GROWTH AND OSCILLATION OF DIFFERENTIAL POLYNOMIALS GENERATED BY COMPLEX DIFFERENTIAL EQUATIONS (original) (raw)
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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.
Publications de l'Institut Mathematique, 2015
We investigate the growth of meromorphic solutions of homogeneous and nonhomogeneous higher order linear differential equations f (k) + k−1 j=1 A j f (j) + A 0 f = 0 (k 2), f (k) + k−1 j=1 A j f (j) + A 0 f = A k (k 2), where A j (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; Belaïdi; Chen; Shen and Xu.