On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point

The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation , where P(z, f) and Q(z, f) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 505–513, April, 2005.     

On the order of growth of solutions of algebraic differential equations

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, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations

We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 356–365, March, 2005.     

On the Growth of Meromorphic Solutions of an Algebraic Differential Equation in a Neighborhood of a Logarithmic Singular Point

We prove that if an analytic function f with an isolated singular point at is a solution of the differential equation P(zlnz, f, f) = 0, where P is a polynomial in all variables, then f has finite order. We study the asymptotic properties of a meromorphic solution with logarithmic singularity.