Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc

Abstract

In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f ′ ′ + A ( z ) f ′ + B ( z ) f = F ( z ) in Δ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values.

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.…”