Hyper-order versus order for solutions of Hayman’s reduction f′(z) = l1(z)f(z) + h(z)

Determine whether the inequality s(f) ≥ o(h) holds for transcendental meromorphic solutions f(z) of the first-order differential equation f′(z) = l1(z)f(z) + h(z), where l1(z) is a rational function and h(z) is a meromorphic solution of h′(z) = y2(z)h(z) + y3(z) with rational functions y2(z) and y3(z). Here s(f) denotes the hyper-order of f, and o(h) denotes the order of h, both defined via Nevanlinna theory. This inequality is known to hold in special cases (e.g., l1 is a nonzero constant and y3 ≡ 0), but its validity in the general setting remains unresolved.

Background

The paper connects its main modulus estimate to the growth of solutions in the context of Hayman’s second-order algebraic differential equation and its first-order reductions. Specifically, when f′(z) = l1(z)f(z) + h(z) and h solves h′(z) = y2(z)h(z) + y3(z) with rational coefficients, certain special cases allow precise determination of hyper-order, such as s(f) = o(h) when l1 is a nonzero constant and y3 ≡ 0.

However, outside these special cases, the relationship between the hyper-order of f and the order of h is unclear. Establishing s(f) ≥ o(h) would generalize the paper’s special-case results and clarify growth behavior for a broad class of solutions arising from Hayman’s framework.