Complex Oscillation Theory of 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> 0. 2000 Mathematics Subject Classification: 34M10, 30D35. 1. Introduction and

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.

Journal of Inequalities in Pure & Applied Mathematics, 2004

In this paper, we study the possible orders of transcendental solutions of the differential equation f (n) + a n−1 (z) f (n−1) + • • • + a 1 (z) f + a 0 (z) f = 0, where a 0 (z) ,. .. , a n−1 (z) are nonconstant polynomials. We also investigate the possible orders and exponents of convergence of distinct zeros of solutions of non-homogeneous differential equation f (n) + a n−1 (z) f (n−1) + • • • + a 1 (z) f + a 0 (z) f = b (z) , where a 0 (z) ,. .. , a n−1 (z) and b (z) are nonconstant polynomials. Several examples are given.

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| 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.

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.

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.

2009

In this paper we investigate the complex oscillation and the growth of some differential polynomials generated by the solutions of the differential equation f ′′ + A1 (z) f ′ + A0 (z) f = F, where A1 (z) , A0 (z) ( 6≡ 0) , F are meromorphic functions of finite order. AMS Mathematics Subject Classification (2000): 34M10, 30D35

Kyoto Journal of Mathematics, 2010

In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation f + A 1 (z)e az f + A 0 (z)e bz f = F, where a, b are complex constants and A j (z) ≡ 0 (j = 0, 1), and F ≡ 0 are entire functions such that max{ρ(A j) (j = 0, 1), ρ(F)} < 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) < 1 (j = 0, 1, 2) generated by solutions of the above equation.

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) < ∞.

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