Abstract
Assume thatf is an integer transcendental solution of the differential equationP n (z, f, f′)=P n−1(z, f, f′, ... f (p)), whereP n andP n−1 are polynomials in all variables, the degree ofP n with respect tof andf′ is equal ton, and the degree ofP n−1 with respect tof, f′, ... f (p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E *, whereE * is a certain set of disks with finite sum of radii, the estimate lnf(z)=z 1/2 (β+o(1)), β∈C, holds forz=re iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re iν)‖=o(r 1/2),r→+∞,r>0,\(r\bar \in \Delta \), where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.
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“L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.
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Mokhon'ko, A.Z., Mokhon'ko, V.D. On the order of growth of solutions of algebraic differential equations. Ukr Math J 51, 76–85 (1999). https://doi.org/10.1007/BF02591916
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DOI: https://doi.org/10.1007/BF02591916