On the relationship between the lower order of coefficients and the growth of solutions of differential equations

Abstract

Some criteria for entire coefficients A(z) and B(z) are given in terms of the lower order forcing the solutions of f + A(z)f + B(z)f = 0 to grow fast. In the literature similar criteria have been published in terms of the usual order. The case when the coefficient A(z) has an asymptotic growth T (r, A) ∼ α log M (r, A), α ∈ (0, 1), outside of an exceptional set is also discussed. Previously, Laine-Wu (2000) and Kim-Kwon (2001) have made use of this asymptotic growth in the case α = 1.

Second order complex differential equations with analytic coefficients in the unit disc

“…Recently, Long, Heittokangas and Ye, in [18], gave a similar result to Theorem 1.1 when the usual orders are replaced with the corresponding lower orders, and they proved the following theorem. Here, µ( f ) denotes the order of growth of f in complex plane which is defined similarly as the order but for "liminf" instead of "limsup".…”

Growth of solutions of second order complex linear differential equations with entire coefficients

“…Motivated by Theorem 1.1, many parallel results written thereafter focus on the case ρ(A) ≥ ρ(B); see, for example, [1,[13][14][15][16]24]. However, in general, the conclusions are false for the case ρ(A) ≥ ρ(B).…”

Lower order and Baker wandering domains of solutions to differential equations with coefficients of exponential growth

“…Note that the lower order can be quite different from the order, since there even exists an entire function of order ρ = ∞ and lower order μ = 0, see [7, p.238]. As we know, there are only some attempts to associate the lower order with complex differential equations, see [14], and references therein. We feel a need to state the results on the lower order separately.…”