Some results on the growth and oscillation of solutions of differential equations withmeromorphic function coefficients of [p,q ]−ϕ order

Abstract

Throughout this work, by using a generalized concept of order called ϕ-order, we have discussed the possibility of extendingsome results about the growth ofmeromorphic so- lutions to linear differential equations of the form: (k) (k−1) 0 A (z) f +A (z) f + · · ·+A (z) f +A (z) f = 0 (3.26) 1 0 k k−1 (k) (k−1) 0 A (z) f +A (z) f + · · ·+A (z) f +A (z) f = F (z) , where A and F aremeromorphic functions of finite [p,q ]−ϕ order. (3.27) 1 0 k k−1 j We have obtained the relationship between the solutions and the meromorphic coeffi- cients in terms of ϕ-order, estimations about the [p,q ]−ϕ order and the [p,q ]−ϕ con- vergence exponent of the solutions to such equations. Now, some open questions and problems are proposed. Problem 1. Can we get the similar result using the (α,β,ν) -order defined in [3]? In other words what can be said about the growth of solutions of the differential equations (3.26) and (3.27) if the coefficients aremeromorphic functions of (α,β,ν) -order? Problem 2. What are the hypothesis on the dominant coefficient that guarantee that the solutions of the above equations have a finite (α,β,ν) -order?