# Mokhon'ko A. Z.

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

Mokhon'ko A. Z., Mokhon'ko O. A.

↓ 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.

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

Mokhon'ko A. Z., Mokhon'ko O. A.

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).

### Deficiency Values for the Solutions of Differential Equations with Branching Point

Mokhon'ko A. Z., Mokhon'ko O. A.

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 *f* ∈ *M* _{ b } with finite order of growth.

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

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*(*z*ln*z*, *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.

### On the order of growth of solutions of algebraic differential equations

Mokhon'ko A. Z., Mokhon'ko V. D.

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 $Δ < ∞$.

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

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

### Meromorphic solutions of first-order differential equations

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

### Meromorphic and algebroidal solutions of functional equations

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