Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 157, pp. 1–8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LINEAR DIFFERENTIAL EQUATIONS WITH ENTIRE COEFFICIENTS HAVING THE SAME ORDER AND TYPE NACERA BERRIGHI, SAADA HAMOUDA Abstract. In this article, we study the growth of solutions to the differential equation f k +(A k−1 (z)e P k-1 (z) e λz m + B k−1 (z))f k−1 + ... +(A 0 (z)e P 0 (z) e λz m + B 0 (z))f =0, where λ ∈ C ∗ , m ≥ 2 is an integer and max j=0,...,k−1 {deg P j (z)} <m, A j ,B j (j =0,...,k − 1) are entire functions of orders less than m. 1. Introduction and statement of results Throughout this paper, we assume that the reader is familiar with the fun- damental results and the standard notation of the Nevanlinna value distribution theory (see [8]). In addition, we use the notation σ 2 (f ) to denote the hyper-order of nonconstant entire function f ; that is, σ 2 (f ) = lim sup r→+∞ log log T (r, f ) log r = lim sup r→+∞ log log log M (r, f ) log r , where T (r, f ) is the Nevanlinna characteristic of f and M (r, f ) = max |z|=r |f (z)| (see [11]). We define the linear measure of a set E ⊂ [0, 2π) by m(E)=  +∞ 0 χ E (t)dt and the logarithmic measure of a set F ⊂ [1, +∞) by lm(F )=  +∞ 1 χ F (t) t dt, where χ H (t) is the characteristic function of a set H. Several authors have studied the particular differential equations f ′′ + e −z f ′ + Q(z)f =0, (1.1) (see [1, 4, 6, 9]). Gundersen [6] proved that if deg Q(z) = 1, then every nonconstant solution of (1.1) is of infinite order. Chen considered the case Q(z)= h(z)e bz , where h(z) is nonzero polynomial and b = −1, see [2]; more precisely, he proved that every nontrivial solution f of (1.1) satisfies σ 2 (f ) = 1. The same paper contains a discussion about more general equations of the type f ′′ + A 1 (z)e az f ′ + A 0 (z)e bz f =0, (1.2) 2000 Mathematics Subject Classification. 34M10, 30D35. Key words and phrases. Linear differential equations; growth of solutions; hyper-order. c 2011 Texas State University - San Marcos. Submitted October 20, 2011. Published November 21, 2011. 1