Meromorphic Coefficients Research Articles

On the Growth of Higher Order Complex Linear Differential Equations Solutions with Entire and Meromorphic Coefficients

We revisit the problem of studying the solutions growth order in complex higher order linear differential equations with entire and meromorphic coefficients of p,q-order, proving how it is related to the growth of the coefficient of the unknown function under adequate assumptions. Our study improves the previous results due to J. Liu - J. Tu - L. Z Shi, L.M. Li - T.B. Cao, and others.

On the Complex Oscillation of Meromorphic Solutions of Nonhomogeneous Linear Differential Equations with Meromorphic Coefficients

In this paper, we study the higher order differential equation f k + B f = H , where B is a rational function, having a pole at ∞ of order n > 0 , and H ≡ 0 is a meromorphic function with finite order, and obtain some properties related to the order and zeros of its meromorphic solutions.

Uniqueness theorems on meromorphic functions and their difference polynomials

ABSTRACT Let f be a nonconstant meromorphic function of finite order, and be a linear difference polynomial in f with small meromorphic coefficients. Some uniqueness theorems related to f sharing three values with are obtained. We also prove that f is identical with its difference operator , if they share three values CM. Some examples are given to show our results are best possible.

GROWTH OF \(\varphi\)–ORDER SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS ON THE COMPLEX PLANE

In this paper, we study the growth of solutions of higher order linear differential equations with meromorphic coefficients of \(\varphi\)-order on the complex plane. By considering the concepts of \(\varphi\)-order and \(\varphi \)-type, we will extend and improve many previous results due to Chyzhykov–Semochko, Belaïdi, Cao–Xu–Chen, Kinnunen.

On Meromorphic Solutions of the Systems of Linear Differential Equations with Meromorphic Coefficients

For systems of linear differential equations whose dimension can be lowered, we estimate 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 the coefficients of the system.

Some properties of meromorphic solutions of linear differential equation with meromorphic coefficients

Some properties of meromorphic solutions of linear differential equation with meromorphic coefficients

On growth of meromorphic solutions of some kind of non-homogeneous linear difference equations

In this paper, we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients. When there are more than one coefficient having the same maximal order and the same maximal type, the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained. Meanwhile, the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.

Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc

In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f ′ ′ + A ( z ) f ′ + B ( z ) f = F ( z ) in Δ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values.

Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let g(f):=∑j=1kbj(z)f(z+cj) be a shift polynomial of f with small meromorphic coefficients bj in the sense of O(rλ+ε)+S(r,f), λ<ρ. Provided α, β, b0 are similar small meromorphic functions, we consider zero distribution of fn(g(f))s−b0, resp. of g(f)−αfn−β.

Iterated exponent of convergence of solutions of linear differential equations in the unit disc

In this paper we investigate the n-iterated exponent of convergence of $$ f^{\left( i\right) }-\varphi $$ where $$f\not \equiv 0$$ is a solution of linear differential equation with analytic and meromorphic coefficients in the unit disc and $$\varphi $$ is a small function of f. By this investigation we can deduce the value distribution of the fixed points of $$f^{\left( i\right) }$$ by taking $$\varphi \left( z\right) =z$$ . This work is an extension to the unit disc and an improvement of recent results in the complex plane by Xu et al. (Adv Differ Equ 2012(114):1–16, 2012) and Tu et al. (Adv Differ Equ 2013(71):1–16, 2013).

A study on algebraic differential equations of Gamma function and Dirichlet series

The paper concerns the problem of algebraic differential independence between the gamma function and the function in a certain class F, which contains Dirichlet L-functions, L-functions in the extended Selberg class and some periodic functions. It is proved that the gamma function and the function in F cannot satisfy a class of algebraic differential equations with meromorphic coefficients ϕ having Nevanlinna characteristic satisfying T(r,ϕ)=o(r) as r→∞.

On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

On the growth of solutions of some higher order linear differential equations with meromorphic coefficients

Monodromization and Difference Equations with Meromorphic Periodic Coefficients

We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrodinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.

Close-to-convex Function Generates Remarkable Solution of 2^nd order Complex Nonlinear Differential Equations

Consider the complex nonlinear differential equation where are complex coefficients, and be a complex function performs non- homogeneous term of given equation.&#x0D; In this paper, we investigated that is a remarkable solution of given equation and belongs to hardy space ; with studying the growth of that solution by two ways ; through the maximum modulus and Brennan’s Conjecture and another by finding the supremum function of a volume of the surface area . Furthermore, we discussed the solution behaviour with meromorphic coefficients properties for given equation.

Eigenmodes in the water-wave problems for infinite pools with cone-shaped bottom

In the framework of the assumptions of the linearized theory of small-amplitude water waves, the eigenfunctions of the point spectrum are studied for boundary-value problems in infinite domains. Special types of three-dimensional infinite water pools characterised by cone-shaped bottoms are considered. By means of an incomplete separation of variables and exploiting the Mellin transform, we reduce construction of the eigenmodes to the study and solution of the problems for some functional difference equations with meromorphic coefficients. The behaviour of the eigenmodes at a singular point of the boundary and the rate of their decay at infinity are also examined.

Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc

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On hyper-order of solutions of higher order linear differential equations with meromorphic coefficients

In this paper, we investigate the growth of meromorphic solutions of the differential equations $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ where $A_{0}(z)\not\equiv0, A_{1}(z), \ldots, A_{k-1}(z)$ and $F(z)\not \equiv0$ are meromorphic functions. A precise estimation of the hyper-order of meromorphic solutions of the above equations is given provided that there exists one dominant coefficient, which improves and extends previous results given by Belaidi, Chen, etc.

Vector Riemann–Hilbert problem with almost periodic and meromorphic coefficients and applications

The vector Riemann–Hilbert problem is analysed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros (rational functions), (ii) periodic poles and zeros, and (iii) an infinite number of non-periodic zeros and poles, are considered. The first case is illustrated by the heat equation for a composite rod with a finite number of discontinuities and a system of convolution equations; both problems are solved explicitly. In the second case, a Wiener–Hopf factorization is found in terms of the hypergeometric functions, and the exact solution of a mixed boundary value problem for the Laplace equation in a wedge is derived. In the last case, the Riemann–Hilbert problem reduces to an infinite system of linear algebraic equations with the exponential rate of convergence. As an example, the Neumann boundary value problem for the Helmholtz equation in a strip with a slit is analysed.

Good Linear Operators and Meromorphic Solutions of Functional Equations

Nevanlinna theory provides us with many tools applicable to the study of value distribution of meromorphic solutions of differential equations. Analogues of some of these tools have been recently developed for difference, q-difference, and ultradiscrete equations. In many cases, the methodologies used in the study of meromorphic solutions of differential, difference, and q-difference equations are largely similar. The purpose of this paper is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. As an example case, we apply our methods to study the growth of meromorphic solutions of the functional equation M(z,f)+P(z,f)=h(z), where M(z,f) is a linear polynomial in f and L(f), where L is good linear operator, P(z,f) is a polynomial in f with degree deg P≥2, both with small meromorphic coefficients, and h(z) is a meromorphic function.

On the growth and the zeros of solutions of higher order linear differential equations with meromorphic coefficients

We investigate the growth of meromorphic solutions of homogeneous and nonhomogeneous higher order linear differential equations f(k) + k-1?j=1 Ajf(j) + A0f = 0 (k ? 2); f(k) + k-1 ?j=1 Ajf(j) + A0f = Ak (k ? 2); where Aj(z)(j=0,1,...,k) are meromorphic functions with finite order. Under some conditions on the coefficients, we show that all meromorphic solutions f ?/0 of the above equations have an infinite order and infinite lower order. Furthermore, we give some estimates of their hyper-order, exponent and hyper-exponent of convergence of distinct zeros. We improve the results due to Kwon, Chen and Yang, Bela?di, Chen, Shen and Xu.