.

The paper explicitly restates the conjecture in this equivalent exponential ratio formulation, clarifying that the central unresolved point is the constancy of α.

References

Therefore Conjecture A can be re-stated as follows: Let $f$ be a non-constant entire function in $\mathbb{C}$ such that $\rho_1(f)\not\in\mathbb{N}\cup{\infty}$ and $a\in\mathbb{C}$. If $f{(1)}-a=e{\alpha}(f-a)$, where $\alpha$ is an entire function in $\mathbb{C}$, then $\alpha$ reduces to a constant, $d$ say and $f(z)$ takes the form $f(z)=c_1e{cz}+a-\frac{a}{c}$, where $c=ed$ and $c_1$ are non-zero constant.

— Bruck conjecture for solutions of first-order partial differential equations in Cm  (2509.02576 - Majumder et al., 25 Aug 2025) in Conjecture B, Section 1 (Introduction)