Growth of solutions of second order linear differential equations with extremal functions for Denjoy's conjecture as coefficients

Abstract

Abstract. The classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f ′′ + A(z) f ′ + B(z) f = 0 are of infinite order is discussed. Some such conditions which involve deficient value, Borel exceptional value and extremal functions for Denjoy's conjecture are obtained.

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).…”

“…The next two lemmas are related with Borel exceptional value. The Lemma 2.6 can be found in [14], in order to reader's convenience, here a proof is given. This completes the proof.…”

On a question of Gundersen concerning the growth of solutions of linear differential equations

“…Motivated by Theorem 1.1, many parallel results written thereafter focus on the case ρ(A) ≥ ρ(B) and B(z) is a transcendental entire function; see, for example, [3,16,18,19,24,25]. Regarding the case of a polynomial B(z), there are many results concerning the growth of solutions of the following special equation…”