Complex oscillation of differential polynomials in the unit

Periodica Mathematica Hungarica, 2013

We consider the complex differential equations f +A 1 (z)f +A 0 (z)f = F and where A 0 ≡ 0, A 1 and F are analytic functions in the unit disc Δ = {z : |z| < 1}. We obtain results on the order and the exponent of convergence of zero-points in Δ of the differential polynomials g f = d 2 f + d 1 f + d 0 f with non-simultaneously vanishing analytic coefficients d 2 , d 1 , d 0. We answer a question posed by J. Tu and C. F. Yi in 2008 for the case of the second order linear differential equations in the unit disc.

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.

2011

In this paper, we study the growth and the oscillation of complex differential equations f + A 1 (z) f + A 0 (z) f = 0 and f + A 1 (z) f + A 0 (z) f = F, where A 0 ≡ 0, A 1 and F are analytic functions in the unit disc ∆ = {z : |z| < 1} with finite iterated p−order. We obtain some results on the iterated p−order and the iterated exponent of convergence of zero-points in ∆ of the differential polynomials g f = d 1 f + d 0 f and g f = d 1 f + d 0 f + b, where d 1 , d 0 , b are analytic functions such that at least one of d 0 (z) , d 1 (z) does not vanish identically with ρ p (d j) < ∞ (j = 0, 1) , ρ p (b) < ∞.

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 $\rho (d_j)&lt;1$ $(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 $\max \left\lbrace \rho (A_j),\, j=0,1,\, \rho (F)\right\rbrace &lt;1.$

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 ∆.

International Journal of Analysis and Applications, 2013

In this paper, we investigate the growth and oscillation of higher order differential polynomial with meromorphic coefficients in the unit disc ∆ = {z : |z| < 1} generated by solutions of the linear differential equation f (k) + A (z) f = 0 (k ≥ 2) , where A (z) is a meromorphic function of finite iterated p−order in ∆. 2010 Mathematics Subject Classification. 34M10, 30D35. Key words and phrases. Iterated p−order, Linear differential equations, Iterated exponent of convergence of the sequence of distinct zeros, Unit disc, Differential polynomials.

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&gt; 0. 2000 Mathematics Subject Classification: 34M10, 30D35. 1. Introduction and

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.

In this article, we give sufficiently conditions for the solutions and the differential polynomials generated by second-order differential equations to have the same properties of growth and oscillation. Also answer to the question posed by Cao [6] for the second-order linear differential equations in the unit disc.