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where k ≥ 1 is an integer, a0(z), a1(z), . . . , ak−1(z) are analytic functions in the unit disc D = {z ∈ C : |z| < 1} and a0(z) 6≡ 0. The theory of complex differential equations in the unit disc has been developed since 1980’s, see [1]. In the year 2000, Heittokangas in [2] firstly investigated the growth and oscillation theory of Equation (1) when the coefficients a0(z), a1(z), . . . , ak−1(z) are analytic functions in the unit disc D by introducing the definition of the function spaces. His results also gave some important tools for further investigations on the theory of meromorphic solutions of Equation (1). In this article, we investigate the growth of solutions of the Equation (1) when the coefficients a0(z), a1(z), . . . , ak−1(z) are analytic in D, and we deal with the case that the coefficients are fast growing in D. To define the order of fast-growth of analytic functions, we define inductively for r ∈ [0,+∞), exp0 r = r, exp1 r = e r and expn+1 r = exp (expn r) , n ∈ N. For all r sufficiently large, we define log0 r = r, log1 r = log r and logn+1 r = log (logn r) , n ∈ N. Also, we need to be familiar with the fundamental results and the standard notations of the Nevanlinna’s theory on the complex plane C and in the unit disc D, for more details on Nevanlinna theory and its applications in complex differential equations in complex plane and in unit disc, we refer to [2–7]. Before stating our main results, we recall definitions and preliminary remarks concerning meromorphic and analytic functions in D. For the definitions and more discussions, we refer the reader to [7–10]. Let p ≥ 1 be an integer and f be a meromorphic function in D. Then, the iterated p-order of f is defined by

Linear differential equations with fast growing coefficients in the unit disc