Growth and oscillation of polynomial of linearly independent meromorphic solutions of second order linear differential equations in the unit disc

Abstract

In this paper, we deal with the growth and oscillation of w = d1f1 + d2f2, where d1, d2 are meromorphic functions of finite iterated p−order that are not all vanishing identically and f1, f2 are two linearly independent meromorphic solutions in the unit disc ∆ = {z ∈ C : |z| < 1} satisfying δ (∞, fj) > 0, (j = 1, 2), of the linear differential equation f + A (z) f = 0, where A (z) is admissible meromorphic function of finite iterated p−order in ∆.

Linear differential equations with analytic coefficients of [p,q]-order in the unit disc

“…L. G. Bernal, L. Kinnunen, J. Tu and T. Long investigated the growth of solutions of (1.1) individually when the coefficients are entire functions of finite iterated order (see [4,21,26]. The properties of the growth of (1.1) also have been studied by J. Heittokangas, T. B. Cao and B. Belaïdi when the coefficients are analytic functions in the unit disc ∆ = {z : |z| < 1} (see [2,3,[5][6][7][8]11,15,17,23]). After that A. El Farissi, B. Belaïdi and Z. Latreuch generalized the results of T. B. Cao and investigated the growth of differential polynomial generated by solutions of second order differential equations in the unit disc see Theorem C).…”

“…for any p ≥ q ≥ 1. By Definition A we have that ρ [1,1] = ρ (f ) (ρ M, [1,1] = ρ M (f )) and ρ [2,1]…”