Complex oscillation of differential polynomials in the unit disc

2013

Abstract. In this paper, we investigate the complex oscillation of the differential polynomial gf = d2f ′′ + d1f ′ + d0f, where dj (j = 0,1,2) are meromorphic functions with finite iterated p−order not all equal to zero generated by solutions of the differential equation f ′′ + A (z) f = 0, where A (z) is a transcendental meromorphic function with finite iterated p−order ρp (A) = ρ> 0. 2000 Mathematics Subject Classification: 34M10, 30D35. 1. Introduction and

2011

In this paper, we investigate the relationship between small functions and differential polynomials $g_{f}(z)=d_{2}f^{\prime \prime }+d_{1}f^{\prime }+d_{0}f$, where $d_{0}(z)$, $d_{1}(z)$, $d_{2}(z)$ are entire functions that are not all equal to zero with $\rho (d_j)<1$ $(j=0,1,2) $ generated by solutions of the differential equation $f^{\prime \prime }+A_{1}(z) e^{az}f^{\prime }+A_{0}(z) e^{bz}f=F$, where $a,b$ are complex numbers that satisfy $ab( a-b) \ne 0$ and $A_{j}( z) \lnot \equiv 0$ ($j=0,1$), $F(z) \lnot \equiv 0$ are entire functions such that $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace <1.$

Hokkaido Mathematical Journal, 2010

This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation f + A 1 (z)e P (z) f + A 0 (z)e Q(z) f = F, where P (z), Q(z) are nonconstant polynomials such that deg P = deg Q = n and A j (z) (≡ 0) (j = 0, 1), F ≡ 0 are entire functions with ρ(A j) < n (j = 0, 1). We also investigate the relationship between small functions and differential polynomials g f (z) = d 2 f + d 1 f + d 0 f , where d 0 (z), d 1 (z), d 2 (z) are entire functions that are not all equal to zero with ρ(d j) < n (j = 0, 1, 2) generated by solutions of the above equation.

Scientific Journals of Rzeszów University of Technology, Series: Journal of Mathematics and Applications, 2014

The main purpose of this paper is to study the controllability of solutions of the differential equation f (k) + A k−1 (z) f (k−1) + • • • + A 1 (z) f ′ + A 0 (z) f = 0. In fact, we study the growth and oscillation of higher order differential polynomial with meromorphic coefficients in the unit disc ∆ = {z : |z| < 1} generated by solutions of the above k th order differential equation.

European Journal of Mathematical Analysis

Throughout this article, we investigate the growth and fixed points of solutions of complex higher order linear differential equations in which the coefficients are analytic functions of [p, q]−order in the unit disc. This work improves some results of Belaïdi [3–5], which is a generalization of recent results from Chen et al. [9].

Acta et Commentationes Universitatis Tartuensis de Mathematica, 2016

We consider the complex oscillation of nonhomogeneous linear differential polynomials g k = k j=0 djf (j) +b, where dj (j = 0, 1,. .. , k) and b are meromorphic functions of finite [p,q]-order in the unit disc ∆, generated by meromorphic solutions of linear differential equations with meromorphic coefficients of finite [p,q]-order in ∆.

In this paper, we deal with the growth and oscillation of w = d1f1 + d2f2, where d1, d2 are meromorphic functions of finite iterated p−order that are not all vanishing identically and f1, f2 are two linearly independent meromorphic solutions in the unit disc ∆ = {z ∈ C : |z| < 1} satisfying δ (∞, fj) > 0, (j = 1, 2), of the linear differential equation

In this paper, we study the complex oscillation of solutions and their derivatives of the di erential equation f00 + A(z) f0 + B (z) f = F (z) ; where A(z) ;B (z) (6 0) and F (z) (6 0) are meromorphic functions of nite iterated p-order in the unit disc
= fz : jzj < 1g.

Acta Universitatis Sapientiae Mathematica, 2010

In this paper, we investigate the relationship between solutions and their derivatives of the differential equation f (k) + A (z) f = 0, k ≥ 2, where A (z) ≡ 0 is an analytic function with finite iterated porder and analytic functions of finite iterated p-order in the unit disc ∆ = {z ∈ C : |z| < 1}. Instead of looking at the zeros of f (j) (z) − z (j = 0, .., k) , we proceed to a slight generalization by considering zeros of f (j) (z) − ϕ (z) (j = 0, .., k), where ϕ is a small analytic function relative to f such that ϕ (k−j) (z) ≡ 0 (j = 0, ..., k), while the solution f is of infinite iterated p-order. This paper improves some very recent results of T. B. Cao and G. Zhang, A. Chen.

Publications de l'Institut Mathematique, 2011

We investigate the complex oscillation of some differential polynomials generated by solutions of the differential equation f + A 1 (z)f + A 0 (z)f = 0, where A 1 (z), A 0 (z) are meromorphic functions having the same finite iterated p-order.

2015

This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation \[ f^{(k)}+A_{k-1}(z) f^{(k-1)}+\cdots +A_1(z) f^{\prime }+A_0(z) f=0, \] where $A_{i}(z)$ $(i=0,1,\cdots ,k-1)$ are meromorphic functions of finite order in the complex plane.

In this paper, we consider some properties on the growth and oscillation of combination of solutions of the linear dierential equation

Electronic Journal of Qualitative Theory of Differential Equations, 2009

In this paper, we investigate the relationship between small functions and differential polynomials g

In this paper, we continue the study of some properties on the growth and oscillation of solutions of linear differential equations with entire coefficients.

Journal of Mathematical Analysis and Applications, 1997

We treat the linear differential equation (∗)f(k)+A(z)f=0, wherek≧2 is an integer andA(z) is a transcendental entire function of order σ(A). It is shown that any non-trivial solution of the equation (∗) satisfies λ(f)≧σ(A), where λ(f) is the exponent of convergence of the zero-sequence off, under the conditionKN̄(r,1/A)≦T(r,A),r∉Efor aK&amp;amp;gt;2kand an exceptional setEof finite linear measure. The second order equationf″+(eP1(z)+eP2(z)+Q(z))f=0, whereP1(z),P2(z) are