On the growth of solutions of second order complex differential equation with meromorphic coefficients

Abstract

We consider the differential equation f'' + Af' + Bf = 0 where A(z) and B(z) ≢ 0 are mero-morphic functions. Assume that A(z) belongs to the Edrei-Fuchs class and B(z) has a deficient value ∞, if f ≢ 0 is a meromorphic solution of the equation, then f must have infinite order.

Growth and value distribution of linear difference polynomials generated by meromorphic solutions of higher-order linear difference equations

“…Recently, there has been an increasing interest in the study on the properties of meromorphic solutions of complex difference equations from the viewpoint of difference analogues of Nevanlinna theory (see [4,6,8]) and among those many good results are obtained for the case of complex linear difference equations (see [4,5,6,11,12,14,16,19,20]). For the case of complex linear differential-difference equations see [1,2,3,13,15,21]. In particular, inspired by the results about the growth and the value distribution of differential polynomials generated by meromorphic solutions of complex linear differential equations, Latreuch and Belaïdi in [11] investigated the growth of linear difference polynomials generated by meromorphic solutions of the second order complex linear difference equation f (z + 2) + a(z)f (z + 1) + b(z)f (z) = 0, (1.1) where a(z) and b(z) are meromorphic functions.…”

“…Note that there is a dominant coefficient b(z) in Theorem 1.1. Chen and Zheng in [13] investigated a special case when ρ(a) = ρ(b) = 1. They considered the homogeneous complex linear difference equation…”

Growth of meromorphic solutions of linear difference equations without dominating coefficients

“…It follows from (9), (12), (14), (15) and (16) that there exist r n ⊂ E \ E 2 ∪ E 3 and θ n ∈ [θ r nφ 0 , θ r n + φ 0 ] such that, for z n = r n e iθ n , we have…”