Afr. Mat. https://doi.org/10.1007/s13370-018-0565-5 Iterated exponent of convergence of solutions of linear differential equations in the unit disc H. Fettouch 1 · S. Hamouda 1 Received: 12 December 2015 / Accepted: 6 February 2018 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018 Abstract In this paper we investigate the n-iterated exponent of convergence of f (i ) − ϕ where f  ≡ 0 is a solution of linear differential equation with analytic and meromorphic coefficients in the unit disc and ϕ is a small function of f . By this investigation we can deduce the value distribution of the fixed points of f (i ) by taking ϕ(z ) = z . This work is an extension to the unit disc and an improvement of recent results in the complex plane by Xu et al. (Adv Differ Equ 2012(114):1–16, 2012) and Tu et al. (Adv Differ Equ 2013(71):1–16, 2013). Keywords Linear differential equations · Exponent of convergence · Growth of solutions · Unit disc Mathematics Subject Classification 34M10 · 30D35 1 Introduction and statement of results Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic function on the complex plane C and in the unit disc D ={z ∈ C :|z | < 1} (see [10, 19]). In addition, for n ∈ N −{0}, the n-iterated order of meromorphic function f (z ) in D is defined by σ n ( f ) = lim sup r →1 − log + n T (r, f ) − log (1 − r ) , B S. Hamouda saada.hamouda@univ-mosta.dz H. Fettouch houari.fettouch@univ-mosta.dz 1 Laboratory of Pure and Applied Mathematics, UMAB University of Mostaganem, Mostaganem, Algeria 123