Vol 86, No 5 (2022)

We obtain estimates for the integrals of derivatives of rational functions inmultiply connected domains in the plane.A sharp order of growth is found for the integral of the modulus of thederivative of a finite Blaschke product in the unit disc.We also extend the results of Dolzhenko about the integrals of thederivatives of rational functions to a wider class of domains, namely, todomains bounded by rectifiable curves without zero interior angles, and showthe sharpness of the results obtained.

In this paper, the problem of topological classification of gradient-like flows without heteroclinic intersections, given on a four-dimensional projective-like manifold, is solved. We show that a complete topological invariant for such flows is a bi-color graph that describes the mutual arrangement of closures of three-dimensional invariant manifolds of saddle equilibrium states. The problem of construction of a canonical representative in each topological equivalence class is solved.

The paper is concerned with solvability in the class of weak solutions of one class ofsemilinear elliptic second-order differential equationson arbitrary closed manifolds. These equations are inhomogeneous analoguesof the stationary Kolmogorov–Petrovskii–Piskunov–Fisher equation, andhave great applied and mathematical value.

In this paper, we study a class of nonlinear integral equationswith a noncompact monotone Hammerstein–Nemytskii operator onthe whole real line. This class of equations is widely usedin various fields of natural science. In particular, such equationsarise in mathematical biology and in the theory of radiative transfer.A constructive existence theorem for a nonnegativenontrivial summable and bounded solution is proved. We also study the asymptotic behavior of the solution at $\pm\infty$. At the end of the paper,specific examples of the indicated equations are given, that satisfy all theconditions of the proved existence theorem. In an important particular case, it is possible to prove a uniqueness theorem in a certainclass of essentially bounded functions.

We study entire solutions (solutions which are entire functions) of differential equations of the form$P(y,y^{(n)})=0$, where $P$ is a polynomial with complex coefficients, $n$ is a natural number.We show that, under some constraints on $P$, all entire solutions of such equations are eitherpolynomials, or functions of the form $e^{-L\beta z}Q(e^{\beta z})$, where $L$ is a nonnegative integer, $\beta$ isa complex number, and $Q$ is a polynomial with complex coefficients.This verifies the well-known A. E. Eremenko's conjecture on meromorphic solutions of autonomousBriot–Bouquet type equations for entire solutions in the nondegenerate case.