2019
Том 71
№ 9

All Issues

Mokhonko A. Z.

Articles: 9
Article (Russian)

On meromorphic solutions of the systems of linear differential equations with meromorphic coefficients

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1227-1240

UDC 517.925.7
For systems of linear differential equations whose dimension can be decreased, we establish estimates for the growth of meromorphic vector solutions. As an essentially new feature, we can mention the fact that no additional restrictions are imposed on the order of growth of coefficients of the system.

Brief Communications (Russian)

On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 139-144

Assume that the coefficients and solutions of the equation $f^{(n)}+p_{n−1}(z)f^{(n−1)} +...+ p_{s+1}(z)f^{(s+1)} +...+ p_0(z)f = 0$ have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients $p_j , j = s+1, . . . ,n−1$, increase slower (in terms of the Nevanlinna characteristics) than $p_s(z)$. It is proved that this equation has at most $s$ linearly independent solutions of finite order.

Brief Communications (Russian)

Malmquist Theorem for the Solutions of Differential Equations in the Vicinity of a Branching Point

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1286–1290

An analog of the Malmquist theorem on the growth of solutions of the differential equation $f' = P(z, f)/Q(z, f)$, where $P(z, f)$ and $Q(z, f)$ are polynomials in all variables, is proved for the case where the coefficients and solutions of this equation have a branching point in infinity (e.g., a logarithmic singularity).

Article (Russian)

Deficiency Values for the Solutions of Differential Equations with Branching Point

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 939–957

We study the distribution of values of the solutions of an algebraic differential equation P(z, f, f′, . . . , f (s)) = 0 with the property that its coefficients and solutions have a branching point at infinity (e.g., a logarithmic singularity). It is proved that if a ∈ ℂ is a deficiency value of f and f grows faster than the coefficients, then the following identity takes place: P(z, a, 0, . . . , 0) ≡ 0, z ∈ {z : r 0|z| < ∞}. If P(z, a, 0, . . . , 0) is not identically equal to zero in the collection of variables z and a, then only finitely many values of a can be deficiency values for the solutions fM b with finite order of growth.

Article (Russian)

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

Mokhonko A. Z.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1489-1502

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.

Article (Russian)

On the order of growth of solutions of algebraic differential equations

Mokhonko A. Z., Mokhonko V. D.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 69–77

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 $\{z: η < \arg z < η+2π\} E∗$, where $E∗$ is some set of disks with the finite sum of radii, the estimate $\ln f(z) = z^{1/2}(β+o(1)),\; β ∈ C$, is true (here, $z=\re i^{φ}, r ≥ r(φ) ≥ 0$, and if $z = \text{re } i^{φ}, r ≥ r(φ) ≥ 0$ and, on a ray $\{z: \arg z=η\}$, the relation $\ln |f(\text{re } i^{η})| = o(r^{1/2}), \; r → +∞,\; r > 0, r \bar \in \Delta$, holds, where $Δ$ is some set on the semiaxis $r > 0$ with mes $Δ < ∞$.

Article (Ukrainian)

An estimate of the modulus of the logarithmic derivative of a function which is meromorphic in an angular region, and its application

Mokhonko A. Z.

Full text (.pdf)

Ukr. Mat. Zh. - 1989. - 41, № 6. - pp. 839-843

Article (Ukrainian)

Meromorphic solutions of first-order differential equations

Mokhonko A. Z.

Full text (.pdf)

Ukr. Mat. Zh. - 1986. - 38, № 6. - pp. 739-744

Article (Ukrainian)

Meromorphic and algebroidal solutions of functional equations

Mokhonko A. Z.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 6. - pp. 780 – 786