Periodica Mathematica Hungarica Vol. 66 (1 ), 2013, pp. 45–60 DOI: 10.1007/s10998-013-9795-3 COMPLEX OSCILLATION OF DIFFERENTIAL POLYNOMIALS IN THE UNIT DISC Zinelaˆ abidine Latreuch 1 , Benharrat Bela¨ idi 2 and Abdallah El Farissi 3 1 Department of Mathematics, Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB) B. P. 227 Mostaganem-(Algeria) E-mail: z.latreuch@gmail.com 2 Department of Mathematics, Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB) B. P. 227 Mostaganem-(Algeria) E-mail: belaidi@univ-mosta.dz 3 Department of Mathematics, Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB) B. P. 227 Mostaganem-(Algeria) E-mail: elfarissi.abdallah@yahoo.fr (Received November 15, 2010; Accepted December 15, 2011) [Communicated by L´aszl´ o Hatvani] Abstract 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. 1. Introduction In this paper, we assume that the reader is familiar with the fundamental results and standard notations of Nevanlinna theory [13, 14, 18, 21] in the unit disc Δ= {z ∈ C : |z | < 1}. In addition, we will use λ(f )(λ 2 (f )) and λ(f )( λ 2 (f )) to denote respectively the exponents (hyper-exponents) of convergence of the zero- sequence and the sequence of distinct zeros of a meromorphic function f , ρ(f ) to Mathematics subject classification numbers : 34M10, 30D35. Key words and phrases: linear differential equations, analytic function, hyper-order, expo- nent of convergence of the sequence of distinct zeros. 0031-5303/2013/$20.00 Akad´ emiai Kiad´o, Budapest c  Akad´ emiai Kiad´o, Budapest Springer, Dordrecht