On the order of growth of solutions of algebraic differential equations - Mokhonko, A. Z.; Mokhonko, V. D. (Ukrains’kyi Matematychnyi Zhurnal, 1999)
Assume that f is an integer transcendental solution of the differential equation P_n(z,f,f′)=P_n−1(z,f,f′,...,f(p))), P_n, P_n−1 are polynomials in all the variables, the order of P_n with respect to f and f′ is equal to n, and the order of P_n−1 with respect to f, f′, ... f(p) is at most n−1. We prove that the order ρ of growth of f satisfies the relation 12 ≤ ρ < ∞. We also prove that if ρ = 1/2, then, for some real η, in the domain {z: η < z < η+2π} E∗, where E∗ is some set of disks with the finite sum of radii, the estimate f(z) = z^1/2(β+o(1)), β ∈ C, is true (here, z= e i^φ, r ≥ r(φ) ≥ 0, and if z = re i^φ, r ≥ r(φ) ≥ 0 and, on a ray {z: z=η}, the relation |f(re i^η)| = o(r^1/2), r → +∞, r > 0, r Δ, holds, where Δ is some set on the semiaxis r > 0 with mes Δ < ∞.